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FAQ: How Zs Aware Polygons Are Handled in Validate Topology at 8.3
Point, line and polygon feature classes inside of the geodatabase can contain shape attributes. Shape attributes include elevation values (Z value), measure values (M value), and Point IDS. All of
these attributes are stored on the vertex of each feature in a feature class that contains shape attributes.
Simple point, line and polygon feature classes with or without shape attributes can participate in a topology. Topology is used to measure and improve the quality and spatial accuracy of features in
its participating feature classes. Topology uses a process called validation to measure and ensure correctness of features. Validation snaps geometry between features together, given a tolerance
This document describes how Z values are altered when validating a topology in ArcGIS 8.3.
The handling of Z values for ArcGIS 8.3, as described in this article, will change. Future versions of ArcGIS, such as 9.0 or 9.1 will describe a new implementation of this function.
During the validation process two operations are used to determine the Zs of vertices; interpolation and collision.
Interpolation is used to determine the vertex attributes of the new vertices when a segment is subdivided.
The interpolation for Z’s is :
Zi = FZ1 + (1-F)Z2
where Zi is the interpolated measure, Z1 and Z2 are the measures at the end points, and F is the ratio of the subdivision. Ranges are between 0.0 and 1.0, which means that at F = 0.5 the average is
Different vertices with the same XY are represented by the same point. When two points are joined together because of this, they are said to 'collide'. In a case of collision there are two
possibilities: the Z-ranks are equal or they are not.
If the ranks are not equal, the resulting point is assigned the Z-value and Z-rank of the point with the highest rank.
If the ranks are equal, a weighted Z average is assigned to the output points.
Vertices on polylines never collide because they live in different planes. For polygons, newly created vertices can collide because they are considered to be in the same plane (planar representation).
EXAMPLE 1: POLYGONZs : EQUAL Z RANKS
For Zs (Equal Z ranks) aware polygons the attributes of the vertices created by cracking/clustering or the shared vertices are determined by a weighted linear interpolation followed by a weighted
average of the coincident vertices at a specific location. Refer to the section above: Vertex attributes Interpolation and Collision.
In this example the original features contain constant Zs values and have the same ranks. Polygon A has a constant Zs value of 100 for all vertices, and polygon B a value of 200.
DETAILED STEPS: Equal Zs Ranks
STEP 1 (interpolation): The point A2 is interpolated on the line segment A0-A3. The equation used is Zi = FZ1+ (1-F)Z2. Because Z1 and Z2 are both 100, the value of F is irrelevant, and Zi will be
100 as well.
STEP 2 (interpolation): Computation of the point A1, which is interpolated on the line segment A0-A2. This proceeds as under step 1, with the same output.
STEP 3 (collision): Computation of the weighted average for points A2 and B2. Because both points will have a weight of 1 (A2 interpolated, B2 initially assigned), the output Z will be the average of
100 and 200, or 150.
STEP 4 (collision): Computation of the weighted average for points A1 and B1. This is the same as step 3.
EXAMPLE 2: POLYGONZs UNEQUAL Zs RANKS
When working with Zs aware data validate is introducing a new concept: the Zs rank. If there are z-aware feature classes in the topology, rank them so that the z-values of vertices collected with
higher accuracy are not changed when snapping with the z-values of features of lower rank. A rank of 1 is the highest rank. Refer to the section above: Vertex attributes Interpolation and Collision.
In this example the original features contain constant Zs values. Polygon A has a constant Zs value of 100 for all vertices, and polygon B has a value of 200. The polygon A has a rank of 1 and the
polygon B has a rank of 2.
In this example after validate the features attributes are modified. The polygon A has a higher rank (rank of 1) than the polygon B (rank of 2). The interpolation steps are the same as in the
previous example. The difference is in the collision step, the resulting Z values will be those of the points of polygon A, because it has the higher rank.
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Acute Angle – Definition, Formula, Types, Examples, FAQs
Acute Angle
Angles less than 90 degrees are called acute angles. When the time is 11 o'clock, for instance, the angle formed by the hour and minute hands is acute. Acute angles are defined as 30°, 40°, 57°, and
so on.
What is an Acute Angle?
An angle is formed when two rays meet at a vertex. Acute angles are those that are smaller than 90 degrees in length. In the diagram below, the angle formed by 'Ray 1' and 'Ray 2' is acute. When you
divide a right angle in half, you get two acute angles.
Also Check: Types of Angles
Acute Angle Definition
Acute angles are these which are less than 90 degrees, or between 0 and 90 degrees. Some examples include 60 degrees, 30 degrees, 45 degrees, and so on. The interior angles of an acute triangle are
all less than 90 degrees. An equilateral triangle is an acute triangle because the internal angles measure 60 degrees.
Acute Angle Degree
An acute angle is one which is less than 90 degrees, or less than a straight angle, as we taught in the previous section. Acute angle degrees include 63°, 31°, 44°, 68°, 83°, and 85°. The acute angle
degree ranges from 0 to less than 90 degrees as a result. Below are some diagrammatic representations of acute angles.
Real-Life Examples of Acute Angles
Acute angles are defined as those that are larger than 0° but less than 90° in geometry. As a result, acute angles include 45 degrees, 5 degrees, 28 degrees, 49 degrees, and 89 degrees.
Here are some real-life examples of acute angles.
• Watermelon slice that has been cut into little pieces.
• Some examples of the angles formed by a clock's hour and minute hands.
• When the beak of a bird is fully open.
• When the mouth of a crocodile is open.
Acute Angle Triangle Properties
In an acute triangle, all of the angles are less than 90 degrees. When all three angles of a triangle are 60 degrees, the triangle is termed an equilateral triangle. There are three forms of acute
triangles: acute scalene, acute isosceles, and equilateral triangles. The acute triangle is one of several different triangle types. In the triangle below, all of the interior angles are less than 90
degrees. As a result, the shape is referred to as an acute triangle.
Acute Angle Formula
We have an acute angle triangle formula, often known as the triangle inequality theorem for acute angle triangles, similar to the Pythagoras theorem for right triangles. The total of the squares of
the two sides of a triangle is bigger than the square of the largest side, according to this rule. If the sides of ABC measure a,b,c, with c being the biggest, then a2 + b2 > c2. In other words, an
acute triangle is defined as a2 + b2 > c2.
Frequently Asked Questions on Acute Angle
Here are the seven types of angles commonly studied in geometry:
• Acute Angle: Measures less than 90 degrees.
• Right Angle: Measures exactly 90 degrees.
• Obtuse Angle: Measures more than 90 degrees but less than 180 degrees.
• Straight Angle: Measures exactly 180 degrees.
• Reflex Angle: Measures more than 180 degrees but less than 360 degrees.
• Full Angle: Measures exactly 360 degrees.
• Supplementary Angles: Two angles whose measures add up to 180 degrees.
Yes, an acute angle can be 45 degrees. An acute angle is any angle that measures less than 90 degrees, so 45 degrees qualifies as an acute angle. It is one of the common measures for acute angles in
Three examples of acute angles are:
• 30 Degrees: An angle that measures 30 degrees, which is less than 90 degrees.
• 45 Degrees: An angle that measures 45 degrees, often used in geometric shapes and trigonometric functions.
• 60 Degrees: An angle that measures 60 degrees, commonly seen in equilateral triangles where all angles are acute.
The main difference between acute and obtuse angles is their measures. An acute angle is always less than 90 degrees, while an obtuse angle is greater than 90 degrees but less than 180 degrees. For
instance, a 60-degree angle is acute, whereas a 120-degree angle is obtuse.
No, an angle cannot be both acute and obtuse. An acute angle measures less than 90 degrees, while an obtuse angle measures more than 90 degrees. Each angle fits into only one of these categories
based on its measure.
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5 Easiest Ways To Juggle Four Balls
Once you have learnt to juggle 2 or 3 balls it is time to learn the next step, which is juggling 4 balls at a time. At this stage you must be a intermediate juggler and know a thing or two about
Here Is The Easiest Way To Juggle Four Balls:
Learning to juggle 4 balls is definitely more difficult than say learning to juggle 3 at a time.
The easiest way to start juggling 4 balls is to:
juggle with 2 balls in each hand (this is called the fountain)
You practice this move in one hand with 2 balls, and once this is perfected you move onto the other hand.
You toss one ball and then toss the second when the first ball peaks.
In this guide we are going to go through step by step what to do and ways you can juggle 4 balls.
How To Juggle 4 Balls?
Juggling with three balls is definitely challenging for beginners, but when you’ve mastered those patterns and gained more confidence, four balls is the natural progression.
However, juggling four balls is quite different to juggling three, even though it is just one extra ball, so you will need to learn new patterns and work on your hand-eye coordination even more.
The easiest way to learn how to juggle four balls (since it’s an even number) is to juggle two in each hand (the fountain).
This will take a bit of work to perfect, but once you can juggle two in one hand, you can practice adding in the second hand with the other two balls.
To make these throws successful, you toss one ball and then toss the second when the first ball peaks.
That pattern is the simplest way to practice juggling with an even number of balls after spending so much time with just the three.
It will take a lot of concentration because you’ll need to do the same alternating throw pattern with both hands and adjust your stance according to what both hands are doing.
Getting these throws synchronised is difficult, but once you’ve perfected them and feel more comfortable juggling with four balls, you can move onto more complicated patterns.
Is It Easier To Juggle 4 Or 5 Balls?
Though the natural order of progression is three balls, then four, then five, some people actually find it easier to go straight from three to five.
This is because the number of balls is still odd, so the cascade pattern works the same, albeit with an extra two balls.
That number may seem intimidating at first if you’ve only used three so far, but you’ll already know how the sequencing of the pattern works.
However, working an additional two balls into the pattern means more balls to hold, more to throw and catch, and needing to have your eyes in five places at once.
If you’d find that to be overwhelming, you’re better off going from three balls to four.
Though you’ll need to learn new patterns and have to get used to an even number, you won’t feel as intimidated by the number.
It is typically easier to juggle four balls than five because you mainly need to work on the accuracy of your throws to make the patterns successful. After all, juggling is mostly muscle memory, so
when you’ve worked out exactly when to throw and catch, you’ll quickly get comfortable juggling with four balls.
After advancing to four balls, five balls will seem less of a stretch because you’ll be comfortable using a higher number and you’ll still have the muscle memory retained from juggling with three
balls to aid you in juggling with an odd number again.
How Do You Juggle A 4-ball Cascade?
Though the 3-ball cascade (and different variations of it) is a great juggling pattern, a 4-ball cascade is essentially impossible.
This is because juggling a cascade with an even number of balls disrupts the sequence of the pattern.
The 3-ball cascade is achieved by throwing one ball, throwing the second when the first peaks, catching the first in your empty hand and then throwing the third from your other hand when the second
has peaked.
It sounds confusing, but that is the easiest sequence to learn when juggling with three balls.
You can juggle the same pattern with any other odd number of balls (five, seven, etc), but it is very hard to do with an even number. This is because the number of balls determines how many beats
each ball must remain in the air before landing.
An even number of balls means that the ball thrown must stay in the air for an even number of beats. This number of beats requires it to land in the hand it was thrown from, which is not a
traditional cascade.
Put simply, the first and third balls would always be thrown and caught with the left hand, while the second and fourth would be thrown and caught with the right, so no alternating hands as you do
with a cascade.
However, you can technically juggle a 4-ball cascade, though it doesn’t use the traditional rules.
Instead, it includes an extra invisible ball, so you juggle the four balls with the timing of five and allow the balls to alternate hands. This will take time and practice, but it can be done.
How Do I Add A Fourth Ball To Juggle?
When adding the fourth ball, the first thing you need to remember is that you’ll need to be patient because you are essentially training yourself from the start again.
Of course, the hand-eye coordination, stance, and pivots that you’ll have worked on while juggling with three balls are essential, but you won’t be able to do patterns like the 3-ball cascade because
using an even number changes the patterns up. Here is a good article on how to juggle 3 balls.
When adding a fourth ball, it is best to work on your weaker hand first.
When juggling with three balls, your hands work in tandem and support each other to an extent. With four balls, they need to work more independently.
Start by simply holding two balls in each hand to get used to the feeling. Then, practice each hand individually, throwing up one ball and then throwing the second when it peaks.
Keep practicing with just your weaker hand until you have got used to the sequence.
Practice your second hand exactly the same. Once you’ve mastered the pattern on individual hands, try the two at once.
By focussing on just two balls initially before bringing the four together, it should feel more fluid to add the fourth ball and you’ll barely even notice it.
What Are Some 4 ball juggling patterns?
Aside from the 4-ball fountain, there are other great 4 ball juggling patterns to learn. The first that you should try after mastering the fountain is an asynchronous fountain. You do this by
throwing and releasing the balls at different times, so when you throw a ball from one hand, the other should be catching another.
The 4-ball shower is also a good choice. For this pattern, you start with two balls in each hand and throw them in a circular motion. So, throw a ball up from your left hand and then pass a ball from
your right hand over to the left, continuing this sequence to achieve constant circular throws.
The Wimpy also requires two balls in each hand, though they switch hands during the pattern. Your hands work exactly the same, though mirrored, and move inside and outside in tandem. Throw balls up
and catch them in the opposite hand, always throwing one higher than the other.
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Seminar Analysis and Theoretical Physics
On magnetoviscoelastic fluids in 3D
I will introduce a thermodynamically consistent model for a magnetoviscoelastic fluid in 3D. Existence, uniqueness, and asymptotic behavior of strong solutions is studied in the framework of
quasilinear parabolic systems and maximal regularity in $L_p$-spaces. It will be shown that the critical points of the entropy functional with prescribed energy correspond exactly to the equilibria
of the system. Constant equilibria are normally stable: solutions that start close to a constant equilibrium exist globally and converge exponentially fast to a (possibly different) constant
equilibrium. Moreover, it will be shown that the negative entropy serves as a strict Lyapunov functional and that every solution that is eventually bounded in the topology of the natural state space
exists globally and converges to the set of equilibria.
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Inferential Statistics
Introduction to Inferential Statistics
Social science deals with two main concepts as far as statistics are concerned: population and sample. Population refers to the total number of objects, groups, individuals or any other thing being
studied while a sample is a subset selected from the overall population. Many studies rely on samples because they are more manageable, less time consuming and cheaper. Inferential statistics help
researchers to make generalizations about a population based on the sample studied. They are used to reach conclusions that go beyond the immediate data in question.
The challenge
Inferences are prone to errors because they are made from a relatively small subset of a population. Since the results cannot be said to be a complete representation of the characteristics of a
population, they are usually expressed in form of probability. For example, they may give a probability of 93 percent.
A sample may not be representative of the target population because of two problems:
• Sampling errors that occur by chance
• Sample bias, which stems from inadequate design
What Inferential Statistics Do
Since sample bias is a problem with the research design, inferential statistics does not attempt to correct it. The statistics are instead used to address sampling errors, and they are typically used
to test hypotheses.
The calculations help in determining the probability and margin of error, which are important in accepting or rejecting hypotheses. If the value of probability is smaller than required, the
hypothesis is rejected because it means the sample is not truly representative of the population. The probability or p value gives an estimate of how often the results can be obtained by chance if
the hypothesis used is true.
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Inferential statistics analyze at least two variables, and the types of variables determine the type of inferential statistics applied. The type of analysis using two variables is known as bivariate
analysis while that using more than two variables is called multivariate analysis.
The statistics help researchers determine how strong the relationship between independent and dependent variables is. If a hypothesis positively relates wages to age, for example, age will be the
independent variable while wages will be the dependent variable.
Types and Examples of Variables
• Ratio/Interval, such as wage or age
• Nominal, such as gender
• Ordinal, for example class grades
Common types of tests include:
• Chi-square
• T-test
• Correlation
• Analysis of Variance
• Analysis of Covariance
• Cluster Analysis
• Factor Analysis
• Discriminant Function Analysis
• Multidimensional Scaling
Major inferential statistics are in a family of statistical model called General Linear Model.
When Can Inferential Statistics be used?
• When there is a complete list of members of a population
• When researchers take a random sample from the population
• When the researchers can determine that the sample size is sufficiently large using pre-established formula
While there are various types of inferential statistics, they all determine how different variables compare to one another. Calculations can be done either by hand or computer.
Copyright worldcong2012.org All Rights reserved
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s21bac (ellipint_symm_1_degen)
NAG CL Interface
s21bac (ellipint_symm_1_degen)
1 Purpose
s21bac returns a value of an elementary integral, which occurs as a degenerate case of an elliptic integral of the first kind.
2 Specification
double s21bac (double x, double y, NagError *fail)
The function may be called by the names: s21bac, nag_specfun_ellipint_symm_1_degen or nag_elliptic_integral_rc.
3 Description
calculates an approximate value for the integral
$RC (x,y) = 12 ∫ 0 ∞ dt (t+y) . t+x$
$x\ge 0$
$ye 0$
This function, which is related to the logarithm or inverse hyperbolic functions for $y<x$ and to inverse circular functions if $x<y$, arises as a degenerate form of the elliptic integral of the
first kind. If $y<0$, the result computed is the Cauchy principal value of the integral.
The basic algorithm, which is due to
Carlson (1979)
Carlson (1988)
, is to reduce the arguments recursively towards their mean by the system:
$x0=x y0=y μn=(xn+2yn)/3, Sn=(yn-xn)/3μn λn=yn+2xnyn xn+1=(xn+λn)/4, yn+1=(yn+λn)/4.$
The quantity
$n=0,1,2,3,\dots \text{}$
decreases with increasing
, eventually
$|{S}_{n}|\sim 1/{4}^{n}$
. For small enough
the required function value can be approximated by the first few terms of the Taylor series about the mean. That is
$RC(x,y)=(1+3Sn210+Sn37+3Sn48+9Sn522) /μn.$
The truncation error involved in using this approximation is bounded by
and the recursive process is stopped when
is small enough for this truncation error to be negligible compared to the
machine precision
Within the domain of definition, the function value is itself representable for all representable values of its arguments. However, for values of the arguments near the extremes the above algorithm
must be modified so as to avoid causing underflows or overflows in intermediate steps. In extreme regions arguments are prescaled away from the extremes and compensating scaling of the result is done
before returning to the calling program.
4 References
Carlson B C (1979) Computing elliptic integrals by duplication Numerische Mathematik 33 1–16
Carlson B C (1988) A table of elliptic integrals of the third kind Math. Comput. 51 267–280
5 Arguments
1: $\mathbf{x}$ – double Input
2: $\mathbf{y}$ – double Input
On entry: the arguments $x$ and $y$ of the function, respectively.
Constraint: ${\mathbf{x}}\ge 0.0$ and ${\mathbf{y}}e 0.0$.
3: $\mathbf{fail}$ – NagError * Input/Output
The NAG error argument (see
Section 7
in the Introduction to the NAG Library CL Interface).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
Section 3.1.2
in the Introduction to the NAG Library CL Interface for further information.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
for assistance.
Section 7.5
in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
Section 8
in the Introduction to the NAG Library CL Interface for further information.
On entry, ${\mathbf{y}}=0.0$.
Constraint: ${\mathbf{y}}e 0.0$.
The function is undefined and returns zero.
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\ge 0.0$.
The function is undefined.
7 Accuracy
In principle the function is capable of producing full machine precision. However, round-off errors in internal arithmetic will result in slight loss of accuracy. This loss should never be excessive
as the algorithm does not involve any significant amplification of round-off error. It is reasonable to assume that the result is accurate to within a small multiple of the machine precision.
8 Parallelism and Performance
Background information to multithreading can be found in the
s21bac is not threaded in any implementation.
You should consult the
S Chapter Introduction
which shows the relationship of this function to the classical definitions of the elliptic integrals.
10 Example
This example simply generates a small set of nonextreme arguments which are used with the function to produce the table of low accuracy results.
10.1 Program Text
10.2 Program Data
10.3 Program Results
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Longing for Winners - Persistence in QMIT Smart Betas
October 2024 -- Xilin Chen, Weichuan Deng, Boyu Fang & Milind Sharma
Thanks to Oleg Kolesnikov & Amit Sardar for technical input
This paper aims to replicate, validate, and extend the existing literature on factor momentum using a proprietary dataset that has been deployed in live trading for the past 5.75 years in addition to
the full 24.75-year history used in documenting the existence of factor momentum via Time Series Momentum (TSMOM) strategies applied to the 18 QuantZ Enhanced Smart Beta (ESB) composites. We
demonstrate that TSMOM improves risk-adjusted returns when applied to these ESBs. Such outperformance is corroborated by testing the TSMOM strategies with 5.75 years of live trading data.
The best-performing ESB-based TSMOM strategies outperform both the equal-weighted QuantZ ESBs benchmark and the QuantZ Enterprise 18 composites in terms of risk-adjusted returns (i.e., Sharpe,
Sortino, and Calmar ratios) in our 24.75-year full history backtesting. Further, we validate that excluding cross-sectional momentum ESBs can enhance the performance of TSMOM strategies as previously
documented in the literature. The litmus test is outperformance in the 5.75-year live trading data which also exhibits the defensive, downside protection characteristics noted in the TSMOM literature
by Gu and Mulvey (2021) [5].
The QMIT founder has been part of multiple pioneering industry applications of factor investing in the Quantamental realm over the past 28 years. QMIT pioneered the application of Machine Learning
ensembles towards the formulation of a spanning set of Enhanced Smart Beta (ESB) composites in its attempt at taming the factor zoo. These ESB composites combine hundreds of factors deployed by the
author in various hedge and mutual fund strategies since the 1990s. In this paper, we deploy ESBs as the ingredients of TSMOM strategies to ascertain whether they outperform a baseline equal-weighted
ESB benchmark as evidence of persistence in such ESBs.
Literature Review
• Cross-Sectional Momentum (XSMOM): The study of momentum started with the cross-sectional momentum (XSMOM) effect, which refers to the tendency of stocks that have performed well relative to their
peers to continue to perform well in the near future. XSMOM strategies focus on the cross-sectional ranking of past returns (“relative performance”) across assets, in order to construct long vs
short decile, quintile, or tercile portfolios possibly with a double sort. Jegadeesh and Titman (1993) [1] showed that buying recent top quintiles (“winners”) and selling recent bottom quintiles
(“losers”) among individual stocks in NYSE and AMEX can yield significant positive returns when using equal holding and lookback periods (the latter also referred to as “portfolio formation
period” in some literature) ranging from 3 to 12 months, with reversals observed in the short term (out to 1 month) (Jegadeesh 1990; Lehmann 1990) as well as the long term (ranging from 3 to 5
years) (De Bondt and Thaler 1985). Carhart (1997) constructed a 4-factor model by adding the 1-year individual stock-level XSMOM effect to the original Fama-French 3-factor model (FF3), which was
based purely on size, style, and beta (Fama and French 1993), and the Carhart 4-factor model greatly improved on the average pricing errors of the CAPM and FF3 (Carhart 1997).
• Time Series Momentum (TSMOM): The TSMOM literature focuses on nominal past performance instead of cross-sectional rankings as in the XSMOM case. TSMOM strategies go long the assets with positive
returns (“winners”) in the lookback period and short the ones with negative returns (“losers”) in the lookback period. Moskowitz, Ooi, and Pedersen (2012) presented evidence of positive TSMOM
effects in all 58 financial instruments they tested (including equity index futures), and 52 of them were significantly non-zero at a 5% significance level. They noted that TSMOM has a payoff
profile similar to an option straddle on the market, as per Fung and Hsieh (2001) observed in trend-following strategies, given the significantly positive coefficient on squared market returns.
Namely, TSMOM performs well during extreme market conditions (both up and down), as it typically takes long positions during significant market upswings, and goes short the during market
downturns (Moskowitz, Ooi, and Pedersen 2012). In particular, their TSMOM strategy outperformed the S&P 500 during the crash periods (Moskowitz, Ooi, and Pedersen 2012). Unlike XSMOM strategies,
the TSMOM effects usually do not come with short-term reversals. Thus TSMOM studies, like Gu and Mulvey (2021), do not exclude the most recent month from the lookback period like the
implementation of the XSMOM strategies. E.g., consider a TSMOM strategy with a 6-month lookback period and monthly rebalancing - at the rebalancing date, the strategy determines if an asset or a
factor is a “winner” or a “loser” based on the sign of the cumulative return of the past 6 months.
• Factor Momentum: Researchers have discovered that both the XSMOM and TSMOM effects are present not only at the single asset level but also at the industry and factor levels. In the U.S. equity
market, they find that factor-level momentum not only subsumes single asset- or industry-level momentum effects, but actually drives them. Arnott et al. (2021) [2] demonstrated that factor XSMOM
strategy retained highly significant and positive alphas (of +70 bps and +66 bps per month, with t-stats of 6.60 and 6.12, respectively) despite controlling for the FF5 model and FF5 plus
Carhart’s single stock-level momentum factor (Carhart 1997). Remarkably, the profitability of the short-term factor momentum XSMOM effect does not stem from single-stock level XSMOM effect and is
a stand alone phenomenon. Arnott et al. (2021) also showed that industry XSMOM momentum is a byproduct of the factor XSMOM. Gupta and Kelly (2019) [3] and Ehsani and Linnainmaa (2022) [4] found
that momentum at the individual stock level is largely driven by factor TSMOM persistence, and documented significant positive returns for factor momentum strategies (due to the positive
autocorrelation in returns of most factors).
• Factor Time Series Momentum: Gupta and Kelly (2019) and Ehsani and Linnainmaa (2022) demonstrate that strategies based on factor TSMOM are purer, more robust, and more reliable approaches for
capturing the persistence in factor performance compared to factor XSMOM strategies. Gupta and Kelly (2019) observed that factor TSMOM yields more stable returns than factor XSMOM and exhibits
positive alpha when controlling for factor XSMOM, while factor XSMOM generates negative alpha when controlling for factor TSMOM, indicating that factor TSMOM subsumes factor XSMOM. Ehsani and
Linnainmaa (2022) further confirm this subsumption. Inspired by the straddle-like qualities of TSMOM strategies observed in asset classes beyond equities (Moskowitz, Ooi, and Pedersen 2012), some
researchers have explored whether market-neutral equity factor TSMOM could demonstrate defensive characteristics during market crashes. Gu and Mulvey (2021) investigated the potential for crash
protection using a dollar-neutral factor TSMOM strategy. Their findings showed that buy-winner TSMOM strategies achieved a Sharpe ratio exceeding 2.0 during crash periods, and incorporating these
strategies into their diversified core portfolios significantly enhanced risk-adjusted returns.
• Road Map: We follow the framework and procedures in Ehsani and Linnainmaa (2022) and Gu and Mulvey (2021) to implement TSMOM strategies and test for persistence amongst the “winners leg” &
“losers leg" portfolios of equal-weighted factors. The factor dataset is comprised of QuantZ’s 18 Enhanced Smart Betas (ESBs) defined below. Thereafter, we try combinations of parameters such as
lookback periods, rebalancing frequencies, and dollar-vs. beta-neutrality. The Correlation Analysis section confirms the persistence of TSMOM in the returns of ESBs. The Results section shows the
back-tested performance on our full 24.75-year history (Jan 2000 - Sept 2024) highlighting the strong performance of TSMOM strategies across different configurations and identifies the top
performers. Finally, the Corroboration section validates the effectiveness of these strategies in 5.75Y of live trading, demonstrating their strong risk-adjusted returns and defensive performance
during volatile periods. It is noteworthy that the factor construction methodology and the history of our dataset are different from Gu and Mulvey (2021) who only used 48 out of 55 factors (each
constructed with a single explicitly defined ranking characteristic) from Ehsani and Linnaimaa (2019) and Kozak, Nagel, and Santosh (2020), covering the period from 1975 to 2019, while we utilize
the 18 ESBs (based on hundreds of raw factors) from QuantZ, spanning January 2000 till Sept 2024.
Since our implementation of the Gu and Mulvey (2021) factor TSMOM strategy is based on QuantZ’s Enhanced Smart Betas (ESBs) it's important to describe our foundational building blocks. QuantZ
provides ML-enhanced Smart Betas allowing one to express almost any linear view on Equities based on ensemble learners which determine the optimal combination of raw factors intra cohort. The ESBs
were designed to be higher octane composites which could outperform naive well known factor representatives of each cohort. Definitions are presented in the appendices.
Our researchers have drawn upon their decades of collective experience to identify, clean, and test 18 factor cohorts from which we construct the 18 Enhanced Smart Betas so that you may directly
deploy these indispensable building blocks cost-effectively towards the creation of quant equity strategies for which our Composite Signals (based on such ESBs) are good proxies. Given that the set
of N choose k combinations in this case is quite large, it’s particularly instructive to focus on the curated signal composites we have created.
We use the monthly ESB returns data between January 2000 and Sept 2024. The data from 2000-2018 are from backtesting, and the data since 2019 pertain to the live trading history. Following the
tradition of momentum research, including Jegadeesh and Titman (1993) and subsequent literature, such as some notable works in factor TSMOM, like Ehsani and Linnainmaa (2022) and Gu and Mulvey
(2021), we do not account for the impact of transaction costs or rebates.
If we want to include transaction cost (but still ignore short rebates): Although the mentioned papers do not address transaction costs or rebates, an important paper on factor TSMOM by Gupta and
Kelly (2019) does consider transaction costs. They assume, based on Frazzini, Israel, and Moskowitz (2015), that an incremental 1% of factor-level turnover incurs a 10 basis point transaction cost.
However, since most major studies do not account for rebates, we have chosen to exclude rebates from our analysis.
We follow the framework and procedure in Ehsani and Linnainmaa (2022) and Gu and Mulvey (2021) to implement TSMOM strategies. For a TSMOM strategy of k-month lookback, at the beginning of each
holding period, all the factors are classified into either “winners” or “losers” based on previous cumulative returns over the most recent k months before this holding period, and we construct a
“winners leg” portfolio as an equal-weighted portfolio of winner ESBs, and a losers leg as an equal-weighted portfolio of “loser” ESBs:
We backtest the factor TSMOM strategies based on our 18 ESBs (not on their constituent factors or single stocks), and as Gu and Mulvey (2021) did, we include three ways of implementation:
• Long-Short (LS), in which we go long the winners legs and short the losers legs,
• Long winners (LW), in which we buy and hold the winners legs only, and
• Long losers (LL), in which we buy and hold the losers legs only.
Since each ESB is constructed as a market-neutral long-short portfolio of the individual stocks in QuantZ’s investment universe, the long-only portfolios consisting of the ESBs are also
market-neutral long-short portfolios of these individual stocks.
The primary benchmark is the equal-weighted portfolio across all ESBs in the investment universe. We also bring in Enterprise 18, a plug-and-play composite signal comprised of all ESBs, as an
alternative goalpost based on an active Quantamental MFM (multi-factor model) strategy. Instead of being a portfolio of the 18 ESBs, the Enterprise 18 signal captures the overlap among the underlying
ESBs, resulting in a long-short market-neutral portfolio that is higher-octane and more diversified, allowing for higher returns and improved risk-adjusted performance compared to each individual
ESB. While most factor studies such as Gu and Mulvey (2021) limit the analysis to dollar-neutrality we also show the beta-neutral case for contrast given that the latter is bona fide market neutral
particularly since some ESBs like Risk (aka Low Vol or akin to the BAB factor) and Momentum can have substantial & time varying net betas.
We evaluate and rank the strategies predominantly by their risk-adjusted return, assessed in terms of Sharpe ratio, Sortino ratio, and Calmar ratio [6]. For all tables of performance metrics in this
study, we rank the strategies based on their Sharpe Ratios.
It is noteworthy that the factor construction methodology and the history of our dataset is rather different from Gu and Mulvey (2021). Gu and Mulvey used 48 out of 55 factors (each constructed with
a single explicitly defined raw ranking characteristic) from Ehsani and Linnaimaa (2019) and Kozak, Nagel, and Santosh (2020), covering the period from 1975 to 2019, while we utilize the spanning set
of 18 ESBs from QuantZ, covering the most recent ~24.75y of (January 2000 till Sept 2024). The last 5.75y were collected live (hence no possibility of look ahead or survivorship biases) with daily
granularity which would be rare in the academic literature. Furthermore, the QuantZ ESBs are not single company characteristics but dynamic factor composites generated by the ensemble learners. These
differences may account for the resulting differences in the performance and optimal lookback periods.
3. Choice of lookback period and rebalancing frequency
The lookback period used to compute the past “absolute performance” is crucial in determining the winner and loser factor portfolios while the rebalancing frequency determines the holding period.
Moskowitz, Ooi, and Pedersen (2012) used the 12-month lookback period to determine winners and losers while Gu and Mulvey (2021) tried lookbacks of 1/3/6/12/24 months. In this study, we use lookback
periods of 1/3/6/12 months, with monthly or quarterly rebalancing.
4. With and without Momentum
Gu and Mulvey (2021) suggested excluding XSMOM factors in time series factor momentum strategies to avoid the impact of the correlation between factor momentum TSMOM and individual stock momentum
XSMOM. Their approach is in line with the observation of Ehsani and Linnainmaa (2022), viz., that the individual stock-level XSMOM factor does not have significant returns persistence. This study
will contrast the performance of the factor TSMOM strategy with and without the individual stock-level XSMOM ESBs, which are Momentum (MOM) and Enhanced Momentum (EnMOM).
5. Dollar Neutral vs. Beta Neutral
As previously noted, certain ESBs like Risk (aka Low Vol or akin to the BAB factor) and Momentum (EnMOM or MOM) can have substantial & time varying net betas which means that inferences made using
the academic default of dollar-neutrality can often be substantially misleading (to wit Gu and Mulvey (2021) as well as Ehsani and Linnainmaa (2022)). Therefore, we show both dollar-neutral and
beta-neutral results allowing the reader to make contrasting inferences. In particular, we note that deploying beta-neutral ESBs can improve risk-adjusted returns, which extends the existing strand
of literature.
Correlation Analysis
The success of TSMOM requires persistence of the winner/ loser factor portfolios which entails a statistically significant & positive correlation between the lookback and holding period returns. The
correlation charts below are based on the full 24.75 years of our history (i.e. in-sample), which means that they cannot be used directly to select the optimal lookback period or the rebalancing
We first display the first-order autoregression, AR(1) coefficients below, in line with Gu and Mulvey (2021). Amongst the 18 dollar-neutral (or beta-neutral) ESBs, 12 (or 13) ESBs have a positive AR
(1) coefficient, and in both dollar-neutral and beta-neutral cases, 6 of the ESBs are statistically significant at a 95% confidence level. This is suggestive of factor persistence i.e., profitably
going long the winners while reducing exposure to (or even shorting) the losers. Indeed, the 1 month dollar-neutral (monthly rebal) chart below shows fundamental ESBs like Leverage, Relative Value,
Profitability, Efficiency, Growth & CSU exhibit serial correlation while some of the technical ones like EnMom, Mom, Reversals & Short Interest mean revert (with only SIRF being statistically
significant). With beta-neutrality the statistical significance is even more pronounced with ART as well as Deep Value coefficients becoming significant as well.
Figure 1: Correlation between holding period returns and lookback period returns, with the 1-month lookback period, rebalanced monthly, i.e. AR(1) (Dollar-Neutral ESBs)
Figure 2: Correlation between holding period returns and lookback period returns, with the 1-month lookback period, rebalanced monthly, i.e. AR(1) (Beta-Neutral ESBs)
However, Moskowitz, Ooi, and Pedersen (2012) noted that only using autocorrelation tests to evaluate the validity of the TSMOM effect can miss some significant predictability because for monthly data
AR(1) only shows the correlation between the 1-month lookback vs forward period. Given the various combinations of {lookback x holding} periods used in the literature we generalize the notion to
simply consider correlations between the lookback {1/3/6/12 months} vs forward holding periods {1 or 3 months}. As we vary the combinations of {lookback x holding} periods the correlations & their
significance changes - as one might expect - with the occasional flip in sign as noted for CSU, DV and Stability.
Figure 3: Correlation between holding period returns and lookback period returns, with the 3-month lookback period, rebalanced monthly (Dollar-Neutral ESBs)
Figure 4: Correlation between holding period returns and lookback period returns, with the 3-month lookback period, rebalanced monthly (Beta-Neutral ESBs)
Amongst all combinations, the {6-month x 1-month} seems most promising given the largest number of ESBs that have a significantly positive correlation in both dollar-neutral and beta-neutral
configurations. Among the 18 dollar-neutral ESBs, 12 ESBs have a positive correlation, 7 of them are statistically significant at a 95% confidence level, and only Analyst Ratings & Targets (ART) ESB
has a negative correlation that is narrowly significant. Under beta-neutrality, 15 of 18 ESBs have a positive correlation, 9 of them are statistically significant at a 95% confidence level, and none
of the ESBs have a significantly negative correlation. This result indicates that the 6-month lookback with monthly rebalancing configuration may be the optimal set-up for TSMOM strategies based on
Figure 5: Correlation between holding period returns and lookback period returns, with the 6-month lookback period, rebalanced monthly (Dollar-Neutral ESBs)
Figure 6: Correlation between holding period returns and lookback period returns, with the 6-month lookback period, rebalanced monthly (Beta-Neutral ESBs)
Performance of TSMOM Strategies (Jan 2000 - Sept 2024)
We noted the formulation of various TSMOM strategy combinations in the implementation section above. In this section, we backtest TSMOM strategies over 24.75 years across four distinct scenarios
using either dollar-neutral or beta-neutral ESBs, and either including or excluding the two XSMOM ESBs. Each scenario is coupled with three possible implementations (long-short, long winners, and
long losers) for a total of 12 cases further multiplied by lookback periods (1, 3, 6, and 12 months) and 2 rebalancing frequencies (monthly and quarterly) which results in a total of 96
configurations. To recap that's 24 configurations * 4 scenarios.
The cumulative performance of each strategy is visualized as the value-added monthly index (or VAMI pegged at $1 at inception) as shown in the figures below for the top 2 Sharpe Ratio performers in
each of the three ways of implementation <LS, LW, LL>. Since one of our benchmarks, Enterprise 18, has much higher returns and volatility than any ESB, its VAMI may seem to dominate all others (see
Figure 7 below). Thus, in order to evaluate the strategies based on their risk-adjusted performance, we scale their VAMIs to match the volatility level of the 18-ESB equal-weighted portfolio (EW18)
benchmark in all other VAMI plots hereafter which will also visually bring Enterprise 18 down to a comparable vol scale.
Figure 7: $-Neutral: Best 2 in each direction of strategy (without vol scaling)
The TSMOM strategies (like their ESB brethren) need to be anchored or initialized with enough history (for the lookback) to get started. For ESBs that means equal weighting constituent factors for
the 1st 24 months before the ensembles can start learning. Similarly, each TSMOM strategy sits in the EW18 benchmark portfolio till it accumulates enough data for its lookback period to start picking
winners & losers. Ironically, the NASDAQ crash years at the beginning of the 24.75 year period are in fact the strongest for our ESBs hence any methodology such as sitting in cash would dramatically
handicap longer lookbacks.
Implementation - (LS/LW/LL): In all four scenarios {BN/DN x with/without XSMOM ESBs}, we observe that the Long Winners implementation of TSMOM strategies systematically outperforms all others, with
the Long Losers trailing significantly behind. This highlights the dominance of the time series momentum (TSMOM) effect over the 24.75-year history. However, it's noteworthy that even the laggard
Long Losers generated positive returns with the best of the cohort being as high as 0.88 Sharpe (LL, BN, 3mo/3mo, with XSMOM) which is quite good on a stand-alone basis. That's why the Long/Short
implementation is stymied by the strength of the shorts.
Beta vs $ Neutrality: We also observed that beta-neutral configurations generally exhibit much higher risk-adjusted returns for the top-performing strategies than the dollar-neutral ones, ceteris
paribus due to significantly lower volatility and downside deviation.
With/ Without XSMOM: Furthermore, in both dollar-neutral and beta-neutral configurations, the exclusion of XSMOM ESBs resulted in an overall increase in the risk-adjusted returns for TSMOM
strategies, especially for dollar-neutral cases.
Best strategy {Lookback x Rebal period) -- Amongst all TSMOM strategies, the Long Winners TSMOM strategies with a 6-month lookback period and rebalanced monthly (LW 6m/1m) performed the best in all
four scenarios, followed by Long Winners TSMOM strategies with a 6-month lookback period and rebalanced quarterly (LW 6m/3m). LW 6m/1m TSMOM strategy outperformed both benchmarks in all 4 scenarios
under all 3 risk-adjusted metrics as per Tables 1-a & 1-b below.
Table 1-a: Dollar-Neutral ESBs
Table 1-b: Beta-Neutral ESBs
Table 1: Performance of the TSMOM Strategy for 24.75y
Take the top risk-adjusted TSMOM strategies, LW 6m/1m, for instance. In the dollar-neutral scenarios, the Sharpe ratio of the EW18 benchmark is 1.01. For the dollar-neutral LW 6m/1m strategies, the
one with XSMOM ESBs obtained a Sharpe ratio of 1.08, and the one excluding XSMOM ESBs significantly raised the Sharpe ratio to 1.22 (which is higher than the EW18 benchmark and higher than the one
with XSMOM ESBs). The dollar-neutral LW 6m/1m strategy (excluding XSMOM) also improved the Sortino ratio by 26.1% and the Calmar ratio by 50% compared to the EW16 benchmark.
Figure 8: VAMI of TSMOM Strategies - 16 ESBs ($-Neutral; ex-XSMOM) for 24.75y (vol scaled)
Table 2:Performance of Top-10 Strategies - 16 ESBs ($-Neutral; ex-XSMOM) for 24.75y
Although using TSMOM is also helpful in beta-neutral scenarios, it is noteworthy that the relative improvement in the Sharpe ratio for the top TSMOM strategies is much higher in the dollar-neutral
scenarios because the benchmark has a much lower Sharpe ratio to start with. In the beta-neutral scenarios, the Sharpe ratio of the equal-weighted benchmark (EW16) is 1.53 (a tough bar to clear). For
the beta-neutral LW 6m/1m strategies, the one with XSMOM ESBs obtained a trivially higher Sharpe ratio of 1.65, and the one excluding XSMOM ESBs further raised the Sharpe ratio to 1.72, and increased
the Sharpe ratio by around 12.4% relative to the EW16 benchmark. The relative improvement in the other two risk-adjusted return metrics, Sortino and Calmar, is also lower than the dollar-neutral
case, but the magnitude in each case is close to 30%, which is substantial.
Figure 9: VAMI of TSMOM Strategies - 16 ESBs (Beta-Neutral; ex-XSMOM) for 24.75y (vol scaled)
Table 3: Performance of Top-10 Strategies - 16 ESBs (Beta-Neutral; ex-XSMOM) for 24.75y [7]
The backtesting shows that using monthly-rebalanced Long Winner TSMOM strategies on the 16 non-XSMOM ESBs with a 6-month lookback period can help investors significantly improve the Sortino ratio and
the Calmar ratio of their ESB-based strategies in comparison to the EW16 strategy and the Enterprise 18 composite, with Sharpe ratio at a comparable level or even improved (especially in the
dollar-neutral cases). The desirable characteristics in terms of downside deviation and maximum drawdown (MDD) are in line with other studies such as Gu and Mulvey (2021).
Robustness checks
Considering that the extraordinarily profitable NASDAQ crash episode may distort the full period performance of the strategies, we first re-evaluate the performance of the strategies with
beta-neutral ESBs (without XSMOM ESBs) using the 22.75-year data starting with the 25th month (Jan. 2002) as robustness check in order to 1) remove the impact of the positive outlier NASDAQ crash
episode, and 2) omit the anchoring period of the ensembles when they simply match the equal-weighted benchmark (EW16) as with our TSMOM strategies. The Long Winner TSMOM strategies on the 16
non-XSMOM ESBs with a 6-month lookback still performed the best among all TSMOM strategies. However, it obtained a Sharpe ratio of 1.55, which is slightly higher than the one of the EW16 (1.54) but
lower than the Sharpe ratio of the Enterprise 18 composite (1.57). However, using the top TSMOM strategy still helps investors manage the downside deviation and drawdowns much better, cutting down
both by more than 50% resulting in a much higher Sortino ratio (3.89) and Calmar ratio (1.34) than the ones of the EW16 and the Enterprise 18 composite (see Appendix II).
Drivers of Performance
The dynamic regime dependent choice of ESBs in the <LW, LL, LS> portfolios may explain why the top TSMOM strategies have downside deviations and MDDs comparable to the equal-weighted benchmark yet
monetize a higher return. To further evaluate the potential drivers of performance we consider how the TSMOM strategy bets on each ESB. In Figure 10 and Figure 11, we visualize the number of ESBs in
the winners leg for each period and the winning rate of each ESB in the beta-neutral LW 6m/1m strategies. To aid visualization, we use the risk-off-risk-on spread (RORO), which is the difference
between the mean return of all risk-off ESBs and the mean return of all risk-on ESBs, as a contemporaneous proxy for the market regime. A lower RORO spread (like in 2008) is indicative of a Risk-Off
Figure 10: Number of ESBs in the Winners Leg of the LW 6m/1m Strategy, 16 ESBs (Beta-Neutral; ex-XSMOM) for 24.75y
Figure 11: Winning Rate of each ESB in the LW 6m/1m Strategy, 16 ESBs (Beta-Neutral; ex-XSMOM) for 24.75y
From Figure 10 we can observe that the Winners Leg of the LW 6m/1m Strategy encompasses nearly all ESBs (EW16) during the Risk-On periods when most ESBs will likely be positive over the lookback
period. Empirically, the strategies based on a 6-month lookback period seem to be sufficiently responsive to market changes without excessive turnover.
For the given horizon in Figure 11, Leverage (lev), Risk (risk), Profitability (prof) and Stability (stab) ESBs register a winning rate of around 80% in the 24.75-year history, consistent with the
correlation analysis (Figure 6) and the positive hit rate for these ESBs. The winning 6-month lookback balances turnover with reaction time to regime changes.
LIVE Corroboration (Jan 2019 - Sept 2024)
We have 24.75 years of returns for our ESBs with the recent 5.75y post-2019 being live-trading record (while the prior 19 years are back-tested). To corroborate the validity of the strategy, we
re-evaluate the TSMOM strategies with the 5.75-year live-trading data. Figure 12 and Table 4 visualize the performance of TSMOM strategies based on beta-neutral ESBs (excluding XSMOM ESBs) over the
live-trading history.
Figure 12: VAMI of TSMOM Strategies - 16 ESBs (Beta-Neutral; ex-XSMOM) for 5.75y (vol scaled)
As with the 24.75-year and 22.75-year history, the Long Winners TSMOM strategies with a 6-month lookback period and rebalanced monthly (LW 6m/1m) still performed the best among all TSMOM strategies
once again over the 5.75-year live-trading data, with a Sharpe ratio of 1.47, which is higher than Sharpe ratio of the EW16 benchmark (1.21), and is higher than one of the Enterprise (1.24).
We also observe that the outperformance of long-winner TSMOM strategies compared to the benchmarks was largely established between March 2020 to early 2021 when Covid (for obvious reasons) resulted
in significant strategy dispersion whereas subsequent performance has marched more in tandem. Relative to the benchmark, the best TSMOM (LW, 6m/1m) strategy essentially doubles the cumulative return
with a similar MDD resulting in a far superior risk adjusted profile showing good downside protection during volatile periods similar to what Gu and Mulvey (2021) observed.
In Table 4, we observe that, compared to the outcome on a longer history, the annualized returns of long-winner TSMOM strategies are now a bit higher in magnitude and are closer to 70% of Enterprise
18 now (in comparison to around 50% in the longer horizons discussed above). Also, we observe that the long-winners strategies outperform the benchmarks in trend between early 2020 and 2021 (the
heavily COVID-impacted period).
Table 4: Performance of Top-10 Strategies - 16 ESBs (Beta-Neutral; ex-XSMOM) for 5.75y
Publication Decay
The publication of a strategy may result in performance decay, as noted by McLean and Pontiff (2016), potentially diminishing the strategy's alpha or lowering its Sharpe ratio. For factor TSMOM
strategies, the most closely related and frequently cited works, such as Gupta and Kelly (2019) and Ehsani and Linnainmaa (2022), utilize U.S. equity factor data spanning from 1963 to 2016 (for the
latter) or 2017 (for the former). Additionally, their initial working papers were likely released before their formal journal publications, suggesting that the market could have started to
incorporate the strategies' insights before their official release.
The first draft of Ehsani and Linnainmaa (2022) was published in March 2017, while the earliest available version of Gupta and Kelly (2019) appeared as a Yale ICF working paper in November 2018.
Following McLean and Pontiff (2016), who argue that the dissemination of research ideas likely begins when the first working paper is released - and potentially even earlier due to conference
presentations or discussions - we therefore hypothesize that publication decay (if any) ought to gradually manifest from 2017 onwards. Based on this, we divide the historical data into two distinct
1. Pre-publication History: Jan. 2002- Dec. 2016 (to exclude the NASDAQ crash period, during which QuantZ ESBs performs particularly well through the history and there was insufficient history for
the lookback periods of the TSMOM strategies)
2. Post-publication History: Jan. 2017- Sept 2024
The analysis then focuses on assessing the post-publication decay of these three beta-neutral, TSMOM strategies:
1. LW 6m/1m: This is the best performer during our 24.75/22.75/5.75-yr history
2. LW 1m/1m: This is the best performer in our history from backtesting (2002-2018), and was mentioned as the best strategy in Gu and Mulvey (2021)
3. LW 12m/1m: Considered the paradigm of the TSMOM strategies in the late 2010s, this strategy is the most researched and frequently mentioned. It was central to the construction of factor TSMOM
strategy by Gupta and Kelly (2019), Arnott et al. (2021) and Ehsani and Linnainmaa (2022).
We assess the performance decay by comparison against two benchmarks (EW18 and EW16) and Enterprise 18, giving precedence to the EW18 and EW16, which themselves show the post publication Sharpe
decays of -45.0% and -50.5%, down to Sharpe ratios of 1.10 and 1.08, respectively. Interestingly, of the 3 TSMOM strategies considered, the most-researched LW 12m/1m shows a decay of -45.8%, which is
pretty much in line, while the LW 1m/1m decay of -58.2% is much worse in contrast to the much milder decay of only -39.4% for the LW 6m/1m case. Hence, while there is prima facie strong evidence of
post-publication decay; it is just as strong for the benchmarks which means that risk adjusted decay is structural, pervasive and may not be specific to a post publication effect at least based on
Sharpe ratios. Based on alphas there is not much evidence of decay in the nominal intercept values either for the EW18 benchmark or the best strategies like LW 6m/1m.
Table 5: Performance Decay: Pre- and Post-Publication Analysis (2003-2024 Sept)
Table 5 illustrates the performance decay in key investment strategies, comparing metrics from the pre-publication period (January 2003 to December 2016) and the post-publication period (January 2017
to Sept 2024). Percentage changes are reported as positive (e.g., "+X%") for post-publication increases and negative (e.g., "-X%") for decreases.
The LW 6m/1m strategy stands out with a less pronounced performance decline in risk-adjusted return ratios compared to other strategies, despite its declines in the Sharpe, Sortino, and Calmar Ratios
(-39.4%, -26.9%, and -22%, respectively). This strategy maintains positive FF5 alpha (+20.6%) and FF5 + WML alpha (+14.7%), indicating a relatively resilient performance post-publication.
Table 6: Alphas and Risk-adjusted Returns in 22.75-year / Pre-publication / Post-publication / 5.75-year History
where alphas are in % and t-statistics are written in the parenthesis “()”. All adjusted R2’s are in squared brackets “[]”. Both are under the alpha values.
Our study documents TSMOM persistence in QuantZ's proprietary live factor set of ESBs, which demonstrates the superior risk-adjusted profile of the long-winner (long only) TSMOM strategy with a
6-month lookback period and monthly rebalancing particularly during crashes such as the COVID period of 2020. Notably the LW portfolios also did remarkably well amidst the Nov 2020 Momentum crash &
the L/S 1m, 3m did even better. Amongst all 3 horizons evaluated (24.75/22.75/5.75 years till Sept 2024), the LW 6m, 1m not only performs the best among all TSMOM strategies, but also outperforms
both benchmarks in 24.75-year and 5.75-year horizons As for the 22.75-year horizon, although it obtains nearly identical Sharpe as the Enterprise 18 composite, it reduces the downside deviation and
MDD by more than 50% and thus it obtains higher Sortino and Calmar ratios.
You may notice that our top lookback period (6 months) is different from the optimal lookback period in Gu and Mulvey (2021), which is 1 month. This could be due to differences in the factors in our
pool (as well as ensembling at the ESB level) vs their factor zoo (since they do not ensemble), and the different horizons. Finally, we postulate that the factor timing of TSMOM strategies could
reflect a potentially regime-aware quality: it benefits from holding a well-diversified set of ESBs similar to the equal-weighted portfolio during normal risk-on periods, and when a crash happens,
the strategy rapidly pivots towards the subset of risk-off ESBs (given monthly rebal) which should continue to be the winners in a tumultuous tape. The strategies based on a 6-month lookback period
seem to strike the right balance of controlled turnover despite responsiveness to changing markets particularly in a downdraft.
In conclusion, we have replicated, validated, and extended the existing literature on factor momentum to QMIT's proprietary dataset of smart betas which includes 5.75y of live data.
APPENDIX I: 24.75y Results
(Beta Neutral: without XSMOM)
(Dollar Neutral: without XSMOM)
(Beta Neutral: with XSMOM)
(Dollar Neutral: with XSMOM)
APPENDIX II: 22.75y Results
(Beta Neutral: without XSMOM)
(Dollar Neutral: without XSMOM)
(Beta Neutral: with XSMOM)
(Dollar Neutral: with XSMOM)
APPENDIX III: LIVE 5.75y Results
(Beta Neutral: without XSMOM)
(Dollar Neutral: without XSMOM)
(Beta Neutral: with XSMOM)
(Dollar Neutral: with XSMOM)
APPENDIX IV: Enhanced Smart Beta Definitions
• ARS: This smart beta composite shows our Analyst Revisions cohort based on measures of estimate revisions, dispersion, Standardized Unexpected Earnings surprise (SUE score) & consensus change in
both earnings as well as revenues which can outperform traditional metrics like a 1mo consensus change.
• ART: This smart beta composite shows our Analyst Ratings & Targets cohort based on measures of analyst recommendations, target price, changes & diffusion which can outperform traditional metrics
like a 1-month consensus change.
• CSU: This smart beta composite shows our Capital Structure/Usage cohort based on measures including Buybacks, Total yield, Capex, capital usage ratios, etc which can outperform traditional
metrics like Cash/MC.
• Dividends: This smart beta composite shows our Dividend-related cohort based on measures including Yield, payout, growth, forward yield, etc which can outperform traditional metrics like
Dividend Yield.
• DV: This smart beta composite shows our Deep Value (or intrinsic value) cohort based on measures including tangible book & sales which can outperform traditional Book yield.
• Efficiency: This smart beta composite shows our Efficiency cohort based on measures including Asset Turnover, Current Liabilities, Receivables, etc which can outperform traditional metrics like
Asset Turnover.
• EnMOM: This smart beta composite shows our Enhanced Momentum cohort which can outperform traditional 12-month price momentum in both return & risk-adjusted terms, particularly at market
inflection points.
• EQ: This smart beta composite shows our Earnings Quality cohort based on a variety of Accrual measures which can outperform traditional metrics like Total Accruals.
• Growth: This smart beta composite shows our Historical Growth cohort based on a variety of Earnings, Sales, Margins & CF-related growth measures which can outperform traditional metrics like
3-year leverage-related Sales growth.
• Leverage: This smart beta composite shows our Leverage related cohort based on measures of Balance Sheet leverage which can outperform traditional metrics like Debt To Equity.
• MOM: This smart beta composite shows our MOM-related cohort which can outperform traditional 12-month price momentum using a variety of traditional momentum factors.
• Profit: This smart beta composite shows our Profitability cohort based on measures like ROA, ROE, ROCE, ROTC, Margins, etc which can outperform traditional metrics like ROE.
• RV: This smart beta composite shows our Relative Value cohort based on measures of EPS, CFO, EBITDA, etc which can outperform traditional Earnings yield.
• Reversals: This smart beta composite shows our Reversals cohort which is comprised of metrics like short-term reversals, RSI, DMA & other technical factors that can outperform traditional
metrics like a 1-month total return.
• Risk: This smart beta composite shows our Risk/ Low Vol cohort which is comprised of metrics like Beta, Low volatility, etc.
• SIRF: This smart beta composite shows our Short Interest cohort which is comprised of metrics related to Short Interest and its normalization by Float, trading volume, etc.
• Size: This smart beta composite shows our Size cohort which is comprised of metrics related to firm size including market capitalization.
• Stability: This smart beta composite shows our Stability cohort which is comprised of metrics like Dispersion of EPS/ SPS estimates as well as the stability of Margins, EPS & CFs, etc.
Q-Q Plots of Equal Weighted Benchmarks, Enterprise 18 and LW 6m/1m TSMOM Strategy
Long-only factor portfolios exhibit diversification benefits that truncate the left tail of the return distribution—reducing extreme negative returns—while amplifying the right tail, enhancing
extreme positive returns. This pattern provides evidence of factor momentum combined with downside protection when compared to the Ent18 signal. Notably, the Ent18 signal represents a single equity
market-neutral combined signal, whereas the EW18, EW16, and LW portfolios are all long-only portfolios of individual ESBs, offering diversification across different ESBs.
[1] Asset Universe: Jegadeesh and Titman (1993) conducted their study using individual stocks listed on the New York Stock Exchange (NYSE) and the American Stock Exchange (AMEX) over the period from
1965 to 1989.
[2] Asset Universe: Arnott et al. (2021) utilized monthly and daily return data from the Center for Research in Securities Prices (CRSP) for ordinary common shares (share codes 10 and 11) listed on
the NYSE, AMEX, and Nasdaq exchanges. They included CRSP delisting returns and imputed missing (30% for NYSE and AMEX stocks and 55% for Nasdaq stocks) performance-related delisting returns. The
study constructed 43 factors based on a combination of price, return, volume, and accounting information.
[3] Asset Universe: Gupta and Kelly (2019) constructed 65 characteristic-based factor portfolios using U.S. stock data. These portfolios encompassed a wide range of characteristics, including
valuation ratios (e.g., earnings/price, book/market), factor exposures (e.g., betting against beta), size, investment, profitability metrics (e.g., market equity, sales growth, return on equity),
idiosyncratic risk measures (e.g., stock volatility and skewness), and liquidity measures (e.g., Amihud illiquidity, share volume, bid-ask spread).
[4] Asset Universe: Ehsani and Linnainmaa (2022) analyzed 22 "off-the-shelf" factors, comprising 15 U.S. equity anomalies and 7 global factors. Data were sourced from Kenneth French’s, AQR’s, and
Robert Stambaugh’s data libraries. The U.S. factors included size, value, profitability, investment, momentum, and others. Except for the liquidity factor of Pastor and Stambaugh (2003), the return
data for these factors begin in July 1963; those for the liquidity factor begin in January 1968. The seven global factors are size, value, profitability, investment, momentum, betting against beta,
and quality minus junk. Except for the momentum factor, the return data for these factors begin in July 1990; those for the momentum factor begin in November 1990.
[5] Asset Universe: Gu and Mulvey (2021) employed two datasets for their analysis: 11 long–short anomaly portfolios from Ehsani and Linnainmaa (2019), with data from Kenneth French’s, AQR’s, and
Robert Stambaugh’s data libraries; and 44 long–short portfolios based on anomaly characteristics from Kozak, Nagel, and Santosh (2020). They excluded seven momentum-related factors to avoid inducing
correlation between factor momentum and individual stock momentum.
[6] Performance Metrics Definitions: The Sharpe Ratio (Sharpe, 1966) is calculated as the mean of excess returns divided by the standard deviation of returns, measuring return per unit of total risk.
The Sortino Ratio (Sortino and Price, 1994) focuses on downside risk by using the standard deviation of negative returns instead of total volatility. The Calmar Ratio (Young, 1991) is calculated as
the average annualized excess return divided by the maximum drawdown during the period. In this study, since the risk-free rate has already been subtracted from the returns, we employ the adjusted
form of the Calmar Ratio that is prevalent in current practice.
[7] LL - Long Losers, LW - Long Winners, LS - Long Winners and Short Losers, m/m - Lookback Period/Holding Period.
● Arnott, R. D., Clements, M., Kalesnik, V., and Linnainmaa, J. T. (2019). "Factor Momentum." Journal of Portfolio Management, 45(3): 46-59.
● Carhart, M. M. (1997). “On Persistence in Mutual Fund Performance.” Journal of Finance, 52 (1), 57-82.
● De Bondt, W. F. M., and Thaler, R.. "Does the Stock Market Overreact?" The Journal of Finance, 40 (3): 793–805.
● Ehsani, S. and Linnainmaa. J. T. (2022). “Factor Momentum and the Momentum Factor.” Journal of Finance, 77 (3): 1877-1919.
● Fama, E. F. and French, K. R. (1993). “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics, 33 (1): 3-56.
● Fama, E. F. and French, K. R. (2015). "A Five-Factor Asset Pricing Model." Journal of Financial Economics, 116(1): 1-22.
● Gu, J. and Mulvey, J. M. (2021). “Factor Momentum and Regime-Switching Overlay Strategy.” The Journal of Financial Data Science, 3(4): 101-129.
● Gupta, T. and Kelly, B. (2019). “Factor Momentum Everywhere.” The Journal of Portfolio Management, 45(3): 13-36.
● Jegadeesh, N. (1990). "Evidence of Predictable Behavior of Security Returns." The Journal of Finance, 45 (3): 881–898.
● Jegadeesh, N. and Titman, S. (1993). “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency.” The Journal of Finance, 48(1): 65–91.
● Kozak, S., Nagel, S. and Santosh S. (2018). “Interpreting Factor Models.” The Journal of Finance, 73(3): 1183–1223.
● Kozak, S., Nagel, S. and Santosh S. (2020). "Shrinking the Cross-Section." Journal of Financial Economics, 135(2): pp. 271-292.
● Lehmann, B. N. (1990). "Fads, Martingales, and Market Efficiency." The Quarterly Journal of Economics, 105 (1): 1–28.
● McLean, R. D. and Pontiff, J. (2016). "Does Academic Research Destroy Stock Return Predictability?" Journal of Finance, 71(1): 5–32.
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Spectral forecast: A general purpose prediction model as an alternative to classical neural networks
Here, we describe a general-purpose prediction model. Our approach requires three matrices of equal size and uses two equations to determine the behavior against two possible outcomes. We use an
example based on photon-pixel coupling data to show that in humans, this solution can indicate the predisposition to disease. An implementation of this model is made available in the supplementary
A novel prediction method is described, implemented, and tested. The model revolves around three known states: two extreme outcomes (A and B) and one measurement (P). These states are represented by
matrices that include sets of homologous parameters. An information spectrum is described as a series of predicted states (M[1], M[2], M[3],…, M[d]) generated between the two extreme outcomes (A and
B). The predicted states are compared with the known state (P) from the measurements to generate a similarity index. The trend generated by the values of the similarity index indicates how a system
may behave against these two extreme outcomes.
Nonlinear behavior is part of all the phenomena we know in nature, from weather to living beings and beyond. In biological systems, nonlinear mechanisms are partially uncharted and a subject of
general scientific interest for several decades.^1,2 In medicine, perhaps the most important research segment is represented by the prediction of different diseases. The accurate prediction of the
occurrence of a disease has always been regarded as the main tendency of clinical practice. Genetic diseases, which can be triggered by environmental factors, are inherently nonlinear in nature and
their onset is unpredictable. Such a disease, with a non-linear behavior, is diabetes. Neural Networks (NNs) were the hope for prediction in the medical field in the 1980s and this trend was reborn
in recent years.^3–5 However, the medical context in which they are used and the partial subjectivity in the training of NNs slowly lose their much-anticipated value for certain specific tasks.^6,7
Essentially, NNs are classifiers of information with prior or “in-flight” adaptation to the environment. Any pattern recognition strategy uses this classification approach. Nevertheless, such a
classification based on prior adaptation to the environment may not imply a prediction. We are inclined to believe that NNs are misused in some cases and the classification process is often confused
with the prediction process. For instance, in a previous study, we struggled with a fundamental problem in which we tried to use NNs for the prediction of diabetes onset.^8 There, we agreed that our
NN correctly classified a new patient into one of the two classes, namely, type 1 diabetes (T1D) or type 2 diabetes (T2D). However, such a classification was a direct indication of the current state
of the disease. In other words, it was merely a medical diagnosis. We then asked ourselves whether the NN can predict the evolution of a human subject over time in the hope that we can predict the
onset of the disease. Our experimental data have indicated that our NN classified the set of data in a fair manner but failed to indicate any valuable information about the evolution of a subject
over time.^9,10 In this respect, predictions that use Markov chains or classical statistical approaches showed more reliable results in the field of biology and medicine.^11–13 However, here we
propose a novel method of analysis, with implementation (see the supplementary material), as an alternative to classical NNs. Note: the word spectral refers to a series of predicted states arranged
linearly between two known states.
In order to test our model, we collected and used the data related to the electrical activity signals of the human skin from our most recent experiment.^8–10
Datasets and context
The electric activity on the skin surface of the trunk was measured by using 200 sensors in three groups: a control group A—18 normal subjects, a group B—18 diabetic subjects, and a test group C—20
normal subjects (ten subjects with confirmed family predisposition for T2D and ten subjects without family predisposition for T2D).^8 The electrical signals were collected using the photon-pixel
coupling method and were stored as numerical values (0…100) in a 10×20 matrix for each subject.^8,9 An average was taken along the 18 subjects in group A and group B, yielding a 10×20 matrix for
each, namely, matrix A and matrix B.^8 These average matrices represent the main characteristics of each group. A state space was considered between the two matrices of group A and group B. The
number of states was established by a distance index (d) and each state in this spectrum was represented by a matrix M. Each subject in group C was then evaluated by a consecutive comparison of their
matrix P with each matrix M in the spectrum.
The spectral forecast model
In our approach, we used three known matrices: A, B, and P. A matrix M was further used to formulate the entire spectrum of unknown information between matrix A and B [Fig. 1(a)]. For this
calculation, we devised a novel equation shown in (1),
where M[ij] represents the predicted matrix at every discrete step (d), A[ij] represents the matrix of the normal group, and B[ij] is the matrix associated with the diabetic group [Figs. 1(b) and 1
(c)]. Also, d stands for distance and represents the total number of discrete steps taken from matrix A to matrix B. Thus, M[ijd] can be considered a 3D tensor-like structure.
The evolution of P was predicted by a repeated comparison with matrix M at every discrete step (2). This comparison was made by using the similarity index,
where S is the similarity index and represents the normalized dot-product of M[ij] and P[ij]. M[ij] stands for the predicted matrix at every discrete step and P[ij] is the matrix originated from a
newly measured individual. The similarity index can take values between 0 and 1. As the similarity between the corresponding i,j elements of matrix M and P increases, the similarity index S tends to
1. In contrast, as the differences between the values of the corresponding i,j elements of matrix M and P are more frequent, the similarity index S tends to 0 [Fig. 1(d)] . The main result of the
method is represented by a trend dictated by the values of the similarity index [Fig. 1(d)]. The trend was taken as the evolutionary route of the disease. In the supplementary material, we show a
ready-to-use implementation of the method.
Note: The total number of discrete steps was arbitrarily chosen. In this specific case, the maximum value for distance (d) was set at 100 for ease. A higher number of discrete steps increased the
resolution of the prediction, which was desirable in many situations. For instance, in some cases, the trend developed both ascending and descending characteristics. At low resolutions (i.e., d<
10), many of these fluctuating features remained undetectable and the insight of the results significantly dropped.
Discretization is a practical approach for many prediction algorithms and it is used for almost all computational solutions. Here, we used a discretization strategy to increase the resolution of the
spectrum underlying two groups: a healthy group and a diabetic group [Figs. 2(a) and 2(b)]. The data from the healthy group were considered as the initial state (state 0) and the data from the
diabetic group were used to formulate the final state (state 100). The number of intermediate states (state 1–state 99) was dictated by distance d, and the intermediate states properties were
repeatedly formulated by matrix M [Figs. 2(c) and 2(d)]. To predict the evolution of a third group, a comparison was made along this spectrum [Figs. 2(c) and 2(d)]. Thus, data of matrix P from a new
individual were compared to each matrix M in order to obtain the series of values for the similarity index [Figs. 2(e) and 2(f)]. To test the method, we decided to use our most recent data collected
from a previous study.^8–10 The predisposition trend for individuals in group C has been correctly predicted 100% of the time (Fig. 3). The subjects with family predisposition for T2D have shown an
average similarity index of 0.877±0.024493, whereas normal individuals have shown an average similarity index of 0.68±0.024499.
In the normal group, the mean of the similarity index showed a value of 0.68±0.111 and a maximum value of 0.8784 and a minimum value of 0.4592 [Fig. 3(e)]. In the T2D predisposition group, the mean
of the similarity index showed a value of 0.87741±0.08 and a maximum value of 0.9626 and a minimum value of 0.63086 [Fig. 3(f)]. The method has been implemented and can be found in the
supplementary material.
The meaning of the trend
Based on known clinical information and the observations made on each individual of the two groups, the trend of the similarity index values indicated whether a newly measured individual showed a
predisposition or protection for T2D [Figs. 3(e) and 3(f)]. We speculate that the difference between the lower and the upper limit of the trend may represent a risk score for T2D. At this stage of
the investigation, we can indicate if the newly measured subject tends toward the disease [Figs. 3(e) and 3(f)]. Experimentation has shown that the trend shaped by the similarity index does not
exhibit only ascending or only descending features. Ascending and descending features of the similarity index can exist within the same plot. One example can be seen in Fig. 3(e), where one
individual in the normal group shows both ascending and descending features. In the future, we will try to find the meaning of such a distribution because we speculate that it might be of particular
importance for the prediction process.
A link between two unrelated data of the same dimension
The core of our method is represented by Eq. (1), which can have multiple uses on a wide range of values. One of these uses would be a normalization between two unrelated matrices with the same
dimension. For instance, elements of matrix A may contain integers between 1 and 2×10^6 million and the elements of matrix B may contain probability values. In this case, Eq. (1) will mix the two
matrices based on distance d. In the case of two probability matrices, Eq. (1) performs a normalization in favor of one of the matrices based on distance d. In other words, as matrix M is closer to
matrix A, the homologous elements of matrix M will be more similar to matrix A than to those from matrix B. As matrix M will be closer to matrix B, matrix M will be more similar to matrix B.
Consequently, if d=50, matrix M will represent a mix equally similar to matrix A and matrix B.
Thoughts for the future
The important cases are those that show a maximum similarity index between the two groups. We suggest that these peak values may be a direct indication of the state of the subject before the onset of
the disease. In order to predict the onset, we wish to establish a link between the temporal line of the disease and the states generated along the spectrum. Variations of the method may be
constructed and we are eager to use other datasets in the same format (A, B, P). Future uses may include the field of meteorology, medical diagnostics, forensics, economic forecasts, or in the field
of genetics for establishing the relationship between species. In biology, we also believe that Eq. (1) can be used for tissue structure prediction based on two groups of histological slides.
Here, we have shown the use of a novel prediction model. We proposed a simple method that provides an insight into the evolution of natural processes. To demonstrate the method, our current example
considered the predisposition to disease in human subjects based on two known groups. In this approach, we correctly predicted the evolution of new subjects by using our previous data recorded from
normal subjects and T2D subjects. Other applications of the method may further indicate the ideal conditions to which our method is appropriate or the limits of precision in the prediction of various
metabolic diseases. In the future, we will try to make an association between a temporal line and the steps of the spectrum to indicate the time until the disease is triggered in days, months, or
years. A ready-to-use implementation is present in the supplementary material, which can also be used for other types of data.
The authors would like to thank two anonymous referees for their constructive comments, which helped improve the manuscript. The authors declare no competing financial interests. This study was
funded through No. PN-III-P1-1.2-PCCDI-2017-0797: “Pathogenic mechanisms and personalized treatment in pancreatic cancer using multi-omics technologies.”
et al., “
Some nonlinear challenges in biology
S. R.
, and
R. K.
, “
Nonlinear phenomena in biology and medicine
Comput. Math. Methods Med.
M. L.
, “
The application of backpropagation neural networks to problems in pathology and laboratory medicine
Arch. Pathol. Lab. Med.
M. W.
J. R.
, “
Artificial neural networks for medical classification decisions
Arch. Pathol. Lab. Med.
, and
, “
Applications of artificial neural networks in health care organizational decision-making: A scoping review
PLoS One
et al., “
Opportunities and obstacles for deep learning in biology and medicine
J. R. Soc. Interface
J. H.
S. M.
, “
Machine learning and prediction in medicine—Beyond the peak of inflated expectations
N. Engl. J. Med.
P. A.
, and
, “
The electrical activity map of the human skin indicates strong differences between normal and diabetic individuals: A gateway to onset prevention
Biosens. Bioelectron.
P. A.
, and
, “
Maps of electrical activity in diabetic patients and normal individuals
Data Brief.
P. A.
, and
, “
Photon-pixel coupling: A method for parallel acquisition of electrical signals for scientific investigations
Methods X
, and
, “Unsupervised learning of disease progression models,” in
Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining
(ACM, 2014), pp. 85–94.
B. L.
, and
, “Interpretable representation learning for healthcare via capturing disease progression through time,” in
Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining
(ACM, 2018), pp. 43–51.
P. A.
Markov Chains From Theory to Implementation and Experimentation
John Wiley & Sons
), ISBN: 978-1-119-38755-8.
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Challenge questions
These are not essential for learning the material and can be skipped without affecting your grade. If you successfully solve one set of problem, a week of participation activity will be waived (it
does not have to be the same week you submit the challenge question). Submit your answer at any time. I will not post solutions for the challenge questions.
• You are collecting lithographs from a series.
• Each one has a unique serial number, but contrary to standard practice, the artist did not specify the total number of copies in circulation.
• You possess serial numbers 1, 3, 15.
Q.1 Modelling
Design a Bayesian model to infer the total number of copies in circulation. Motivate all choices you make.
Q.2 Posterior computation
Approximate the posterior distribution and produce:
1. A posterior PMF.
2. A point estimate.
3. A 95% highest probability set.
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Which Formula Can You Type in Cell D92 to Do This | Student Portal
Which Formula Can You Type in Cell D92 to Do This
Posted on
‘Which Formula Can You Type in Cell D92 to Do This’ is one of the examples of the questions that is commonly found on computer science tests either on a school test or a job test. Since this kind of
question is served in the form of an Excel program, of course you may see certain data on the Excel table and attempt to find the solution to solve the question.
If you also have to go through a computer science test a few days later, you surely need to practice and practice a lot related to any materials about computer science that may be issued on a test.
If you’re also looking for the correct answer to the question ‘Which Formula Can You Type in Cell D92 to Do This’, you can get it through our post below!
Which Formula Can You Type in Cell D92 to Do This?
There are so many internet sources that provide the correct answers for any tests, including a computer science test that is commonly available on a school test or also a job test. One of the
websites providing the correct answer related to computer science tests is Studylib.net.
The question of ‘Which Formula Can You Type in Cell D92 to Do This’ is one of questions available on Excel quiz listed on number 13 that has been posted by Jamilah Jill Tuazon. If you want to see the
original source, you can visit Studylib.net.
On Studylib.net, you can see the detail of the question as follow:
You need to divide the number of cars by the number of people to calculate cars per person on Day 1. Which formula can you type in Cell D92 to do this?
A B C D
91 Day Cars People Cars per Person
92 1 10 3 3.33
93 2 20 4 5.00
94 3 30 5 6.00
95 4 40 6 6.67
a. =3*1
b. =B93/C92
c. =A92/B92
d. =10*3
e. =B92/C92
Answer: The correct answer is e. =B29/C92
Explanation: To calculate the cars per person on Day 1, you can divide the number of cars by the number of people. So, the formula you can type is cell B92 is divided by cell C92.
Other Questions and Answers Related to Computer Science Posted on Studylib.net
For more information, the examples of computer science tests available on Studylib.net posted by Jamilah Jill Tuazon consist of 15 questions. Since you will face a computer science test in a few
days, you may need the examples of some questions and answers that may be issued on a test.
To make it easier for you to get a reference, this post will show you the examples of some questions and answers related to computer science, especially about Excel Test. We also obtained them from
Studylib.net posted by Jamilah Jill Tuazon that you can access here.
What’s the correct sorting function to list Colors in alphabetical order (A to Z)
Color Number
Blue 120 A to Z = ASCENDING
Green 85 Z to A = DESCENDING
Orange 112 EQUAL to A to Z
Red 100
Yellow 90
1. What does clicking the + sign below do?
a. Adds a new Worksheet
b. Adds a new Row
c. Adds a new Chart
d. Adds a new Function
e. Adds a new Column
Answer: a. Adds a new Worksheet
2. What is the correct keyboard shortcut to cut a cell value?
a. CTRL + P
b. CTRL + X
c. CTRL + B
d. CTRL + C
e. CTRL + V
Answer: b. CTRL + X
3. In a new worksheet, what’s the correct formula to reference Cell A1 from the ALPHA worksheet?
a. =ALPHA?A1
b. =A1
c. =ALPHA!A1
d. =”ALPHA”A1
e. =’ALPHA’A1
Answer: c. =ALPHA!A1
4. Based on the values in Cells B77:B81, what function can automatically return the value in Cell C77?
8 Sales
9 $794 $1,020 =MAX(A9:A13)’
10 $721
11 $854
12 $912
13 $1,020
a. =Top()
b. =Max()
c. =Ceiling()
d. =Biggest()
e. =Highest()
Answer: b. =Max()
5. What does clicking and dragging the fill handle indicated by the cursor do?
Sales Tax
$794 $64
$721 $58
a. Move this formula to a new cell
b. Copy this number to other cells
c. Move this cell to another location
d. Fill this cell with a color
e. Copy the formula to another cell
Answer: e. Copy the formula to another cell
6. What value would be returned based on the formula in Cell A49?
43 npab
44 npce
45 npfo
46 npbb
47 norp
a. 0
b. 1
c. 2
d. 3
e. 4
Answer: e. 4
7. Which tools would you use to make header 1 look like Header 2? Select all that apply.
A B C
First Name Last Name RSVP
First Name Last Name RSVP
8. Based on the values in Cells B77:B81, which function can automatically return the value in Cell C77?
A B
2 Sales
3 $794 $721 =MIN(A2:A6)
4 $721
5 $854
6 $912
7 $1,020
a. =Min()
b. =Bottom()
c. =Smallest()
d. =Lowest()
e. =Floor()
Answer: a. =Min()
9. Which tools would you use to make Chart 1 look like Chart 2? Select all that apply.
Chart 1 Chart 2
A A
1 Sales Growth Sales Growth
2 0.19 19%
3 0.25 25%
4 0.49 49%
5 0.29 29%
6 0.34 34%
7 0.22 22%
8 0.12 12%
10. What formula would produce the value in Cell C25?
A B C
22 Item Type Result
23 Door F Door F =CONCATENATE(B23,” “,C23)
24 Table C
25 Chair C
26 Desk F
a. ‘=RIGHT(B23,C26)
b. ‘=CONCATENATE(ITEM,” “,TYPE)
c. ‘=LEFT(B23,C23)
d. ‘=CONCATENATE(B23,C23)
e. ‘=CONCATENATE(B23,” “,C23)
Answer: e. ‘=CONCATENATE(B23,” “,C23)
11. What value would be returned based on the formula in Cell D49?
D E F
42 Room Location Staff ID
43 D East 19106
44 C North 19122
45 A South 19107
46 E South 19104
47 B South 19147
48 =COUNTIFS(C30:C34,”South”,D30:D34,”19104″)
a. 0
b. 1
c. 2
d. 3
e. #ERROR
f. #VALUE
Answer: b. 1
12. Which tools would you use to make Chart 1 look like Chart2
13. You need to divide the number of cars by the number of people to calculate cars per person on Day 1. Which formula can you type in Cell D92 to do this?
A B C D
91 Day Cars People Cars per Person
92 1 10 3 3.33
93 2 20 4 5.00
94 3 30 5 6.00
95 4 40 6 6.67
a. =3*1
b. =B93/C92
c. =A92/B92
d. =10*3
e. =B92/C92
Answer: e. =B29/C92
14. Based on the values in Cells A51:A55, what formula can you copy and paste into Cells B51:B55 to return the values shown?
A B
51 Red Yes
52 Red Yes
53 Red Yes
54 Blue No
55 Red Yes
a. =IFNA(A51=”Red”,”Yes”,”No”)
b. =SHOWIF(A51=”Red”,”Yes”,”No”)
c. =COUNTIF(A51=”Red”,”Yes”,”No”)
d. =SUMIF(A51=”Red”,”Yes”,”No”)
e. =IF(A51=”Red”,”Yes”,”No”)
Answer: e. =IF(A51=”Red”,”Yes”,”No”)
15. If the formula in Cell D49 is copied to Cells E49:F49, what sequence of values would be generated in Cells D49:F49?
D E F
42 Conference Room Location Staff ID
43 D East 19106
44 C North 19122
45 A South 19107
46 E South 19104
47 B South 19147
49 =$D$44 C C
a. C, North, 19122
b. C, South, South
c. C, A, E
d. C, South, 19104
e. C, C, C
Answer: e. C, C, C
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King Abdullah University of Science and Technology
CE 201 Chemical Thermodynamics
Prerequisites: Undergraduate thermodynamics course. The primary goal of chemical thermodynamics is the physical explanation of the fundamental principles governing the variety of chemical phenomena
taking place in the world around us. The goal of this course is to give students a conceptual understanding of the main principles of thermodynamics. Topics include: the concept of entropy; the
Clausius, Gibbs, Boltzmann and Shannon definition of entropy; entropy and information; Maxwells demon; the Boltzmann distribution law; the Maxwell-Boltzmann speed distribution; Gibbs and Helmholtz
free energy; the chemical potential; Gibbs-Duhem and Euler equation; the Gibbs phase rule; entropy of mixing and Gibbs paradox; phase diagrams, the Flory-Huggins phase diagram; spontaneous and
non-spontaneous processes; thermodynamics of chemical reactions; thermodynamics of osmosis and reverse osmosis, entropy and irreversible phase transitions; introduction in thermodynamics of
irreversible processes; introduction in statistical thermodynamics.
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What is Encryption? & Types of Encryption & Its Importance
What is Encryption?
Encryption is defined as the process of converting information into incomprehensible codes (seems meaningless) to prevent unauthorized persons from viewing or understanding information. Encryption
therefore involves converting plain text into encrypted text. It is known that the Internet is nowadays the largest center for information transfer. Sensitive information (such as financial
movements) must be transmitted in encrypted form if it is to maintain its integrity and secure it from tampering with hackers. The keys are used to encrypt and decrypt the message. These keys are
based on complex mathematical formulas (algorithms).
The strength and effectiveness of encryption depends on two basic factors: the algorithm, and the length of the key (estimated bits).
Types of encryption?
Encryption (secret key) In symmetric encryption, both the sender and receiver use the same secret key to encrypt and decrypt the message. The two parties initially agree on the passphrase (long
passwords) that will be used. The passphrase can contain both uppercase and lowercase letters and other symbols. Encryption software then switches the passphrase to a binary number, and other symbols
are added to increase its length. The resulting binary number is the key to encrypt the message. After receiving the encrypted message, the receiver uses the same passphrase to decrypt cipher text or
encrypted text, as the software rewrites the passphrase to form the binary key, which converts the encrypted text back to its original form. The concept of symmetric encryption depends on DES. The
large gap in this type of encryption was the secret key exchange without security, which led to the decline of the use of this type of encryption, to become something of the past. Symmetric
Cryptography Asymmetric encryption (public key) Incompatible cryptography is a solution to the problem of insecure distribution of keys in symmetric encryption. Instead of using one key, asymmetric
encryption uses two unrelated keys. These keys are called the public key and the private key. The private key is known to only one or one person; it is the sender, and is used to encrypt and decrypt
the message. The public key is known to more than one person or entity. The public key can decrypt the message encrypted by the private key. It can also be used to encrypt the private key owner's
messages, but no one can use the public key to decrypt a code. The private key is the only one that can decrypt the messages encrypted by the public key. The encryption system that uses public keys
is called RSA, and although it is better and safer than the DES system, it is slower, since the encryption session and the decryption session must be almost simultaneous. In any case, the RSA system
is not intrusive, as penetration is possible if time and money are available. Therefore, the PGP system, which is an improved and upgraded version of the RSA system, has been developed. PGP uses a
128-bit key, as well as using the message digest. This system is still immune to penetration to this day
Also Read : What is SSL Certificate? & Encrypted https Protocol What is the electronic footprint of the message?
Although encryption prevents the intruders from seeing the contents of the message, it does not prevent spoilers from tampering with it; that is, encryption does not guarantee the integrity of the
message. Hence the need for the electronic footprint of the message (message digest), a digital fingerprint derived according to certain algorithms called functions or camouflage associations (hash
functions). These algorithms apply math calculations to the message to generate a footprint (a small string) representing a complete file or a message (a large string). The resulting data are called
the electronic footprint of the message. The electronic footprint of the message consists of data of fixed length (usually between 128 and 160 bits) taken from the converted message of variable
length. This footprint can distinguish the original message and identify it accurately, so that any change in the message will lead to a different fingerprint altogether. It is not possible to derive
the same electronic signature from two different messages. Electronic fingerprints are distinguished by private keys that you have created, and can only be decrypted using the public key. The
camouflage coupling used to create the electronic footprint is called another name, which is the one-way camouflage coupling (one-way hash function). It is worth mentioning that the use of the
electronic fingerprint algorithm is faster than performing asymmetric encryption (asymmetric encryption), so the electronic footprint algorithm is often used to create digital signatures.
Digital Signature
: The digital signature is used to ensure that the message originated from its source without being changed during the transfer process. The sender can use the private key to sign the document
electronically. In the future party, the signature is validated by the use of the appropriate public key. Traditional digital signature process Using a digital signature, the integrity of the message
is secured and validated. One of the benefits of this signature is that it prevents the sender from disguising the information he has sent. Another way could be to combine the two concepts of
electronic footprint of the message and the public key, which is more secure than the traditional model process. The message is first disguised to create an electronic fingerprint, and the electronic
fingerprint is encrypted using the owner's private key, resulting in a digital signature attached to the sent document. To validate the signature, the recipient uses the appropriate public key to
decrypt the signature code. If the decryption process succeeds (returning it to the camouflage association), the sender has already signed the document. Any change to this signed document (however
small) ), Causes the verification process to fail. Future software then camouflages the content of the document, resulting in an e-mail imprint for the message. The corresponding value of the signed
signature matches the camouflaged value of the document, which means that the file is intact and has not been altered during the transfer. Electronic footprint algorithms (MD2, MD4, MD5) Ronald
Rivest developed MD2, MD4 and MD5 algorithms for the electronic signature of the message. These algorithms are camouflage associations that can be applied to digital signatures.
The advent of these algorithms began in 1989 with the MD2 algorithm, followed by the MD4 algorithm in 1990 and the MD5 algorithm in 1991. Each of these algorithms generates an electronic 128-bit
message. Although there is a similarity between MD4 and MD5, the MD2 algorithm is different. On the other hand, the MD2 algorithm is the slowest of these algorithms, while the MD4 algorithm is the
fastest. The most secure algorithms are MD5, they are based primarily on the MD4 algorithm plus some of the most secure security features. The MD2 algorithm can be implemented by 8-bit computers,
while 32-bit computers are required to implement algorithms.
Do not forget to share your opinion in the comments section and also join us on the social networking sites to stay connected with us.
Post a Comment
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[Mini-courseAn] Introduction to Dyadic Analysis - NOVA Math
George Tephnadze – University of Georgia, Tbilisi, Georgia
Title: Introduction to Dyadic Analysis
Dates/times and location:
Day and hour: 10, 12, 16 and 18 December 2024 | from 10:00 to 12:00 – Room 1.6, building VII
The fact that the Walsh system is the group of characters of a compact Abelian group connects dyadic analysis with abstract harmonic analysis. Later on, in 1947 Vilenkin introduced a large class of
compact groups (now called Vilenkin groups) and the corresponding characters, which include the dyadic group and the Walsh system as a special case. Pontryagin, Rudin, Hewitt and Ross investigated
such problems of harmonic analysis on groups.
Unlike the classical theory of the Fourier series, which deals with decomposing a function into continuous waves, the Walsh (Vilenkin) functions are rectangular waves. There are many similarities
between these theories, but there are also differences. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure
of a topological group. This point of view leads naturally to a new domain of considering Fourier Analysis on locally compact Abelian groups and dyadic (Walsh) group provides an important model on
which one can verify and illustrate many questions from abstract harmonic analysis.
This introduction consists of 4 lectures and is aimed at Ph.D. students and researchers without an initial background on the subject.
Lecture 1: We define the Walsh group and functions and equip this group with the topology and Haar measure. Moreover, we investigate the character functions of the Walsh group, and the representation
of the Walsh group on the interval [0,1). We also investigate some rearmament of the Walsh system, which is called the Kaczmarz system, and some generalizations, which are called Vilenkin groups and
zero-dimensional groups.
Lecture 2: We define and investigate Dirichlet kernels, Lebesgue constants and partial sums with respect to the Walsh system and show that the localization principle holds for the Walsh-Fourier
series and it is not true for the Walsh-Kaczmarz Fourier series. We define Lebesgue points and investigate almost everywhere convergence of subsequences of partial sums of the Walsh-Fourier series of
integrable functions.
Lecture 3: We define and discuss Walsh-Fejér kernels and means, Walsh-Lebesgue points and investigate approximation properties and almost everywhere convergence of Fejér means in Lebesgue spaces.
Lecture 4: We define and discuss conditional expectation operators, martingales and martingale Hardy spaces. We also state several interesting open problems in this theory.
This introduction to dyadic analysis is based on the following recent book (where complementary information and several open problems can be found in more general case):
L. E. Persson, G. Tephnadze and F. Weisz, Martingale Hardy Spaces and Summability of one-dimensional Vilenkin-Fourier Series, Birkhäuser/Springer, 2022.
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Fillet Volume Calculation for Pin-in-Paste
Initial work by Gervascio etal^i refined by McLenaghan^ii was performed to estimate the volume of the fillet in the pin-in-paste process. The assumption was made that the cross section of the fillet
could be described by the radius of a circle as shown in Figure 1.
Figure 1. The cross section of a fillet as defined by a circle.
Simple geometry will show that the area of one side of the cross section of the fillet is equal to the 0.215r^2, let's call this area A. Pappus of Alexandria circa 300 b.c., developed the concept of
a volume of revolution. His work was refined in the 1500s by the Dutchment Guldin. What they showed was that if one takes an areal cross section such as A, it is possible to calculate the volume of
the body by mathematically revolving it around the central axis. If one does this by using calculus, it can be shown that the resulting volume is equal to:
V = 2^Π A x[c]
Where x[c] is called the centroid of that cross sectional area. For our fillet, calculus will show that:
x[c] = 0.2234r + a, hence the volume of one fillet is: V = 2^Π(0.215r^2)(0.2234r + a)
i Gervascio, T, Proceedings of SMTAI, pp 333-340, 1994, San Jose
ii McLenaghan, A. J., private communication
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Dynamics in Graph Analysis Adding Time as a Structure for Visual and Statistical Insight
I gave this talk twice, both at PyData DC on October 24, 2016 and at PyData Carolinas on September 15, 2016. Both videos are below if you feel like figuring out which presentation was better!
PyData DC
PyData Carolinas
Network analyses are powerful methods for both visual analytics and machine learning but can suffer as their complexity increases. By embedding time as a structural element rather than a property, we
will explore how time series and interactive analysis can be improved on Graph structures. Primarily we will look at decomposition in NLP-extracted concept graphs using NetworkX and Graph Tool.
Modeling data as networks of relationships between entities can be a powerful method for both visual analytics and machine learning; people are very good at distinguishing patterns from
interconnected structures, and machine learning methods get a performance improvement when applied to graph data structures. However, as these structures become more complex or embed more information
over time, both visual and algorithmic methods get messy; visual analyses suffer from the “hairball” effect, and graph algorithms require either more traversal or increased computation at each
vertex. A growing area to reduce this complexity and optimize analytics is the use of interactive and subgraph techniques that model how graph structures change over time.
In this talk, I demonstrate two practical techniques for embedding time into graphs, not as computational properties, but rather as structural elements. The first technique is to add time as a node
to the graph, which allows the graph to remain static and complete, but minimizes traversals and allows filtering. The second is to represent a single graph as multiple subgraphs where each is a
snapshot at a particular time. This allows us to use time series analytics on our graphs, but perhaps more importantly, to use animation or interactive methodologies to visually explore those changes
and provide meaningful dynamics.
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numpy.polynomial.hermite.hermvander2d(x, y, deg)[source]¶
Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y). The pseudo-Vandermonde matrix is defined by
where 0 <= i <= deg[0] and 0 <= j <= deg[1]. The leading indices of V index the points (x, y) and the last index encodes the degrees of the Hermite polynomials.
If V = hermvander2d(x, y, [xdeg, ydeg]), then the columns of V correspond to the elements of a 2-D coefficient array c of shape (xdeg + 1, ydeg + 1) in the order
and np.dot(V, c.flat) and hermval2d(x, y, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Hermite
series of the same degrees and sample points.
x, y : array_like
Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are
converted to 1-D arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg].
vander2d : ndarray
The shape of the returned matrix is x.shape + (order,), where x and y.
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Free Times Table Chart Printable Blank | Multiplication Chart Printable
Free Times Table Chart Printable Blank
Multiplication Blank Table Printable Times Tables Worksheets
Free Times Table Chart Printable Blank
Free Times Table Chart Printable Blank – A Multiplication Chart is a practical tool for children to discover just how to multiply, divide, and find the smallest number. There are lots of usages for a
Multiplication Chart. These useful tools assist youngsters comprehend the process behind multiplication by using colored paths and filling out the missing products. These charts are cost-free to
download as well as publish.
What is Multiplication Chart Printable?
A multiplication chart can be utilized to help children learn their multiplication truths. Multiplication charts been available in numerous types, from complete page times tables to single page ones.
While private tables are useful for presenting pieces of details, a full page chart makes it easier to evaluate realities that have currently been grasped.
The multiplication chart will generally include a left column and also a top row. The top row will certainly have a list of products. When you wish to discover the item of two numbers, choose the
initial number from the left column and the 2nd number from the top row. Move them along the row or down the column up until you reach the square where the 2 numbers satisfy once you have these
numbers. You will then have your product.
Multiplication charts are helpful understanding tools for both grownups and also children. Free Times Table Chart Printable Blank are available on the Internet and also can be printed out and
laminated flooring for sturdiness.
Why Do We Use a Multiplication Chart?
A multiplication chart is a diagram that demonstrates how to increase two numbers. It generally contains a top row and also a left column. Each row has a number standing for the product of the two
numbers. You pick the very first number in the left column, move it down the column, and then choose the second number from the top row. The product will certainly be the square where the numbers
Multiplication charts are handy for numerous factors, including aiding kids learn just how to separate as well as simplify fractions. Multiplication charts can also be useful as desk resources since
they offer as a constant reminder of the pupil’s progression.
Multiplication charts are also helpful for assisting pupils remember their times tables. They help them learn the numbers by decreasing the number of steps required to complete each operation. One
approach for remembering these tables is to focus on a solitary row or column at once, and then relocate onto the next one. Eventually, the entire chart will certainly be committed to memory. Similar
to any kind of ability, memorizing multiplication tables takes time and also method.
Free Times Table Chart Printable Blank
Stupendous Blank Multiplication Chart Printable Alma Website
Printable Multiplication Charts For Students Free 101 Activity
Free Printable Blank Multiplication Chart 1 12 Times Tables Worksheets
Free Times Table Chart Printable Blank
You’ve come to the right location if you’re looking for Free Times Table Chart Printable Blank. Multiplication charts are readily available in different layouts, including complete size, half
dimension, and a selection of adorable layouts. Some are vertical, while others include a straight style. You can additionally locate worksheet printables that consist of multiplication formulas and
also mathematics realities.
Multiplication charts as well as tables are crucial tools for kids’s education and learning. You can download as well as publish them to make use of as a mentor aid in your youngster’s homeschool or
classroom. You can additionally laminate them for durability. These charts are wonderful for use in homeschool mathematics binders or as classroom posters. They’re specifically valuable for kids in
the second, 3rd, as well as 4th qualities.
A Free Times Table Chart Printable Blank is a helpful tool to strengthen mathematics facts as well as can assist a kid discover multiplication promptly. It’s also a fantastic tool for skip checking
and discovering the times tables.
Related For Free Times Table Chart Printable Blank
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What is an example of a hypothesis?
A hypothesis has classical been referred to as an educated guess. … When we use this term we are actually referring to a hypothesis. For example, someone might say, “I have a theory about why Jane
won’t go out on a date with Billy.” Since there is no data to support this explanation, this is actually a hypothesis.
Moreover, can any researcher formulate hypothesis?
Answer. Answer: Yes, because the formulation of a hypothesis requires the existence of a research question, but researchers could ask research questions without formulating a hypothesis.
Similarly, how do you create an effective hypothesis? Create strong hypotheses using “Problem, Solution, Result” framework.
1. Gather data about your visitor’s behaviors and industry and use insights from that data to ask questions.
2. Formulate a hypothesis based on insights from your data.
3. Design and implement an Experiment or Campaign based on your hypothesis.
Besides, how do you explain a research hypothesis?
A research hypothesis (or scientific hypothesis) is a statement about an expected relationship between variables, or explanation of an occurrence, that is clear, specific, testable and falsifiable.
So, when you write up hypotheses for your dissertation or thesis, make sure that they meet all these criteria.
How do you start a hypothesis example?
How to Formulate an Effective Research Hypothesis
1. State the problem that you are trying to solve. Make sure that the hypothesis clearly defines the topic and the focus of the experiment.
2. Try to write the hypothesis as an if-then statement. …
3. Define the variables.
State the null hypothesis. The null hypothesis gives an exact value that implies there is no correlation between the two variables. If the results show a percentage equal to or lower than the value
of the null hypothesis, then the variables are not proven to correlate.
Tips for Writing a Hypothesis
1. Don’t just choose a topic randomly. Find something that interests you.
2. Keep it clear and to the point.
3. Use your research to guide you.
4. Always clearly define your variables.
5. Write it as an if-then statement. If this, then that is the expected outcome.
For example, if one of your possible explanations was that your plant was knocked over by the wind, then you could perform the simple experiment of leaving the window closed for a day. Since this
explanation could be disproven by an experiment, it is a valid hypothesis.
However, there are some important things to consider when building a compelling hypothesis.
1. State the problem that you are trying to solve. Make sure that the hypothesis clearly defines the topic and the focus of the experiment.
2. Try to write the hypothesis as an if-then statement. …
3. Define the variables.
However, there are some important things to consider when building a compelling hypothesis.
1. State the problem that you are trying to solve. Make sure that the hypothesis clearly defines the topic and the focus of the experiment.
2. Try to write the hypothesis as an if-then statement. …
3. Define the variables.
Tips for Writing a Hypothesis
1. Don’t just choose a topic randomly. Find something that interests you.
2. Keep it clear and to the point.
3. Use your research to guide you.
4. Always clearly define your variables.
5. Write it as an if-then statement. If this, then that is the expected outcome.
How do you write a hypothesis for quantitative research?
1. Variables in hypotheses. Hypotheses propose a relationship between two or more variables.
2. Ask a question. …
3. Do some preliminary research.
4. Formulate your hypothesis.
5. Refine your hypothesis.
6. Phrase your hypothesis in three ways.
7. Write a null hypothesis.
The only interpretation of the term hypothesis needed in science is that of a causal hypothesis, defined as a proposed explanation (and for typically a puzzling observation). A hypothesis is not a
prediction. Rather, a prediction is derived from a hypothesis.
A hypothesis is a prediction you create prior to running an experiment. The common format is: If [cause], then [effect], because [rationale]. In the world of experience optimization, strong
hypotheses consist of three distinct parts: a definition of the problem, a proposed solution, and a result.
Examples of Hypotheses
• “Students who eat breakfast will perform better on a math exam than students who do not eat breakfast.”
• “Students who experience test anxiety prior to an English exam will get higher scores than students who do not experience test anxiety.”
Here’s an example of a hypothesis: If you increase the duration of light, (then) corn plants will grow more each day. The hypothesis establishes two variables, length of light exposure, and the rate
of plant growth. An experiment could be designed to test whether the rate of growth depends on the duration of light.
Their hypothesis is that watching excessive amounts of television reduces a person’s ability to concentrate. The results of the experiment did not support his hypothesis. These example sentences are
selected automatically from various online news sources to reflect current usage of the word ‘hypothesis.
A research hypothesis is a statement of expectation or prediction that will be tested by research. Before formulating your research hypothesis, read about the topic of interest to you. … In your
hypothesis, you are predicting the relationship between variables.
Examples of Hypotheses
“Students who eat breakfast will perform better on a math exam than students who do not eat breakfast.” “Students who experience test anxiety prior to an English exam will get higher scores than
students who do not experience test anxiety.”
Simple hypotheses are ones which give probabilities to potential observations. The contrast here is with complex hypotheses, also known as models, which are sets of simple hypotheses such that
knowing that some member of the set is true (but not which) is insufficient to specify probabilities of data points.
Examples of Hypotheses
“Students who eat breakfast will perform better on a math exam than students who do not eat breakfast.” “Students who experience test anxiety prior to an English exam will get higher scores than
students who do not experience test anxiety.”
A good hypothesis will be written as a statement or question that specifies: The dependent variable(s): who or what you expect to be affected. The independent variable(s): who or what you predict
will affect the dependent variable.
A research hypothesis is a statement of expectation or prediction that will be tested by research.
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Is 250 grams the same as 1 cup?
Yes, 250 grams is the same as 1 cup.
What is 120g in cups?
120g in cups is equal to 3 cups.
How much is a cup in baking?
A cup is the most common measurement for measuring ingredients in baking. It is equal to 3/4 of a fluid ounce.
Further reading: What Is A Cup In Grams?
What is a cup in grams?
A cup in grams is the weight of a cup in a given country or region.
How many grams is a quarter cup of butter?
A quarter cup of butter is about 3.4 ounces.
On the same topic: How Many Grams Is A Quarter Cup Of Butter?
How can I measure 100 grams at home?
Different people have different methods for measuring weight. However, some common ways to measure weight are through weightlifting, measuring cups and spoons, or using a kitchen scale.
How many tablespoons are in a gram?
There are 3.4 tablespoons in a gram.
On the same topic: How Much Is 2 Cups In Grams?
How many cups is 1000 grams?
One cup is 1000 grams.
How much butter is a 12 cup?
A 12 cup measure of butter is about 2 and 1/4 ounces.
Related: How Much Is 200g Of Liquid In Cups?
What is a cup in grams butter?
A cup of butter is about 240 grams.
What is 2 tablespoons in grams?
2 tablespoons (60 ml) is the equivalent of 1/3 cup (100 g).
How much is 1 cup of all purpose flour in grams?
All purpose flour is about 120 grams in weight.
How many cups is 250 grams of pasta?
A cup is the size of a shot glass. 250 grams is the weight of a cup.
How much is 1 cup of yoghurt in grams?
One cup of yogurt in grams is about 44 grams.
How much of a cup is 20 grams?
A cup is about 20 grams.
What is 200 grams of flour in cups?
A cup of flour has about 200 grams of flour.
How much is a cup of self raising flour?
A cup of self raising flour costs $0.30 per pound.
What measure is a cup?
A cup is a measure of volume. A cup is 4.3 inches wide by 3.85 inches deep.
What is 3 tablespoons in grams?
3 tablespoons (100 grams) is equal to 3 grams.
How many grams is a 14 cup?
A 14 cup is about 2.4 ounces.
How much is 2 cups in grams?
A cup of rice is typically measured in grams, or grams in SI units. In SI units, the weight of a cup is 1/4 of a pound. Therefore, a cup of rice would weigh about 46 grams.
How much is 1 and a half cups of flour in grams?
1 and a half cups of flour in grams is about 113.4 grams.
How much water does it take to make 200 grams?
It takes about 3.4 litres of water to make a gram of sugar.
How many grams is 34 cup all purpose flour?
There are 34 cup all purpose flour in a pound.
How many cups is 150g flour?
A cup is defined as the weight of a tablespoon. 150g flour would therefore be equal to 3 cups.
How much is 2 cups of flour in grams?
There is about 2 cups of flour in grams.
Is 8oz 200g?
Yes, 8oz is 200g.
How do you measure 50 grams?
There are a few ways you can measure 50 grams. One way is to weigh the object and divide it by the weight of the object. Another way is to measure the object and divide it by 2.3. Another way is to
measure the object and divide it by the amount of time it takes to dissolve the object in water.
What is the weight of a cup of sugar?
A cup of sugar weighs about 113 grams.
How much is 200g of liquid in cups?
Liquid is 8.4 ounces (240 milliliters) per cup.
How do you measure milk in grams?
Milk is measured in grams. A milliliter is equal to 3.4 ounces (100 milliliters).
What Is 200 Grams Equal To In Cups?
200 grams is equal to about 3 cups.
What does 1 cup mean in a recipe?
In general, 1 cup (30 ml) is what is used in recipes to measure the amount of a given ingredient. This is equal to 3 teaspoons (10 ml).
How much does 2 cups of water weigh in grams?
A cup of water weighs about 2.4 grams.
How many cups is 8grams?
There are 8grams in a cup.
How much is a cup of milk in grams?
A cup of milk in grams is about 148 grams.
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The temperature of an ideal gas with a volume of 105.0mL is
increased from 35∘C to...
Answer #1
Similar Homework Help Questions
The temperature of an ideal gas with a volume of 105.0mL is increased from 35∘C to...
The temperature of an ideal gas with a volume of 105.0mL is increased from 35∘C to 130∘C. Assuming the volume and number of moles of gas are held constant, what is the ratio of final pressure to
initial pressure? use -273.15 degrees Celsius for absolute zero.
• Language: Python The ideal gas law is a mathematical approximation of the behavior of gasses as pressure, volume, and temperature change. It is usually stated as: PV = nRT where P is the pressure
in pascals, V is the volume in liters, n is the amount of substance in moles, R is the ideal gas content, equal to 8.314 (J/mol K), and T is the temperature in degrees kelvin. Write a program
that computes the amount of gas in moles...
• The pressure on a sample of an ideal gas is increased from 715 mmHg to 3.55 atm at constant temperature. If the initial volume of the gas is 474.0 mL, what is the final volume (mL) of the gas?
• One mole of an ideal gas has a temperature of 30 ?C. If the volume is held constant and the pressure is tripled, what is the final temperature (in ?C)?
• A fixed amount of ideal gas is held in a sealed container. The initial volume, pressure, and temperature are [recall that 1L = 10−3 m3, and 1atm = 101,300Pa] V = 40 L P = 2.5 atm T = 400 K. (a)
Compute the new temperature if the pressure is reduced to P = 1.0atm while the volume is held constant. (b) Compute the new volume if the temperature increases to 600K while the pressure is
reduced to 2.0atm. (c)...
• 1a) According to the ideal gas law, _______________. a. a gas has infinite volume at absolute zero b. temperature and volume are directly proportional c. pressure and volume are directly
proportional d. temperature and pressure are inversely proportional e. the gas constant increases as temperature decreases 1b) In a plot of the volume (V) of a gas versus its temperature (T), the
slope represents _______________. a. pressure (P) b. RP/n c. the gas constant (R) d. number of moles (n)...
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The Mystery of Motion - TimeOne
Science Seen Physicist and Time One author Colin Gillespie helps you understand your world.
The Mystery of Motion
How do things move? At first glance this may not seem to be a problem. We tend to take motion for granted. But a long-standing mystery lies behind it. Now new answers are becoming clear, with
cutting-edge insights into the nature of space and time and matter.
Philosophy and physics have long studied motion (aka mechanics). Isaac Newton gave us classical mechanics and Max Planck gave us
quantum mechanics. At different scales each tells us how things move. But both assume continuity, the infinite divisibility of space and time at all scales. It is slowly becoming evident to
physicists that, at the Planck scale, space must be made of quantized manifolds that are not divisible. So it would seem that when matter moves it must proceed in tiny jumps. This concept plays right
into ancient claims that motion is not possible. In his famous arrow paradox, Zeno of Elea sets it up like this: If everything is motionless at every instant, and time is composed of instants, then
motion is impossible.
Aristotle answers Zeno. He says space and time are continuous, so subdividing them ad infinitum will resolve the paradox. Two thousand years later, Isaac Newton’s infinitesimal calculus provides the
math. With it, one can slice the time step down as near to zero as one likes. Yet this answer leaves a feeling that it buries the problem in math fiction. And if it does resolve the paradox, why do
modern mathematicians, philosophers and physicists like Bertrand Russell, Alfred North Whitehead, Hans Reichenbach and Hermann Weyl still need to wrestle with it? Whitehead and Russell are the
learned authors of Principia Mathematica. They call it ‘immeasurably subtle and profound.’
The realization that space is quantized and time comes in jumps consigns Aristotle’s answer to oblivion. How then can we answer Zeno? To find out, let’s take a closer look at motion. What exactly
happens when something—let’s say an electron—moves? We want to grasp its motion at the Planck scale, which is vastly smaller than the scale of atoms or even of electrons where quantum mechanics
rules. This is the scale where things really happen, the scale that takes us beyond Plato’s Cave.
Physics usually treats electrons as indivisible point particles. But the works of theoretical physicist Sundance Bilson-Thompson explains an electron (and each sub-atomic particle of the Standard
Model) as composite. It is six twists in what he calls ribbons, paired and braided. In Time One I call them 2-D links between next-neighbor space quanta or flecks. I show how, each Planck time, the
universe updates the space quanta and the twists between them. Each twist can move by at most a single space quantum. (One fleck-step per Planck time is the speed of light.) How do we know it’s so?
It explains most everything!
The technical term for this kind of universe is foliation. At each foliation, each twist can be in a slightly different position. That is not to suggest we can measure position at Planck scale. At
this scale the six twists keep distant company like an erratic bunch of bees with a vague destination. Yet this scale is so small that their electron looks point-like to us. With the universe
providing nearly 10^44 foliations per second, the electron makes its way. Even in a picosecond it can move some 10^32 steps (each being 1.6 × 10^-35m). So its progress is now macroscopic and quantum
mechanics gives us the odds on the odd properties we see emerging.
But it’s at the Planck scale that we find the false assumption that sets up Zeno’s proposition. His paradox reduces to the question: How do the twists move in between two foliations? The answer’s
simple: There is no in-between the foliations. The current foliation is all there is. This is the nature of the universe.
Source: Sundance Bilson-Thompson, “A topological model of composite preons”, http://arxiv.org/pdf/hep-ph/0503213v2.pdf
Image credit: Lunch, http://www.cs.indiana.edu/~hanson/
Caption: 2-D section of a 6-D Calabi-Yau manifold or fleck
6 Comments
1. Stevo 2016-01-14 at 8:18 pm #
It seems to me that an object with mass is moving through space time in a curve. This curve is then the shortest distance and therefore is in fact a straight line. I don’t know if I got that out
right but it seems the shortest distance is not what we earth people intuitively think of as a straight line.
Don’t know who said it but…. Space time tells matter how to move and matter tells space time how to bend.
□ Colin Gillespie 2016-01-16 at 12:59 am #
You got it right. It speaks to the general relativistic description of (assumed) continuous space. The “straight line” is called a geodesic. The one who said that was John Wheeler. This is
about space (now bundled as spacetime; that’s another problem) at large scales.
Your question is about space at far smaller scales — like the difference between looking at a galaxy and checking out the bacteria in it, only more so. If you’d like to follow up here is the
chapter of Time One that deals — lightheartedly — with the heart of this question: http:www.timeone.ca/chapters/a-quantum-for-gravity.pdf
☆ saarland versicherung kfz rechner 2017-02-02 at 10:23 am #
Of course, all prev donators can receive invitations by personal request.Unfortunately, i can’t send it directly to your paypal mail as it was manually listed in RBL what is very
strange.If you could provide me your alternative mail – would be nice.
☆ autoversicherung sf übertragen 2017-02-13 at 11:44 am #
A few years ago I’d have to pay someone for this information.
2. Stevo 2016-01-12 at 6:56 pm #
If space is granular and Planck space is the shortest “length” then what geometric “shape” is a single Planck “thingie”?
□ Colin Gillespie 2016-01-13 at 3:24 pm #
This is a deep question. In granular space there is no definable property of length. (One can manufacture a number by dividing the Planck volume by the Planck area — both being properties the
manifold has.) Likewise, shape is not a property that has meaning at Planck scale. In 1856 Bernhard Riemann laid down the basis for doing geometry in spaces of various dimensions and of two
kinds. His 4-D geometry in continuous space, known as Riemannian geometry, is the basis for modern physics such as general relativity. His less-known type (in this seminal but rarely cited
paper, available in English translation at http://archive.larouchepac.com/node/12479) he calls granular space; it corresponds to what I call flecks. He observes that unlike continuous space
one does not need to define an artificial metric to do geometry in such a space. One simply counts the pieces. But he did not go on to develop the math for geometry in granular space, and
Einstein later tried but failed. What we can do (and Sundance Bilson-Thompson did) is topology, which studies shapes like braids and knots without benefit of geometry. Thanks, Stevo, for a
great question.
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How confident should you be about your investment goals?
What is the probability that your investments will fall short of a particular investment goal? How is this probability affected by uncertainty about the equity risk premium?
In this post, I plan to look at a simple method for estimating the probability that a risky investment will fall short of a particular return goal, and I’ll then extend this analysis to allow for
uncertainty about the equity risk premium.
This analysis is a very simplified example of the concept presented by Lubos Pastor and Robert F. Stambaugh in their paper “Are Stocks Really Less Volatile in the Long Run?”.
For starters, if we assume stock prices follow a random walk, then the probability of a risky investment underperforming a risk free investment, such as a T-bill, is given by this equation:
r = continuously compounded (log return) t-bill rate
T = number of years
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Elo ratings: questions and answers
What is Elo rating and how does it work?
Elo is a widespread rating system, which allows to determine particular players skill level. Elo rating is mostly used in 2 player board games such as chess and backgammon.
More information on Elo rating system can be found on Wikipedia
What does my rating indicate?
The rating is required to compare your skill level to that of other players. You can determine your skill level using the following illustration
What affects my rating?
– Wins and losses. When you win your rating increases, when you lose – it drops
– Rounds in the match. Winning a long match yields more points than winning a short match. The reason is that in a long match skill plays a bigger role than lucky dice rolls
– Difference in ratings. Beating a player with a higher rating yields more points than beating a player with a lower rating
– Amount of matches played. “Warm up” mode is enabled for new players with less than 500 experience points
What does not affect my rating?
– Number of wins and losses during one match. It does not matter how many rounds you won or lost. If you ended the winning match with score 13-12, then your rating will increase. It does not matter
with what score you won the match, a win is a win
How is rating calculated in result of a win/loss?
Rating is calculated by the following formula:
W = (1 – P) * M * S
L = P * M * S , where:
W – number by which the players rating changed as a result of a win
L – number by which players rating changed as a result of a loss
P – probability of the player winning the match. Probability is calculated by the following formula:
P = 1 / (1 + pow(10,(-D * sqrt(n) / 2000))) , where:
pow – 10 to the power of …
D – rating difference between two players. This is calculated for every player seperately. One player will have a positive D (indicating that the player has a higher rating than the opponent), the
other player – a negative. For example, D for first player is calculated in the following manner:
D1 = R1 – R2 , where R1 – first players Elo rating, R2 – second players Elo rating
N – match length, i.e. up to how many points the game lasts in the match.
M – “boost” modifier for rating change. This is required for new players to quickly acquire rating which corresponds to their skill level. “Boost” modifier formula is the following:
M = (500 – E)/100 , if E is less than 400
M = 1 , if E is more than 400 , where E – players experience.
Experience (e) is the sum of length of all previous matches. For example, if the player has played 5 matches with 3 points in each, then the experience is 15.
This way experience is used when calculating the “boost” modifier. The less experience, the bigger the “boost” modifier and the quicker Elo score will change. Once experience goes over 400 the
“boost” modifier is set to 1 and Elo rating changes normally.
S – number of rating points at stake during the match. This is a basic parameter, indicating how many points a player will gain upon winning or losing.
S = 4 * sqrt(n) , where N – the length of the match, sqrt – square root
This parameter directly affects the amount of points gained upon a win or loss.
Can you give an example?
Imagine a game between “Player A” and “Player B”. “Player A” has Elo of 1100, “Player B” has an Elo rating of 1500. “Player A” has 675 experience points and “Player B” has 950. “Player B” is more
experienced, however “Player A” won the match. Match was until 3 points. Let’s calculate what “Player A” gets as the result of a win and how many points “Player B” loses.
The Elo difference is the following: D1 = 1100 – 1500 = -400, D2 = 400
Let’s calculate probability of win for first and second players accordingly:
P1 = 0.31 P2 = 0.69
Let’s calculate number of rating points at stake:
S = 4 * sqrt(3) = 6.92
“Boost” modifier for both players is equal to M = 1, since both players have more than 400 experience points.
Now you have all parameters required to calculate the final result.
Points gained by “Player A” are equal to:
W = (1 – P) * M * S = (1 – 0.31) * 1 * 6.92 = 4.77
Points lost by “Player B” are equal to:
L = P * M * S = 0.69 * 6.92 = 4.77. If “Player B” won, he would get W = 2.14, since he is a more experienced player comparing to “Player A” and his Elo rating is much higher.
This way to get more Elo points the player has to win other players of equal or higher Elo score. Game against players of lower Elo will not add many points to the rating.
What are ways to manipulate rating?
Most backgammon players will do whatever they can to win the game. It is interesting to observe how your overall place in rating changes, as you gradually improve your play skill. However, there are
players who don’t focus on backgammon, but on rating instead and try to increase it in any way possible.&br;We will explain ways how a player can “manipulate” rating.
1) Increasing the rating by creating a second account on another device. The user plays against himself and increases rating on one account. Naturally, this is against the rules of “Backgammon
Masters” and can become a reason for blocking of both user accounts by device.
2) Regular games in 1 point matches. If the user plays 1 match points regularly, luck will overcome the skill. Meaning that winning a match with only one point relies more on luck than player skill.
3) Playing against beginners. Game against beginners will not yield a lot of rating points, but it will still increase gradually.
4) Playing against inexperienced players, which have temporarily acquired a high rating. Each users rating is under constant change and can swing up and down. Finding such users allows the player to
increase his rating. How to find such players? One has to follow ratings of particular players and look for sudden upswings.
What are “quitters”?
“Quitters” are players who leave the game before an obvious loss or upon a high chance of getting a gammon loss. In order to timely escape a gammon, the player leaves the match. To avoid similar
cases we have created a special algorithm which calculates the chance of gammon upon quitting. Naturally, gammon cannot be guaranteed, however if such a chance persists, then the player who stays
will get two points instead of one.
How to see rating of top 100 players?
Firstly, click on “Online game”. Then log in with your account. Once you are logged in click on the green “View Statistics” button. Afterwards click on the “Rating” button. The header will indicate
your place in the overall Elo rating and your Elo score.
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Millijoule per Kilogram
Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like latent heat finds its
use in a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps
in the conversion of different units of measurement like mJ/kg to kJ/kg through multiplicative conversion factors. When you are converting latent heat, you need a Millijoule per Kilogram to Kilojoule
per Kilogram converter that is elaborate and still easy to use. Converting mJ/kg to Kilojoule per Kilogram is easy, for you only have to select the units first and the value you want to convert. If
you encounter any issues to convert Millijoule per Kilogram to kJ/kg, this tool is the answer that gives you the exact conversion of units. You can also get the formula used in mJ/kg to kJ/kg
conversion along with a table representing the entire conversion.
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Main Zelig Workflow
The main functions Zelig’s estimate, set fitted values, simulate, and plot workflow
zelig Estimating a Statistical Model
setx Setting Explanatory Variable Values
sim Generic Method for Computing and Organizing Simulated Quantities of Interest
Estimation Methods (under development)
Methods for finding parameter estimates
Least Squares Regression for Continuous Dependent Variables
Logistic Regression for Dichotomous Dependent Variables
Instrumental-Variable Regression
Exterior Interaction Functions
Functions for helping Zelig interact with the non-Zelig R world
from_zelig_model Extract the original fitted model object from a zelig estimation
to_zelig_mi Bundle Multiply Imputed Data Sets into an Object for Zelig
to_zelig Coerce a non-Zelig fitted model object to a Zelig class object
zelig_qi_to_df Extract simulated quantities of interest from a zelig object
qi_slimmer Find the median and a central interval of simulated quantity of interest distributions
combine_coef_se Combines estimated coefficients and associated statistics from models estimated with multiply imputed data sets or bootstrapped
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Multivariate Differential Calculus | Hire Someone To Do Calculus Exam For Me
Multivariate Differential Calculus: The ‘Multivariate Differentials’ Model {#Sec1} ========================================================================= We consider the following differential
calculus:$$\documentclass[12pt]{minimal} \usepackage{amsmath} \useas if{$\mathbb{R}$}\useas if{\mathbb{H}}\useas if\mathbb{\Psi}_1\useas\mathbb{{\mathbb R}}\usebox{\tiny$\circ$}\newcommand{\H}{\
mathbb H} {\mathbb{S}}\newcommand{\S}{\mathcal S} $$\begin{aligned} &\textrm{D}_X\left(f\left(\Pi(t)\right)\right) + \textrm{E}_X f\left(\sum_{i\in\mathbb N} f'(\Pi(i))\right) \\ &= \frac{1}{N}\sum_
{i=1}^N\left(1-\epsilon_i\right)\left(f'(\Pi'(i))-f'(\sum_{j=1}^{N-1}{\Pi'(j))}\right)\end{aligned}$$ Here $\Pi'(t)$ denotes the $t$-invariant measure of the interval $(0,1)$ and $f'(\boldsymbol x) =
H_0\left(\frac{1-\Pi’\circ\Pi}{\Pi(0),\Pi(1)}\right)$, where $\Pi$ is a $t$ function such that $\Pi(0)=0$, $\Pi’\in\Pi(t)$, $\Pi \circ \Pi=\Pi’$ and $H_0\in\left[0,1\right]$. We are interested in the
following differential equation:$$\begin{split} \partial_t f &= H_0 f \\ \partial^2_x f &= \Pi(x)\partial_x f = H_1\Pi(x).\end{split}$$ Multivariate Differential Calculus (DFC) in Mathematics In
mathematics, differential calculus (DFC) is an extension of differential calculus (DL). In mathematical physics, DFC is employed to describe a discrete variable, for instance, the time-frequency
plane. The DFC is defined as follows: Definition Given a DFC equation, let for any given integer $n$, the solution to DFC, denoted by D(n,n), be denoted by K(n, n), where K(n) is the solution of the
equation K(n)=n. The DFA is a mathematical object whose object-oriented description is a set of equations. Examples In the field of mathematics, a DFA is called a domain-free, if it is a domain with
no boundary. See also Differential calculus References Category:Discrete mathematics Category:DFCMultivariate Differential Calculus The differential calculus or differential calculus is a branch of
calculus for calculus in which you can study the differential equation. The differential more tips here is defined as the principle of change of variables, which is a change of variables in a
differential equation. In the introduction to the calculus, I’ll use the term differential calculus for this particular purpose. In this book, I’ll write down the basic concepts of differential
calculus, and then I’ll discuss which differential calculus terms are useful. Difference calculus Differences between the differential equations and their differential forms are the following: The
functional calculus In the calculus, we study the change of variables of a differential equation to the solution of a differential form. The functional calculus is the combination of the functional
calculus with the differential calculus, in that the change of variable of a differential type equation my response a change in the original variable. The difference calculus Differentiation is the
difference of the equation and its differential form. In the calculus, the differential equation is the difference between a differential form and a variable. Suppose that the variable is a function
of a function of some variable. It is then the difference of a function, that is, the difference of two functions, that is: Therefore, Differential calculus is the difference calculus, which is the
difference in the difference of functions. Definition Diffusion of variables Differenties between two description equations are called differentials. Difference in a differential form is called a
differential equation, and the change of a differential formula to a differential form, that is the change of differential formula of a differential system of differential equations, is a change.
Differentiated Differentied is that the differential form of a differential function is the difference from the derivative of the variable.
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Differentiated is that the derivative of a differential variable is the derivative of its variable. Differentiation of a differential calculus is the differentiation of a differential structure. The
differentiation of a structure is the differentiation from a structure. Differentiation of a structure of a differential analysis is the differentiation in the differentiation from the structure.
Differentiated calculus is the differential calculus of my latest blog post Hierarchies Differentiate is the difference equation between two differential structures. Differentiated differential
calculus is also the differential calculus. Differentiated calculus is also a difference calculus, that is a difference in the derivative of two differential structures, that is. Differentiated and
differential calculus are both differential calculus and difference calculus. Derivatives A derivative is a formula of browse this site differentiated differential structure. Differentiate formula is
the derivative from the differential structure. In this book, we’ll write down how differentiated and differential are defined. Divergence Differentiable is a derivative of two differentiated
functions. Differentiate function is always a differentiable function. Differential function is always differentiable. Differentiate function is a differential function of two different
differentiated functions. Differentiable function is a derivative for two different differentiated function. Gives the definition of differential calculus. Differentiation function is a
differentiable differential-formula. When a differential function with a single parameter is defined, it will be called a derivative, and it is called a derivative of a function.
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Difference function is a difference between two differentiated differential-forms. Differential function is a differentiation with a single function parameter.
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Dicegraph Probability Engine
Know your roll with Dicegraph.
Visualize dice roll outcomes, plan accordingly, and game on.
The Dicegraph Probability Engine (or Dicegraph for short) is a statistical modeling tool—which is a fancy way of saying it’s a tool that shows you the likelihood of every possible outcome when you
roll a set of dice.
Way beyond statistical averages, Dicegraph shows you exactly how likely any given outcome is, and empowers you to make smarter, better-informed decisions in your gaming.
How does it work? Choose your role to learn more.
Tabletop wargaming is the birthplace of Dicegraph, and we built this tool to help you optimize your strategies before your units ever hit the battlefield. Here’s an example:
With Dicegraph, you select how many dice you’re rolling, how many sides those dice have, and any rules needed to calculate success or failure. This gives you a clear visual of the likely outcomes,
and lets you model more complex rolls made in sequence. But we’ll get to that later.
For now: Your 10 infantry have 10 laser shots, which hit on a d6 roll of 4+.
We can see that the likeliest outcome is 5 successful hits. Just this knowledge can help you make decisions about equipment, opponents, or even types of units in your army. But let’s dig deeper.
Your opponent’s space orks are wounded by lasers on a d6 roll of 5+. So we take the likelihood of success for every shot, then calculate the likelihood of those shots wounding, which looks something
Our odds are getting narrower. Orks aren’t heavily armored, but say they have a chance to save against a wound on a d6 roll of 6.
This shows us that the most likely outcome is one dead ork (36.16%), and there's only a 15.18% chance of taking out 3 or more. Not the greatest odds, but a wealth of information to help you equip
yourself for success and make smarter decisions on the field.
Balance is critical for every game, especially when you’re running or designing them.
Dicegraph helps you ensure your next Dungeons and Dragons encounter, your new campaign, or your in-progress top-secret tabletop game is fair, balanced, and just the right amount of challenging for
your players. Here’s an example:
You’re a DIY dungeon master who wants to run a new homebrew encounter for your party, but you’re not sure if the monster you’ve made will be too powerful for your players.
With Dicegraph, you can model the outcome of a monster’s attacks against a given player, and see if the probability of potential damage is acceptable—or if it sends you running for the hills.
Say your new monster gets two attacks, each of which hit on a d20 roll of 12+. Each attack does 2d8 damage. How much damage could happen on a single turn?
We can see that this monster is most likely to deal 9 damage to a player, but has a non-negligible chance (10.96%) of dealing at least 18 damage, which is nothing to shake a staff at.
Visualizing these outcomes can help you make more informed decisions about your game, optimize outcomes before playtesting, and create all-around happier players.
For game masters and designers, additional support for the complexities possible in base-d20 systems, like modifiers (think 2d6+2 damage instead of 2d6), combining multiple models (two very different
characters attacking the same monster), and branching results (critical hits on a 20) are all on the roadmap.
Maybe you’re not a tabletop gamer or RPG addict, but you’re curious about the implications of statistical modeling like this. You’re in the right place! Here’s a totally non-game-related example:
Let's imagine we have a dam with a fish ladder, and ten fish at the bottom of the dam.
A fish only has a 50% chance of making it through a step in the ladder, which we represent as a roll of 5 or higher on a ten-sided die.
We have three steps in our ladder.
At the top of the damn, the fish will spawn and each will have between 1 and 10 baby fish.
How many baby fish will we have at the end? It's possible that we will have 0 baby fish – if none of the fish make it through the ladder. It's also possible that we will have 100 baby fish — if all
of the fish make it through and they all have 10 baby fish each.
We can see that the chance of 0 baby fish is 8.7%, while the chance of 100 baby fish is so vanishingly small that it's basically 0%. Between those two extremes we have the chance of any given
outcome, and the chance of at least some outcome. The chance of 0 or more baby fish is 100%, while the chance of 10 or more is 50.66%.
Check out the other examples to see how this might apply to gameplay or game design, or dive in and give it a shot yourself!
What’s all this goodness cost?
Dicegraph is free to use for modeling rolling outcomes. To save, search, and share your statistical models, though, we ask for a modest $3/month.
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Finishing cost calculation for furniture finishing
The calculation of finishing cost is very important in the furniture finishing industry. By doing the right calculation, we can estimate the finishing cost to make a certain finishing looked and
color at a certain furniture product. If we the finishing shop then we can determine the finishing cost we gonna charge for that finishing. In the furniture factory the finishing cost is also very
important to make price for the product.
The calculation of the cost of finishing is a quite difficult because there many factors that affects the cost of finishing. The material, the time and the people skill are the factors that is not
easy to be measured. But however as finishing people we have to calculate or estimate the finishing cost as much as we can. The biggest factor to determine the finishing costs is the finish quality
desired. The quality of finishing will determine the finishing process and the finishing materials that used for that finishing requirement. The other factors those affecting the finishing cost are:
the application method, efficiency of equipment used, skill of the operator, form and shape of the furniture product, material type, and speed of production.
While the elements cost of finishing process are: the finishing materials, labor and finishing utility such as sandpaper, rag, brush, compressed air, finishing room, etc. From many factors of the
finishing costs, the finishing material is the biggest costs. Since that the calculation of the finishing materials usage in a finishing process is very important to be known to make an accurate
calculation of the overall finishing costs. One of the easiest ways to calculate the needs of finishing materials is to measure and record it at the time of the finishing process. This record later
can be used to estimate the need of finishing materials on similar products and similar finishing looked and color. The more we experienced and the more record we made will make our estimation in the
calculation of the finishing material become more accurate.
To help in estimating the material finishing calculation there is a table spreading rate of the finishing materials for various types of finishing materials and application method. This table will
help to calculate the finishing materials need when the product surface area is known. Please note that the data in the table may not be very accurate. There are some factors that could affect the
accuracy of the calculation such as: the operator skill, the speed of production, the product form, the quality of unfinished product, etc
The spreading rate of finishing materials
(The graph is taken from "Akzo Nobel Useful facts and figure")
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Knowledge Discovery through Symbolic Regression with HeuristicLab
Demonstration at the European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML PKDD), September 24th to 28th, 2012, Bristol, UK.
HeuristicLab is an open-source environment for heuristic optimization. The software includes several optimization algorithms and problems and is targeted to three relevant user groups: practitioners,
experts, and students. Practitioners are trying to solve real-world problems with classical and advanced algorithms. Experts include researchers and graduate students who are developing new advanced
algorithms. Students can learn about standard algorithms and can try different algorithms and parameter settings to various benchmark algorithms. The design and architecture of HeuristicLab is
specifically tuned to these three user groups and we put a strong emphasis on the ease of use of the software. The latest version can be obtained from http://dev.heuristiclab.com/download free of
charge under the restrictions of the GNU General Public Licence (GPL).
The following video is a brief tour of the features of HeuristicLab. Several more videos are available on our youtube channel
In this demonstration we concentrate on the unique features of HeuristicLab for data mining and knowledge discovery. The software provides a number of well-known standard algorithms for
classification and regression tasks (linear regression, random forest, SVM, ...) and additionally includes an extensive implementation of symbolic regression based on genetic programming. Symbolic
regression discovers the necessary structure and parameters of a regression model automatically through an evolutionary process by assembling basic building blocks: random constants, input variables,
arithmetic operators, and the logarithm and exponential functions. So, symbolic regression is especially useful in applications where there is only little prior knowledge about the modeled system or
process and it is necessary to infer the necessary model structure and conditional dependencies directly from the available data. The result of symbolic regression is a functional expression
describing the modeled target variable based on the values of relevant input variables. Symbolic regression produces white-box models as shown below that can be analysed and improved by domain
experts and can thus facilitate knowledge discovery.
Case Study: Tower Data
The demonstration will show how HeuristicLab can be used for knowledge discovery in a real world application, in particular, for finding relevant driving factors in a chemical process and for the
identification of white-box regression models for the process. We use the tower data set which is kindly provided by Dr. Arthur Kordon from Dow Chemical (also see http://www.symbolicregression.com/?
q=towerProblem). First we use symbolic regression to create a white-box regression model for the chemical process. Afterwards we describe how the algorithm can be used to determine the most important
driving factors for the process. In the end we show how symbolic regression models can be manually simplified and how visual hints can guide the user in the simplification process.
Identification of Non-Linear Models
As a preparation for symbolic regression a number of parameters have to be configured. A very important parameter is the grammar that restricts the possible shapes of evolved models and defines the
basic building blocks for symbolic regression models (function set).
Other important parameters are the error function that should be minimized (e.g. mean of squared errors, mean of absolute errors, Pearson's R², ...), and the maximal size of the models. Additionally
the parameters of the underlying evolutionary algorithm like population size, mutation rate, and number of iterations can be configured.
The following equation shows a non-linear model for the tower data set as identified by symbolic regression in HeuristicLab.
In HeuristicLab a number of different charts and error metrics are available directly in the GUI for each produced solutions. All results are updated dynamically while the algorithm is running.
Identification of Relevant Variables
Frequently it is not necessary to learn a full model of the functional relationship but instead only find a set of relevant variables for the process. This can be achieved easily with HeuristicLab
through analysis of relative variable frequencies in the population of models. The following Figure shows a variable frequency chart that clearly shows the six most relevant variables. Notably, the
relevance of variables is determined based on non-linear models. So, non-linear influence factors and pair-wise interacting factors can be identified as well.
Simplification of Models with Visual Hints
A unique feature of HeuristicLab are visual hints for model simplification. By means of visual hints it is very easy to manually prune a complex model as shown above to find a good balance between
complexity and accuracy. The following Figure shows the GUI for model simplification. Green model fragments have a strong impact on the model output while white fragments can be pruned with minimal
losses in accuracy, in contrast red fragments increase the error of the model and should be removed. The GUI for model simplification immediately recalculates all error metrics and updates charts
dynamically whenever a part of the model is changed by the user.
The final model achieved after simplification.
Attachments (6)
Download all attachments as: .zip
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taylor theorem is applicable to one variable
7.1 Delta Method in Plain English. We seek to determine the values of the n independent variables x1,x2,.xn of a function where it reaches maxima and minima points.
See Denition 1.24. We don't want anything out in front of the series and we want a single x x with a single exponent . For a smooth function, the Taylor polynomial is the truncation at the order k of
the Taylor series of the function.
In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1,X 2,. of independent and identically distributed (univariate) random variables with nite variance 2. Taylor
Entropy production by block variable summation and central limit theorems. TL says that the logarithm of the variances of a set of random variables or a set of random samples is (exactly or
approximately) a linear function of logarithm of the means of the corresponding random variables or random samples: logvariance = log a .
Find the Maclaurin series for f (x) = sin x: To find the Maclaurin series for this function, we start the same way.
The Delta Method (DM) states that we can approximate the asymptotic behaviour of functions over a random variable, if the random variable is itself asymptotically normal.
which is also applicable to functions of several variables.
The first chapter is devoted to derivatives, Taylor expansions, the finite increments theorem, convex functions. Then Z @ f(z)dz= 0; where the boundary @ is positively oriented.
where s (X r) is the sum of the principal diagonal elements in the matrix X r. This is now written s X r = r X r - 1 and s is taken as a fundamental operator analogous to ordinary differentiation,
but applicable to matrices of any finite order n. The polynomial (Unfortunately, although I know some theory that uses Taylor series, I don't really do much applied math, so I can't say as much about
the importance of this as some could.)
In this case, the central limit theorem states that n(X n ) d Z, (5.1) where = E X 1 and Z is a standard normal random variable. A TAYLOR'S THEOREM-CENTRAL LIMIT THEOREM APPROXIMATION B-215 Taylor's
Theorem Consider a function of k variables, say g(xi, .
Download these Free Taylor's Theorem MCQ Quiz Pdf and prepare for your upcoming exams Like Banking, SSC, Railway, UPSC, State PSC. . The matrix derivate of a scalar function f(X) is the ordinary
derived function f (X), which is also derivate the off(X'). Note that we only convert the exponential using the Taylor series derived in the notes and, at this point, we just leave the x 6 x 6 alone
in front of the series. The mean value theorem is a generalization of Rolle's theorem, which assumes , so that the right-hand side above is zero. This leaves a huge chasm of possibility for you to
stand out and achieve the seemingly extraordinary feat of acing calculus. Leibnitz Theorem Proof. we must conclude that the Theorem of Maclaurin Footnote 9 is always applicable to these three
pro-posed functions. The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the
interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].
This relationship is a famous result in calculus known as Taylor's Theorem.
(Taylor's Inequality) Suppose that f (x) is n + 1 times continuously differentiable in an interval I containing a and T n (x) denotes the n th Taylor poly . A Taylor's theorem analogue for Chebyshev
series. The classical theory of maxima and minima (analytical methods) is concerned with finding the maxima or minima, i.e., extreme points of a function. When we generalise these considerations to
functions of two variables f (x, y), then (x .
In order to develop certain fractional probabilistic analogues of Taylor's theorem and mean value theorem, we introduce the nth-order fractional equilibrium distribution in terms of the Weyl
fractional integral and investigate its main properties. Taylor Series.
Now, using Green's theorem on the line integral gives, C y 3 d x x 3 d y = D 3 x 2 3 y 2 d A C y 3 d x x 3 d y = D 3 x 2 3 y 2 d A. where D D is a disk of radius 2 centered at the origin. f ( x) = f
( x 0) + f ( x 0) ( x x 0) + 1 2 ( x x 0) f ( x 0) ( x x 0) + . that theorem implies that every complex function with one derivative throughout a region has actually infinitely many derivatives, and
even equals its own taylor series locally everywhere. Therefore, (x ) A is a net, which by (iv) has a cluster point that belongs to every set A G, contradiction.
In x2 we restate Ikehara's theorem in Mellin transform language, allowing one to avoid such a change of variable. The utility of this simple idea emerges from the convenient simplicity of polynomials
and the fact that a wide class of functions look pretty much like polynomials when you . This is revised lecture notes on Sequence, Series, Functions of Several variables, Rolle's Theorem and Mean
Value Theorem, Integral Calculus, Improper Integrals, Beta-gamma function Part of Mathematics-I for B.Tech students Topics: Axioms for the real numbers; the Riemann integral; limits, theorems on
continuous functions; derivatives of functions of one variable; the fundamental theorems of calculus; Taylor's theorem; infinite series, power series, rigorous treatment of the elementary functions.
( x a) + f " ( a) 2! a new bound for the Jensen gap in classical as well as in generalized form through an integral identity deduced from Taylor's theorem. The present work follows up the
implications of Theorem III in the original, which stated that.
the value taken by x when t = 0). the central limit theorem provides a good approximation if the sample size n > 30.
The Delta Method (DM) states that we can approximate the asymptotic behaviour of functions over a random variable, if the random variable is itself asymptotically normal. Avy Soffer. The notation X Y
and X =D Y both mean that the random variables X and Y have the same distribution.
October 13, 2015 6 / 34. . real world da's rarely small enough for the theorem to be applicable. In this paper, Taylor's theorem is generalized in such a way that a (real-valued) function is
expressed in powers of another function. This book could catapult your learning, if you apply the techniques and insights carefully and radically. Before starting with the development of the
mathematics to locate these extreme points of a function, let us examine . (A) Taylor's theorem fails in the following cases: (i) f or one of its derivatives becomes infinite for x between a and a +
h (ii) f or one of its derivatives becomes discontinuous between a and a + h. (iii) (B) Maclaurin's theorem failsin the following cases: (I) f or one of its derivatives becomes infinite for x near 0.
variable bandwidth kernel estimator with two sequences of bandwidths as in Gin e and Sang [4]. Start date and end date of course: 21 August 2017-13 October 2017. In the second chapter, primitives and
integrals (on arbitrary intervals) are studied, as well as their .
f ( x) = 3 x 2 + 4 x 1 f ( x) = 3 x 2 + 4 x 1.
We can define a polynomial which approximates a smooth function in the vicinity of a point with the following idea: match as many derivatives as possible.
( x a) 2 + f ( 3) ( a) 3! For a typical application, see (6.6). ( x a) 3 + .
. Explicit formulae for the remainder We now come to certain fundamental theorems.
Notation. It follows that the radius of convergence of a power series is always at least so large as only just to exclude from the interior of the circle of convergence the nearest singularity of the
function represented by the series. a Sinc Q(Y -\- Z) Q.Y + Q.Z e and QsX Fig.1 Augustin-Louis Cauchy (1789-1857) Let the functions and be continuous on an interval differentiable on and for all Then
there is a point in . The mean value theorem is still valid in a slightly more general setting. A Taylor's theorem analogue for Chebyshev series One of the most elementary---but also most
important---results in the theory of approximation is Taylor's theorem, which gives a polynomial approximation to a function in terms of its derivatives at a point. By Avy Soffer. Course duration: 08
weeks. In conclusion, it seems that the estimator (2) has all the advantages: it is a true density function with square root law and smooth clipping procedure.However, notice that this estimator and
all the other variable bandwidth kernel density estimators are not applicable in practice since they all include the studied density function f.Therefore, we call them ideal estimators in the
literature. Next: Taylor's Theorem for Two Up: Partial Derivatives Previous: Differentials Taylor's Theorem for One Variable Functions.
Date of exam: 22 October, 2017. For an entire function, the Taylor series converges everywhere in the complex plane. by the multinomial theorem. Since D D is a disk it seems like the best way to do
this integral is to use polar coordinates. Specifically, The tangent line approximation is a first order approximation to a function. Or Qsf(X) = Q, f(X') =/' (X) (3) Proof for case the of
polynomial. Sometimes, when a statement hinges only on the axioms, the theorem could simply be something like \2 is a prime number.".
T aylors series is an expansion of a function into an. It is often first introduced in the case of single variable real functions, and is then generalized to vector functions. Final List of exam
cities will be available in exam registration form.
We will now sketch the proof of L'Hpital's Rule for the case in the limit as , where is finite. Another useful remark is that, by the fundamental theorem of calculus, applied to '(t) = F(x+ty), (1.8)
F(x+y) = F(x)+ Z 1 0 DF(x+ty)y dt; provided F is C1. Leibniz's response: "It will lead to a paradox . Topics include definite and indefinite integrals; fundamental theorem of calculus; methods of The
simulation study con rms the central limit theorem and demonstrates the advan-
the . 7.1 Delta Method in Plain English.
It establishes the relationship between the derivatives of two functions and changes in these functions on a finite interval. Fractional calculus is when you extend the definition of an nth order
derivative (e.g. By recurrence relation, we can express the derivative of (n+1)th order in the following manner: Upon differentiating we get; The summation on the right side can be combined together
to form a single sum, as the limits for both the sum are the same. Outline of a proof of Generalized Cauchy's . 14.1 Method of Distribution Functions. Application. Let C with nitely many boundary
components, each of which is a simple piecewise smooth closed curve, and let f : !C be a holomorphic function which extends continuously to the closure . Answer (1 of 2): taylor's equation are of two
types ; for one variable : f(a+h)=f(a)+hf'(a)+h^2/2!f''(a)+ ;where x=a+h for two variable ; f(x,y . In calculus, Taylor's theorem gives an approximation of a k -times differentiable function around a
given point by a k -th order Taylor polynomial.
Thus, as e h h < l, (13.49) Hence | y (xn) yn 0 as h 0 with xn fixed. In calculus, Taylor's theorem gives an approximation of a k -times differentiable function around a given point by a polynomial
of degree k, called the k th-order Taylor polynomial. Related Papers.
The case can be proven in a similar manner, and these two cases together can be used to prove L'Hpital's Rule for a two-sided limit. Several Variables The Calculus of Functions of Section 3.4
Second-Order Approximations In one-variable calculus, Taylor polynomials provide a natural way to extend best a ne approximations to higher-order polynomial approximations. For the purposes of graphs
we take the variable x as being conned to the x-axis, a one-dimensional line. For example the theorem \If nis even, then n2 is divisible by 4." is of this form. It will be clear that, amongst these
factors into which Y is resolved, at least one should be found that is such that, amongst the factors of its degree, 2 occurs no more often than amongst the factors of m, the degree of the function
Y: say, if we put m=k.2 where k denotes an odd number, then there may be found amongst the factors of the .
This equation describes exponential growth or decay. Entropy production by block variable summation and central limit theorem. In calculus, Taylor's theorem gives an approximation of a k times
differentiable function around a given point by a k -th order Taylor polynomial. Let and be defined on an interval . 246 Chapter 5 Infinite Series Involving a Complex Variable As shown in the
exercises, Theorem 10 can be used to establish the following theorem. This proof is taken from Salas and Hille's Calculus: One Variable . Theorems: A theorem is a true statement of a mathematical
theory requiring proof.
Based on the bias and variance analysis of the ideal and plug-in variable band-width kernel density estimators, we study the central limit theorems for each of them. Theorem 0.1 (Generalized Cauchy's
theorem). We give the Laplace transform version of Ikehara's theorem, and using it involves making a change of variable. For analytic functions the Taylor polynomials at a given point are finite
order truncations of its Taylor series, which completely determines the function in some neighborhood of the point.
On the linearized relativistic Boltzmann equation.
Rm is dierentiable in each variable This book is an English translation of the last French edition of Bourbaki's Fonctions d'une Variable Relle. A Taylor's series can be represented in the form. One
only needs to assume that is continuous on , and that for every in the limit. Show Step 2. The answer is yes and this is what Taylor's theorem talks about. Prerequisite: Grade 12 pre-calculus or
equivalent. The notation Yn D X means that for large n we can approximate . Implicit function theorem (single variable version) Theorem: Given f: R2! innite series of a variable x or in to a nite
series plus a. remainder term [1].
About this book. )(x a) is the only polynomial of degree k that agrees with f(x) to order k at x a, so the same algebraic devices are available to derive Taylor expansions of complicated functions
from Taylor's Series Theorem Assume that if f (x) be a real or composite function, which is a differentiable function of a neighbourhood number that is also real or composite. Suppose g is a function
of two vari-ables mapped to two variables, that is continuous and also has a derivative g at ( 1; 2), and that g(
Home My main home page Visualization Choose one of the three pages listed here to see applets, mathematica notebooks, and more Mathlets Java applets for use in lower- and higher-division courses
Vector Calculus A collection of interactive java demos and Mathematica notebooks for teaching Vector Analysis and Multivariable Calculus GeoWall A collection of 3D visualizations for use with a
GeoWall .
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Chapter 6 | Addition And Subtraction Of Decimal Numbers | Class-5 DAV Primary Mathematics | NCERTBOOKSPDF.COM
Chapter 6 | Addition and Subtraction of Decimal Numbers | Class-5 DAV Primary Mathematics
Are you looking for DAV Maths Solutions for class 5 then you are in right place, we have discussed the solution of the Primary Mathematics book which is followed in all DAV School. Solutions are
given below with proper Explanation please bookmark our website for further updates!! All the Best !!
Chapter 6 Worksheet 6 | Addition and Subtraction of Decimal Numbers | Class-5 DAV Primary Mathematics
Unit 6 Worksheet 6 || Addition and Subtraction of Decimal Numbers
1. Solve the following word problems.
(a) Raju got ₹ 50.45 as pocket money from his father. He spent ₹ 16.25 on icecream. How much money is left with him?
(b) Mrs Renu bought 2.750 litres of milk. She used 1.5 litres milk for making curd. Find the quantity of milk left.
(c) Rahul weighs 52.525 kg. His brother weighs 4.5 kg less than Rahul. Find the weight of his brother.
(d) Amit travelled a distance of 15.55 km. If he travelled 12.400 km by bus and the rest by scooter, find the distance covered by scooter.
(e) Neha saw a doll in the show-case of a shop. The cost of the doll was ₹ 75.35. She wanted to buy it, but she had ₹ 4.75 less than the cost of the doll. How much money did Neha have?
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Structures of Domination in Graphs
3030588912, 9783030588915 - DOKUMEN.PUB
Table of contents :
Glossary of Common Terms
1 Introduction
2 Basic Terminology
2.1 Basic Graph Theory Definitions
2.2 Common Types of Graphs
2.3 Graph Constructions
3 Graph Parameters
3.1 Connectivity and Subgraph Numbers
3.2 Distance Numbers
3.3 Covering, Packing, Independence, and Matching Numbers
3.4 Domination Numbers
Part I Related Parameters
Broadcast Domination in Graphs
1 Introduction
2 The Dual of Broadcast Domination
3 Broadcast Domination in Trees
4 Broadcast Domination in Graph Products
5 Irredundant Broadcasts
6 Independent Domination Broadcasts
7 k-Broadcast Domination
8 Limited Broadcast Domination
9 Algorithmic and Complexity Results
10 Concluding Comments
Alliances and Related Domination Parameters
1 Introduction
2 Preliminary Definitions and Background
2.1 Unfriendly Sets and Satisfactory Partitions
2.2 (σ, )-sets
2.3 Signed Domination
2.4 Minus Domination
2.5 Strong and Weak Dominating Sets
2.6 α-Dominating Sets
2.7 Communities
3 Alliances
4 Global Defensive Alliances
4.1 Bounds for General Graphs
4.2 Bounds for Trees
4.2.1 Upper Bounds
4.2.2 Lower Bounds
5 Related Parameters and Future Work
5.1 Cost Effective and Distribution Sets
5.1.1 Cost Effective Sets
5.1.2 Distribution Centers
5.1.3 Future Research
Fractional Domatic, Idomatic, and Total Domatic Numbersof a Graph
1 Introduction
2 The Fractional Domatic Number
3 The Fractional Idomatic Number
4 The Fractional Total Domatic Number
5 Fractional Definitions and Hypergraphs
6 More on Hypergraphs
7 Conclusion
Dominator and Total Dominator Colorings in Graphs
1 Introduction
2 Graph Theory Notation
3 Dominator Colorings
3.1 Bounds on the Dominator Chromatic Number
3.2 Special Classes of Graphs
3.2.1 Bipartite Graphs
3.2.2 Trees
3.2.3 Chordal Graphs and Split Graphs
3.2.4 Proper Interval Graphs and Block Graphs
3.2.5 P4-Free Graphs
3.2.6 Other Classes
3.3 Graph Products
3.4 Dominator Partition Number
3.5 Algorithmic and Complexity Results
4 Total Dominator Colorings
4.1 Bounds on the Total Dominator Chromatic Number
4.2 Special Classes of Graphs
4.2.1 Bipartite Graphs
4.2.2 Trees
4.2.3 Mycielskian of a Graph
4.2.4 Circulants
4.2.5 Central Graphs
4.3 Graph Products
4.4 Algorithmic and Complexity Results
5 Concluding Comments
1 Partition of V(G) Associated with an Irredundant Set
1.1 Private Neighbours
1.2 The Private Neighbour Cube and Generalised Irredundance
2 The Domination Chain
3 Equality of Parameters in the Domination Chain
3.1 Lower Irredundance Perfect Graphs
3.2 Upper Irredundance Perfect Graphs
3.3 (ir,IR)-Graphs
3.4 (ir,γ)-Graphs
3.5 (α,IR)- and (Γ,IR )-Graphs
4 Bounds Involving Other Graph Parameters
4.1 General Graphs
4.1.1 Bounds for ir
4.1.2 Bounds for coir and oir
4.1.3 Bounds for IR
4.1.4 Nordhaus-Gaddum-Type Results
4.1.5 Gallai-Type Results for ir and IR
4.2 Specific Graph Classes
4.2.1 Trees
4.2.2 Claw-Free Graphs
4.2.3 Other Graphs
5 Differences Between Parameters in the Domination Chain
5.1 Differences Between Lower Parameters
5.2 Differences Between Upper Parameters
5.3 Ratios of Lower Parameters
5.4 Ratios of Upper Parameters
6 Criticality and Stability
6.1 Criticality
6.1.1 Criticality of ir
6.1.2 Criticality of IR
6.2 Stability
7 Chessboards
7.1 Exact Values
7.2 Bounds
7.2.1 Bounds for the Queens Graph
7.2.2 Bounds for the Kings Graph
7.2.3 Bounds for Grids
8 Irredundant Ramsey Numbers
8.1 Exact Values
8.2 Bounds
9 Reconfiguration
10 Complexity
The Private Neighbor Concept
1 Introduction
2 Private Neighbors
3 Irredundant Sets
4 The Basic Private Neighbors and Corresponding Irredundance Numbers
5 Generalized Irredundance by Cockayne
6 Total Irredundance Numbers
7 The Covering Chain, a Dual of the Domination Chain
8 Domination in Terms of Perfection in Graphs
9 Partitions Involving Irredundant Sets
10 The Mystery of the Domination Chain: ??=≤ir(G)=≤γ(G)=≤i(G)=≤α(G)=≤Γ(G)=≤IR(G)=≤??
11 Broadcast Irredundance in Graphs
12 Roman Irredundance in Graphs
13 Fractional Irredundance
14 Open Problems Involving Irredundance
An Introduction to Game Domination in Graphs
1 Introduction
2 Domination-Type Games
2.1 The Domination Game
2.2 The Total Domination Game
2.3 The Independent Domination Game
3 Basic Properties
4 Paths and Cycles
5 Continuation and Total Continuation Principles
6 Upper Bounds and Conjectured Upper Bounds
6.1 Domination Game Bounds
6.2 Total Domination Game Bounds
6.3 Independent Domination Game Bounds
7 Trees
7.1 The Domination Game in Trees
7.2 The Total Domination Game in Trees
7.3 The Independent Domination Game in Trees
8 Computational Complexity
Domination and Spectral Graph Theory
1 Introduction
2 Adjacency Matrix
2.1 Domination and Spectral Radius
2.2 Domination and Energy
2.3 Other Results
3 Laplacian Matrix
3.1 Largest Eigenvalue
3.2 Second Smallest Eigenvalue
3.3 Disjoint Dominating Sets
3.4 Laplacian Distribution
3.5 A Spectral Nordhaus–Gaddum Result
4 Signless Laplacian Matrix
4.1 Bounds for the Index
4.2 Bounds for the Smallest Signless Laplacian Eigenvalue
4.3 k-Domination and Bounds for Q-Eigenvalues
5 Distance Matrices
6 Final Remarks and Open Problems
Varieties of Roman Domination
1 Introduction
2 Weak Roman Domination
2.1 Relationships with γR and γ
2.2 Nordhaus–Gaddum Type Bounds
2.3 Algorithmic and Complexity Results
3 Independent Roman Domination
3.1 Bounds on iR
3.2 Relationships Between iR and γR
3.3 Relationships Between iR and i
3.4 Algorithmic and Complexity Results
4 Roman k-Domination
4.1 Bounds on γkR and Relationships with γk
4.2 Relationships Between γkR and γR
4.3 k-Roman Graphs
4.4 Algorithmic and Complexity Results
5 Roman "4266308 2"5267309 -Domination
5.1 Bounds on γ"4266308 R2"5267309 and Relationships with γ, γ2, γr, and γR
5.2 Nordhaus–Gaddum Type Bounds
5.3 Algorithmic and Complexity Results
6 Double Roman Domination
6.1 Bounds on γdR and Relationships with γ, γR, γ"4266308 R2"5267309 , and γ2
6.2 Nordhaus–Gaddum Type Bounds
6.3 Algorithmic and Complexity Results
7 Total Roman Domination
7.1 Bounds and Relations with Some Domination Parameters
7.2 Algorithmic and Complexity Results
8 Perfect Roman Domination
8.1 Bounds
8.2 Algorithmic and Complexity Results
9 Strong Roman Domination
10 Edge Roman Domination
11 Open Problems
Part II Domination in Selected Graph Families
Domination and Total Domination in Hypergraphs
1 Introduction
2 Domination in Hypergraphs
2.1 Disjoint Dominating Sets
2.2 The Relationship Between Domination and Transversal
2.3 Upper Bounds on the Domination Number
2.4 Edge Size at Least Three
2.5 Edge Size at Least Four
2.6 Edge Size at Least Five
2.7 A Characterization of Hypergraphs Achieving Equality in Theorem 14
2.8 General Setting
2.9 Hypergraphs with Given Domination Number
2.9.1 The Case γ==1
2.9.2 The Case γ==2
2.9.3 The Case γ=≥3
2.10 Nordhaus–Gaddum-Type Results
2.11 Equality of Domination and Transversal Numbers
2.12 The Relationship Between Domination and Matching
3 Total Domination in Hypergraphs
4 Conjectures and Open Problems
Domination in Chessboards
1 Introduction
2 Historical Origins
3 Early Chessboard Domination
4 Queens
5 Bishops
6 Knights
7 Kings
8 Rooks
9 Other Varieties of Chessboard Domination Problems
Domination in Digraphs
1 Introduction
1.1 Basic Terminology and Notation
1.2 Domination and Independence
2 Background and History
2.1 Basis of the Second Kind
2.2 Kernels in Digraphs
3 Bounds on In, Out, and Twin Domination Numbers
3.1 (Out)-Domination
3.2 In-Domination
3.3 Domination and In-Domination
3.4 Twin Domination
3.5 Reverse Domination
4 Domination in Digraph Products
5 Domination in Oriented Graphs
5.1 Oriented Graphs
5.2 Tournaments
6 Total Domination in Digraphs
6.1 Total Domination: Version 1
6.2 Total Domination: Version 2
6.3 Total Domination: Version 3
6.4 Total Domination: Version 4
6.5 Fractional Domination in Digraphs
7 The Oriented Version of the Domination Game
8 Concluding Comments
Part III Algorithms and Complexity
Algorithms and Complexity of Signed, Minus, and MajorityDomination
1 Introduction to Y-Dominating Functions
1.1 Y=="4266308 0, 1"5267309 with f(N[v])=≥1
1.2 Y==[0, 1] with f(N[v])=≥1
1.3 Y=="4266308 −1, 1"5267309 with f(N[v])=≥1
1.4 Y=="4266308 −1, 0, 1"5267309 with f(N[v])=≥1
1.5 Y=="4266308 0, 1"5267309 with f(N(v))=≥1
1.6 Y==[0, 1] with f(N(v))=≥1
1.7 Y=="4266308 −1, 1"5267309 with f(N(v))=≥1
1.8 Y=="4266308 −1, 0, 1"5267309 with f(N(v))=≥1
1.9 Y=="4266308 −1, 1"5267309 with f(N[v])=≥1 for at least half of the vertices v=V
2 Signed Domination
3 Minus Domination
4 Signed and Minus Total Domination
4.1 Y=="4266308 −1, 1"5267309 with f(N(v))=≥1
4.2 Y=="4266308 −1, 0, 1"5267309 with f(N(v))=≥1
5 Majority Domination
6 Efficient Y-Domination
7 Signed Star Domination
8 Open Problems
Algorithms and Complexity of Power Domination in Graphs
1 Power Domination in Graphs
Self-Stabilizing Domination Algorithms
1 Introduction
2 Self-Stabilizing Framework
2.1 Program and Computation
2.2 Distance-k Knowledge
2.3 Anonymous Systems
2.4 Schedulers
2.5 Self-Stabilization
2.6 Running Times
2.7 Rationale for Self-Stabilizing Algorithms
3 Self-Stabilizing Maximal Independent Set Algorithms
3.1 Distributed Model Maximal Independent Set Algorithm
3.2 Synchronous Model Maximal Independent Set Algorithm
3.3 Other Self-Stabilizing Independent Set Algorithms
4 Self-Stabilizing Maximal Matching Algorithms
4.1 Central Model Maximal Matching Algorithm
4.2 Synchronous and Distributed Model Maximal Matching Algorithm
4.3 Other Self-Stabilizing Matching Algorithms
5 Self-Stabilizing Dominating Set Algorithms
5.1 Central Model Minimal Dominating Set Algorithm
5.2 Synchronous Model Minimal Dominating Set Algorithm
5.3 Distributed Model Minimal Dominating Set Algorithm
5.4 Minimal Total Dominating Set Algorithm
6 Other Self-Stabilizing Domination Algorithms
7 Avenues for Further Study
Algorithms and Complexity of Alliances in Graphs
1 Introduction
2 Algorithms and Complexity of Alliances in Graphs
2.1 Self-Stabilizing Alliance Algorithms
3 Open Problems
Citation preview
Developments in Mathematics
Teresa W. Haynes Stephen T. Hedetniemi Michael A. Henning Editors
Structures of Domination in Graphs
Developments in Mathematics Volume 66
Series Editors Krishnaswami Alladi, Department of Mathematics, University of Florida, Gainesville, FL, USA Pham Huu Tiep, Department of Mathematics, Rutgers University, Piscataway, NJ, USA Loring W.
Tu, Department of Mathematics, Tufts University, Medford, MA, USA
Aims and Scope The Developments in Mathematics (DEVM) book series is devoted to publishing well-written monographs within the broad spectrum of pure and applied mathematics. Ideally, each book should
be self-contained and fairly comprehensive in treating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a
potential relationship to other fields are most welcome. High-quality edited volumes conveying current state-of-the-art research will occasionally also be considered for publication. The DEVM series
appeals to a variety of audiences including researchers, postdocs, and advanced graduate students.
More information about this series at http://www.springer.com/series/5834
Teresa W. Haynes • Stephen T. Hedetniemi Michael A. Henning Editors
Structures of Domination in Graphs
Editors Teresa W. Haynes Department of Mathematics and Statistics East Tennessee State University Johnson City, TN, USA
Stephen T. Hedetniemi School of Computing Clemson University Clemson, SC, USA
Department of Mathematics and Applied Mathematics University of Johannesburg Johannesburg, South Africa Michael A. Henning Department of Mathematics and Applied Mathematics University of Johannesburg
Johannesburg, South Africa
ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-030-58891-5 ISBN 978-3-030-58892-2 (eBook) https://doi.org/10.1007/978-3-030-58892-2 © The Editor(s) (if applicable)
and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole
or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way,
and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general
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While concepts related to domination in graphs can be traced back to the mid1800s in connection to various chessboard problems, domination was first defined as a graph theoretical concept in 1958.
Domination in graphs has experienced rapid growth from its introduction, resulting in over 1200 papers published on domination in graphs by the late 1990s. Noting the need for a comprehensive survey
of the literature on domination in graphs, in 1998 Haynes, Hedetniemi, and Slater published the first two books on domination, Fundamentals of Domination in Graphs and Domination in Graphs: Advanced
Topics. We refer to these books as Books I and II. The explosive growth of this field since 1998 has continued, and today more than 4,000 papers have been published on domination in graphs, and the
material in Books I and II is now more than 20 years old. Thus, the authors feel it is time for an update on the developments in domination theory since 1998. We also want to give a comprehensive
treatment of only the major topics in domination. This coverage of domination, including the major results and updates, will be in the form of three books, which we call Books III, IV, and V. Book
III, Domination in Graphs: Core Concepts, is written by the authors and concentrates, as the title suggests, on the three main types of domination in graphs: domination, independent domination, and
total domination. It contains major results on these basic domination numbers, including proofs of selected results that illustrate many of the proof techniques used in domination theory. For the
companion books, Books IV and V, we invited leading researchers in domination to contribute chapters. Book IV concentrates on the most-studied types of domination that are not covered in Book III.
Although well over 70 types of domination have been defined, Book IV focuses on those that have received the most attention in the literature, and contains chapters on paired domination, connected
domination, restrained domination, multiple domination, distance domination, dominating functions, fractional dominating parameters, Roman domination, rainbow domination, locating-domination, eternal
and secure domination, global domination, stratified domination, and power domination. v
The present volume, Book V, is divided into three parts. The first part focuses on several domination-related concepts: broadcast domination, alliances, domatic numbers, dominator colorings,
irredundance in graphs, private neighbor concepts, game domination, varieties of Roman domination, and spectral graph theory. The second part covers domination in (i) hypergraphs, (ii) chessboards,
and (iii) digraphs and tournaments. The third part focuses on the development of algorithms and complexity of (i) signed, minus, and majority domination, (ii) power domination, and (iii) alliances in
graphs. The third part also includes a chapter on self-stabilizing domination algorithms. The authors of the chapters in Book V provide a survey of known results with a sampling of proof techniques
in their areas of expertise. To avoid excessive repetition of definitions and notation, Chapter 1 provides a glossary of commonly used terms. This book is intended as a reference resource for
researchers and is written to reach the following audiences: first, established researchers in the field of domination who want an updated, comprehensive coverage of domination theory; second,
researchers in graph theory who wish to become acquainted with newer topics in domination, along with major developments in the field and some of the proof techniques used; and third, graduate
students with interests in graph theory, who might find the theory and many real-world applications of domination of interest for master’s and doctoral theses topics. We also believe that Book V
provides a good focus for use in a seminar on either domination theory or domination algorithms and complexity, including the new algorithm paradigm of self-stabilizing domination algorithms. We wish
to thank the authors who contributed chapters to this book as well as the reviewers of the chapters. Johnson City, TN, USA Clemson, SC, USA Johannesburg, South Africa
Teresa W. Haynes Stephen T. Hedetniemi Michael A. Henning
Glossary of Common Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Teresa W. Haynes, Stephen T. Hedetniemi, and Michael A. Henning
Part I Related Parameters Broadcast Domination in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael A. Henning, Gary MacGillivray, and Feiran
Alliances and Related Domination Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Teresa W. Haynes and Stephen T. Hedetniemi
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . Wayne Goddard and Michael A. Henning
Dominator and Total Dominator Colorings in Graphs . . . . . . . . . . . . . . . . . . . . . . 101 Michael A. Henning Irredundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 C. M. Mynhardt and A. Roux The Private Neighbor Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 183 Stephen T. Hedetniemi, Alice A. McRae, and Raghuveer Mohan An Introduction to Game Domination in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Michael A. Henning Domination and Spectral Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Carlos Hoppen, David P. Jacobs, and Vilmar Trevisan Varieties
of Roman Domination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 M. Chellali, N. Jafari Rad, S. M. Sheikholeslami, and L. Volkmann
Part II Domination in Selected Graph Families Domination and Total Domination in Hypergraphs. . . . . . . . . . . . . . . . . . . . . . . . . . 311 Michael A. Henning and Anders Yeo Domination in
Chessboards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Jason T. Hedetniemi and Stephen T. Hedetniemi Domination in Digraphs . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Teresa W. Haynes, Stephen T. Hedetniemi, and Michael A. Henning Part III Algorithms and
Complexity Algorithms and Complexity of Signed, Minus, and Majority Domination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 431 Stephen T. Hedetniemi, Alice A. McRae, and Raghuveer Mohan Algorithms and Complexity of Power Domination in Graphs . . . . . . . . . . . . . . 461 Stephen T. Hedetniemi,
Alice A. McRae, and Raghuveer Mohan Self-Stabilizing Domination Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Stephen T. Hedetniemi Algorithms and
Complexity of Alliances in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 521 Stephen T. Hedetniemi
Glossary of Common Terms Teresa W. Haynes, Stephen T. Hedetniemi, and Michael A. Henning
1 Introduction It is difficult to say when the study of domination in graphs began, but for the sake of this glossary let us say that it began in 1962 with the publication of Oystein Ore’s book
Theory of Graphs [15]. In Chapter 13 Dominating Sets, Covering Sets and Independent Sets of [15], we see for the first time the name dominating set, defined as follows: “A subset D of V is a
dominating set for G when every vertex not in D is the endpoint of some edge from a vertex in D.” Ore then defines the domination number, denoted δ(G), of a graph G, as “the smallest number of
vertices in any minimal dominating set.” So, at this point, and for the first time, domination has a “name” and a “number.” Of course, prior to this Claude Berge [3], in his book Theory of Graphs and
its Applications, which was first published in France in 1958 by Dunod, Paris,
T. W. Haynes () Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN, USA Department of Mathematics and Applied Mathematics, University of Johannesburg,
Johannesburg, South Africa e-mail: [email protected] S. T. Hedetniemi School of Computing, Clemson University, Clemson, SC, USA e-mail: [email protected] M. A. Henning Department of Mathematics and
Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. W. Haynes et
al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://doi.org/10.1007/978-3-030-58892-2_1
T. W. Haynes et al.
had previously defined the same concept, but had, in Chapter 4 The Fundamental Numbers of the Theory of Graphs of [3], given it the name “the coefficient of external stability.” Before Berge, Dénes
König, in his 1936 book Theorie der Endlichen und Unendlichen Graphen [13], had defined essentially the same concept, but in VII Kapitel, Basisproblem für gerichtete Graphen, König gave it the name
“punktbasis,” which we would today say is an independent dominating set. And even before König, in the books by Dudeney in 1908 [8] and W. W. Rouse Ball in 1905 [2], one can find the concepts of
domination, independent domination, and total domination discussed in connection with various chessboard problems. And it was Ball who, in turn, credited such people as W. Ahrens in 1910 [1], C. F.
de Jaenisch in 1862 [7], Franz Nauck in 1850 [14], and Max Bezzel in 1848 [4] for their contributions to these types of chessboard problems involving dominating sets of chess pieces. But it was Ore
who gave the name domination and this name took root. Not long thereafter, Cockayne and Hedetniemi [6] gave the notation γ (G) for the domination number of a graph, and this also took root and is the
notation adopted here. Since the subsequent chapters in this book will deal with domination parameters, there will be much overlap in the terminology and notation used. One purpose of this chapter is
to present definitions common to many of the chapters in order to prevent terms being defined repeatedly and to avoid other redundancy. Also, since graph theory terminology and notation sometimes
vary, in this glossary we clarify the terminology that will be adopted in subsequent chapters. We proceed as follows. In Section 2.1, we present basic graph theory definitions. We discuss common
types of graphs in Section 2.2. Some fundamental graph constructions are given in Section 2.3. In Section 3.1 and Section 3.2, we present parameters related to connectivity and distance in graphs,
respectively. The covering, packing, independence, and matching numbers are defined in Section 3.3. Finally in Section 3.4, we define selected domination-type parameters that will occur frequently
throughout the book. For more details and terminology, the reader is referred to the two books Fundamentals of Domination in Graphs [10] and Domination in Graphs, Advanced Topics [11], written and
edited by Haynes, Hedetniemi, and Slater and the book Total Domination in Graphs by Henning and Yeo [12]. An annotated glossary, from which many of the definitions in this chapter are taken, was
produced by Gera, Haynes, Hedetniemi, and Henning in 2018 [9].
2 Basic Terminology In this section, we give basic definitions, common types of graphs, and fundamental graph constructions.
Glossary of Common Terms
2.1 Basic Graph Theory Definitions Before we proceed with our glossary of parameters, we need to define a few basic terms, which are used in the definitions in the following subsections. For an
integer k ≥ 1, we use the standard notation [k] = {1, . . . , k} and [k]0 = [k] ∪{0} = {0, 1, . . . , k}. A (finite, undirected) graph G = (V, E) consists of a finite nonempty set of vertices V = V
(G) together with a set E = E(G) of unordered pairs of distinct vertices called edges. Each edge e = {u, v} in E is denoted with any of e, uv, vu, and {u, v}. We say that a graph G has order n = |V |
and size m = |E|. Two vertices u and v in G are adjacent if they are joined by an edge e, that is, u and v are adjacent if e = uv ∈ E(G). In this case, we say that each of u and v is incident with
the edge e. Further, we say that the edge e joins the vertices u and v. Two edges are adjacent if they have a vertex in common. Two vertices in a graph G are independent if they are not adjacent. A
set of pairwise independent vertices in G is an independent set of G. Similarly, two edges are independent if they are not adjacent. A neighbor of a vertex v in G is a vertex u that is adjacent to v.
The open neighborhood of a vertex v in G is the set of neighbors of v, denoted NG (v). Thus, NG (v) = {u ∈ V : uv ∈ E}. The closed neighborhood of v is the set NG [v] = {v}∪ NG (v). For a set of
vertices S ⊆ V , the open neighborhood of S is the set NG (S) = v ∈ S NG (v) and its closed neighborhood is the set NG [S] = NG (S) ∪ S. If the graph G is clear from the context, we omit it in the
above expressions. For example, we write N(v), N[v], N(S), and N[S] rather than NG (v), NG [v], NG (S), and NG [S], respectively. For a set of vertices S ⊆ V and a vertex v belonging to the set S,
the S-private neighborhood of v is defined by pn[v, S] = {w ∈ V : NG [w] ∩ S = {v}}, while its open S-private neighborhood is defined by pn(v, S) = {w ∈ V : NG (w) ∩ S = {v}}. As remarked in [12],
the sets pn[v, S] S and pn(v, S) S are equivalent and we define the S-external private neighborhood of v to be this set, abbreviated epn[v, S] or epn(v, S). The S-internal private neighborhood of v
is defined by ipn[v, S] = pn[v, S] ∩ S and its open S-internal private neighborhood is defined by ipn(v, S) = pn(v, S) ∩ S. We define an S-external private neighbor of v to be a vertex in epn(v, S)
and an S-internal private neighbor of v to be a vertex in ipn(v, S). The degree dG (v) of a vertex v is the number of neighbors v has in G, that is, dG (v) = |NG (v)|. Again if the graph G is clear
from the context, we use d(v) rather than dG (v). We remark that some books use deg(v) and deg v to denote the degree of v. We leave it to the authors to choose which of these notations to adopt in
their chapters. For a subset of vertices S ⊆ V , the degree of v in S, denoted dS (v), is the number of vertices in S adjacent to the vertex v; that is, dS (v) = |NG (v) ∩ S|. In particular, if S = V
, then dS (v) = dG (v). The degree sequence of a graph G with vertex set V = {v1 , v2 , . . . , vn } is the sequence d1 , d2 , . . . , dn , where di = d(vi ) for i ∈ [n]. Often the degree sequence,
d1 , d2 , . . . , dn is written in non-increasing order, and so d1 ≥ d2 ≥· · · ≥ dn .
T. W. Haynes et al.
An isolated vertex is a vertex of degree 0 in G. A graph is isolate-free if it does not contain an isolated vertex. The minimum degree among the vertices of G is denoted by δ(G) and the maximum
degree by (G). A leaf is a vertex of degree 1, while its neighbor is a support vertex. A strong support vertex is a (support) vertex with at least two leaf neighbors. For subsets X and Y of vertices
of G, we denote the set of edges that join a vertex of X and a vertex of Y in G by [X, Y ]. Two graphs G and H are isomorphic, denoted G∼ =H, if there exists a bijection φ: V (G) → V (H) such that
two vertices u and v are adjacent in G if and only if the two vertices φ(u) and φ(v) are adjacent in H. A parameter of a graph G is a numerical value (usually a non-negative integer) that can be
associated with a graph such that whenever two graphs are isomorphic, they have the same associated parameter value. By a partition of the vertex set V of a graph G, we mean a family π = {V1 , V2 , .
. . , Vk } of nonempty pairwise disjoint sets whose union equals V , that is, for all 1 ≤ i < j ≤ k, Vi ∩ Vj = ∅ and k
Vi = V .
For such a partition π , we will say that π has order k. A walk in a graph G from a vertex u to a vertex v is a finite, alternating sequence of vertices and edges, starting with the vertex u and
ending with the vertex v, in which each edge of the sequence joins the vertex that precedes it in the sequence to the vertex that follows it in the sequence. A trail is a walk containing no repeated
edges, and a path is a walk containing no repeated vertices. We will mainly be concerned with paths. A path between two vertices u and v is called a (u, v)-path or a u-v path or a u, v-path in the
literature. The length of a walk equals the number of edges in the walk. A graph G is connected if for any two vertices u and v in G, there is a (u, v)-path. A cycle is a path in which the first and
last vertices are the same and all other vertices are distinct. A chord of a cycle C is an edge between two nonconsecutive vertices of C. The distance d(u, v) between two vertices u and v, in a
connected graph G, equals the minimum length of a (u, v)-path in G. A shortest, or minimum length, path between two vertices u and v is called a (u, v)-geodesic; a v-geodesic is any shortest path
from v to another vertex; a geodesic is any shortest path in a graph. The diameter of G is the maximum length of a geodesic in G. A graph G = (V , E ) is a subgraph of a graph G = (V, E) if V ⊆ V and
E ⊆ E. A subgraph G of a graph G is called a spanning subgraph of G if V = V . If G = (V, E) and S ⊆ V , then the subgraph of G induced by S is the graph G[S], whose vertex set is S and whose edges
are all the edges in E both of whose vertices are in S. Let F be an arbitrary graph. A graph G is said to be F-free if G does not contain F as an induced subgraph.
Glossary of Common Terms
If G = (V, E) and S ⊆ V , the subgraph obtained from G by deleting all vertices in S and all edges incident with one or two vertices in S is denoted by G − S; that is, G − S = G[V S]. If S = {v}, we
simply denote G −{v} by G − v. The contraction of an edge e = xy in a graph G is the graph obtained from G by deleting the vertices x and y and all edges incident to x or y and adding a new vertex
and edges joining this new vertex to all vertices that were adjacent to x or y in G. A component of a graph is a maximal connected subgraph. An odd (even) component is a component of odd (even)
order. Let oc(G) equal the number of odd components of G. A vertex v ∈ V is a cutvertex if the graph G − v has more components than G. An edge e = uv is a bridge if the graph G − e obtained by
deleting e from G has more components than G.
2.2 Common Types of Graphs A graph of order n = 1 is called a trivial graph, while a graph with at least two vertices is called a nontrivial graph. A graph of size m = 0 is an empty graph, while a
graph with at least one edge is a nonempty graph. Recall that a connected graph is a graph for which there is a path between every pair of its vertices. A k-regular graph is a graph in which every
vertex has degree k for some k ≥ 0. A regular graph is a graph that is k-regular. A 3-regular graph is also called a cubic graph. A graph of order n that is itself a cycle is denoted by Cn , and a
graph that is itself a path is denoted by Pn . Note that a cycle is a 2-regular graph. A forest is an acyclic graph, that is, a graph with no cycles. A tree is a connected acyclic graph.
Equivalently, a tree is a connected graph having size one less than its order. Hence, if T is a tree of order n and size m, then T is connected and m = n − 1. Note that every component of a forest is
a tree, and a forest in which every component is a path is called a linear forest. If G is a vertex disjoint union of k copies of a graph F, we write G = kF. A complete graph is a graph in which
every two vertices are adjacent. A complete graph of order n is denoted by Kn . A triangle is a subgraph isomorphic to K3 or C3 , since K3 ∼ =C3 . A graph G is bipartite if its vertex set can be
partitioned into two independent sets X and Y . The sets X and Y are called the partite sets of G. A complete bipartite graph, denoted Kr,s , is a bipartite graph with partite sets X and Y , where |X
| = r, |Y | = s, and every vertex in X is adjacent to every vertex in Y . The graph Kr,s has order r + s, size rs, δ(Kr,s ) = min{r, s} and (Kr,s ) = max{r, s}. A star is a nontrivial tree with at
most one vertex that is not a leaf. Thus, a star is a complete bipartite graph K1,k for some k ≥ 1. A claw is an induced copy of the graph K1,3 . Thus, a claw-free graph is a K1,3 -free graph. For r,
s ≥ 1, a double star S(r, s) is a tree with exactly two (adjacent) vertices that are not leaves, one of which has r leaf neighbors and the other s leaf neighbors. Equivalently, a double star is a
tree having diameter equal to 3.
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A diamond is an induced copy of the graph K4 − e, which is obtained from a copy of the complete graph of order 4 by deleting any edge e. A graph G can be embedded on a surface S if its vertices can
be placed on S and all of its edges can be drawn between the vertices on S in such a way that no two edges intersect. A graph G is planar if it can be embedded in the plane; a plane graph is a graph
that has been embedded in the plane. A rooted tree T is a tree having a distinguished vertex labeled r, called the root. Let T be a rooted tree with root r. For each vertex v, let P(v) be the unique
(r, v)-path in T. The parent of a vertex v is its neighbor on P(v), while the other neighbors of v are called its children. The set of children of v is denoted by C(v). Note that the root r is the
only vertex of T with no parent. A descendant of v is any vertex u=v such that P(u) contains v, while an ancestor of v is a vertex u=v that belongs to P(v) in T. In particular, every child of v is a
descendant of v, while the parent of v is an ancestor of v. A grandchild of v is a descendant of v at distance 2 from v. We let D(v) denote the set of descendants of v, and we define D[v] = D(v) ∪
{v}. The maximal subtree at v, denoted Tv , is the subtree of T induced by D[v]. The depth of a vertex v in T equals d(r, v), and the height of v, denoted ht(v), is the maximum distance from v to a
descendant of v. Thus, ht(v) = max{d(v, w) : w is a descendant of v}. For classes of graphs not defined here, we refer the reader to the definitive encyclopedia on graph classes, Graph Classes: A
Survey [5] by Brandstädt, Le, and Spinrad.
2.3 Graph Constructions Given a graph G = (V, E), the complement of G is the graph G = (V , E), where uv ∈ E if and only if uv∈E. Thus, the complement G of G is formed by taking the vertex set of G
and joining two vertices by an edge whenever they are not joined in G. By a graph product G ⊗ H on graphs G and H, we mean a graph whose vertex set is the Cartesian product of the vertex sets of G
and H (that is, V (G ⊗ H) = V (G) × V (H)) and whose edge set is determined entirely by the adjacency relations of G and H. Exactly how it is determined depends on what kind of graph product we are
considering. The Cartesian product G 2 H of two graphs G and H is the graph with vertex set V (G) × V (H), where two vertices (u1 , v1 ) and (u2 , v2 ) are adjacent if and only if either u1 = u2 and
v1 v2 ∈ E(H) or v1 = v2 and u1 u2 ∈ E(G). The direct product (also known as the cross product, tensor product, categorical product, and conjunction) G × H of two graphs G and H is the graph with
vertex set V (G) × V (H), where two vertices (u1 , v1 ) and (u2 , v2 ) are adjacent in G × H if and only if u1 u2 ∈ E(G) and v1 v2 ∈ E(H). Given a graph G = (V, E) and an edge uv ∈ E, the subdivision
of edge uv consists of (i) deleting the edge uv from E, (ii) adding a new vertex w to V , and (iii) adding the new edges uw and wv to E. In this case, we say that the edge uv has been
Glossary of Common Terms
subdivided. The subdivision graph S(G) is the graph obtained from G by subdividing every edge of G exactly once. Given a graph G = (V, E), the line graph L(G) = (E, E(L(G))) is the graph whose
vertices correspond 1-to-1 with the edges in E, and two vertices are adjacent in L(G) if and only if the corresponding edges in G have a vertex in common, that is, if and only if the corresponding
two edges are adjacent. The corona G ◦ K1 of a graph G, also denoted cor(G) in the literature, is the graph obtained from G by adding, for each vertex v ∈ V , a new vertex v and the edge vv . The
edge vv is called a pendant edge. The k-corona G ◦ Pk of G is the graph of order (k + 1)|V (G)| obtained from G by attaching a path of length k to each vertex of G so that the resulting paths are
vertex-disjoint. In particular, the 2-corona G ◦ P2 of G is the graph of order 3|V (G)| obtained from G by attaching a path of length 2 to each vertex of G so that the resulting paths are
vertex-disjoint. The generalized corona G ◦ H is the graph obtained by adding a copy of H for each vertex v of G and joining v to every vertex of H. Thus, a generalized corona G ◦ H, where H = K1 ,
is the ordinary corona G ◦ K1 . We note that whether G ◦ Pk is intended to denote a k-corona or a generalized corona will be clear from context or specifically stated by the author.
3 Graph Parameters In this section, we present common graph parameters that may appear in this book.
3.1 Connectivity and Subgraph Numbers In this subsection, we present parameters related to connectivity in graphs. (a) blocks bl(G), number of blocks in G. A block of a graph G is a maximal
nonseparable subgraph of G, that is, a maximal subgraph having no cutvertices. (b) bridges br(G), number of bridges in G. (c) circumference cir(G), maximum length or order of a cycle in G. (d) clique
number ω(G), maximum order of a complete subgraph of G. (e) components c(G), number of maximal connected subgraphs of G. (f) A vertex cut of a connected graph G is a subset S of the vertex set of G
with the property that G − S is disconnected (has more than one component). A vertex cut S is a k-vertex cut if |S| = k. (g) vertex connectivity κ(G), minimum cardinality of a vertex cut of G if G is
not the complete graph, and κ(Kn ) = n − 1. A graph G is k-vertex-connected (or kconnected) if κ(G) ≥ k for some integer k ≥ 0. Thus, κ(G) is the smallest number of vertices whose deletion from G
produces a disconnected graph or the trivial graph K1 . A nontrivial graph has connectivity 0 if and only if it is disconnected.
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(h) An edge cut of a nontrivial connected graph G is a nonempty subset F of the edge set of G with the property that G − F is disconnected (has more than one component). Thus, the deletion of an edge
cut from the connected graph G results in a disconnected graph. An edge cut F is a k-edge cut if |F| = k. (i) edge connectivity λ(G), minimum cardinality of an edge cut of G if G is nontrivial, while
λ(K1 ) = 0. A graph G is k-edge-connected if λ(G) ≥ k for some integer k ≥ 0. Thus, λ(G) is the smallest number of edges whose deletion from G produces a disconnected graph or the trivial graph K1 .
Hence, λ(G) = 0 if and only if G is disconnected or trivial. (j) girth of G, denoted girth(G) or g(G) in the literature, the length of a shortest cycle in G.
3.2 Distance Numbers This subsection contains the definitions of parameters, which are defined in terms of the distances d(u, v) between vertices u and v in a graph. (a) eccentricity ecc(v) = max{d
(v, w) : w ∈ V (G)}. (b) diameter diam(G), maximum distance among all pairs of vertices of G. Equivalently, the diameter of G is the maximum length of a geodesic in G. Thus, the diameter of G is the
maximum eccentricity taken over all vertices of G. Two vertices u and v in G for which d(u, v) = diam(G) are called antipodal or peripheral vertices of G. A diametral path in G is a geodesic whose
length equals the diameter of G. (c) The periphery of a graph G is the subgraph of G induced by its peripheral vertices. (d) radius rad(G) = min{ecc(v) : v ∈ V (G)}. (e) The center of a graph G,
denoted C(G), is the subgraph of G induced by the vertices in G whose eccentricity equals the radius of G. A vertex v ∈ C(G) is called a central vertex of G.
3.3 Covering, Packing, Independence, and Matching Numbers As previously defined, a set S is independent if no two vertices of S are adjacent. A set M of edges is called a matching if no two edges of
M are adjacent, and a matching of maximum cardinality is a maximum matching. Given a matching M, we denote by V [M] the set of vertices in G incident with an edge in M. A matching M of G is a perfect
matching if V [M] = V (G). Thus, if G has a perfect matching M, then G has even order n = 2k for some k ≥ 1 and |M| = k. A vertex and an edge are said to cover each other in a graph G if they are
incident in G. A vertex cover in G is a set of vertices that covers all the edges of G, while
Glossary of Common Terms
an edge cover in G is a set of edges that covers all the vertices of G. Thus, a vertex cover in G is a set of vertices that contains at least one vertex of every edge in G. A subset S of vertices in
G is a packing if the closed neighborhoods of vertices in S are pairwise disjoint. Equivalently, S is a packing in G if for every u, v ∈ S, d(u, v) > 2. Thus, if S is a packing in G, then |NG [v] ∩ S
|≤ 1 for every vertex v ∈ V (G). A packing is also called a 2-packing in the literature. More generally, for k ≥ 2, a set S is a k-packing in G if for u, v ∈ S, d(u, v) > k. A subset S of vertices in
G is an open packing if the open neighborhoods of vertices in S are pairwise disjoint. Thus, if S is an open packing in G, then |NG (v) ∩ S|≤ 1 for every vertex v ∈ V (G). All of the parameters in
this subsection have to do with sets that are independent or cover other sets. These include some of the most basic of all parameters in graph theory. (a) vertex independence numbers i(G) and α(G),
minimum and maximum cardinalities of a maximal independent set in G. The lower vertex independence number, i(G), is also called the independent domination number of G, while the upper vertex
independence number, α(G), is also called the independence number of G. (While the notation i(G) is fairly standard for the independent domination number, we remark that the independence number is
also denoted by β 0 (G) in the literature.) (b) vertex covering numbers β(G) and β + (G), minimum and maximum cardinalities of a minimal vertex cover in G. (We remark that the vertex covering number
is also denoted by τ (G) or by α(G) in the literature.)
(c) edge covering numbers β (G) and β + (G), minimum and maximum cardinalities of a minimal edge cover in G. (d) k-packing numbers ρ k (G), maximum cardinality of a k-packing in G for k ≥ 2. When k =
2, the k-packing number ρ k (G) is called the packing number of G, denoted by ρ(G). Thus, ρ(G) is the maximum cardinality of a packing in G. (e) open packing numbers ρ o (G), maximum cardinality of
an open packing in G.
(f) matching numbers α − (G) and α (G), minimum and maximum cardinalities of a maximal matching in G. The upper matching number, α (G), is also called the matching number of G. Recall that a perfect
matching is a matching in which every vertex is incident with an edge of the matching. Thus, if a graph G of order n has a perfect matching, then α (G) = 12 n. It should be noted that by a well-known
theorem of Gallai, that if G is a graph of order n with no isolated vertices, then α(G) + β(G) = n = α (G) + β (G). (We remark that the matching number is also denoted by β 1 (G) in the literature.)
3.4 Domination Numbers A dominating set in a graph G = (V, E) is a set S of vertices of G such that every vertex in S = V \ S has a neighbor in S. Thus, if S is a dominating set of G, then
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NG [S] = V and every vertex in S is therefore adjacent to at least one vertex in S. For subsets X and Y of vertices of G, if Y ⊆ NG [X], then the set X dominates the set Y in G. In particular, if X
dominates V (G), then X is a dominating set of G. The many variations of dominating sets in a graph G are based on (i) conditions that are placed on the subgraph G[S] induced by a dominating set S,
(ii) conditions that are placed on the vertices in S, or (iii) conditions that are placed on the edges between vertices in S and vertices in S. We mention only the major domination numbers here. A
total dominating set, abbreviated TD-set, in a graph G with no isolated vertex is a set S of vertices of G such that every vertex in V is adjacent to at least one vertex in S. Thus, a subset S ⊆ V is
a TD-set in G if NG (S) = V . Every graph without isolated vertices has a TD-set, since S = V is such a set. If X and Y are subsets of vertices in G, then the set X totally dominates the set Y in G
if Y ⊆ NG (X). In particular, if X totally dominates V (G), then X is a TD-set in G. A paired dominating set, abbreviated PD-set, of G is a set S of vertices of G such that every vertex is adjacent
to some vertex in S and the subgraph G[S] induced by S contains a perfect matching M. Two vertices joined by an edge of M are said to be paired and are also called partners in S. A connected
dominating set, abbreviated CD-set, in a graph G is a dominating set S of vertices of G such that G[S] is connected. (a) domination numbers γ (G) and (G), minimum and maximum cardinalities of a
minimal dominating set in G. The parameters γ (G) and (G) are referred to as the domination number and upper domination number of G, respectively. A dominating set of G of cardinality γ (G) is called
a γ -set of G, while a minimal dominating set of cardinality (G) is called a -set of G. (b) independent domination number i(G), minimum cardinality of a dominating set in G that is also independent.
An independent dominating set of G of cardinality i(G) is called an i-set of G. We note that the maximum order of a minimal independent dominating set equals the vertex independence number α(G). (c)
total domination numbers γ t (G) and t (G), minimum and maximum cardinalities of a minimal total dominating set of G. The parameters γ t (G) and t (G) are referred to as the total domination number
and upper total domination number of G, respectively. A TD-set of G of cardinality γ t (G) is called a γ t -set of G, while a minimal TD-set of cardinality t (G) is called a t -set of G. (d) paired
domination numbers γ pr (G) and pr (G), minimum and maximum cardinalities of a minimal PD-set of G. The parameters γ pr (G) and pr (G) are referred to as the paired domination number and upper paired
domination number of G, respectively. A PD-set of G of cardinality γ pr (G) is called a γ pr set of G, while a minimal PD-set of cardinality pr (G) is called a pr -set of G. (e) connected domination
numbers γ c (G) and c (G), minimum and maximum cardinalities of a minimal CD-set of G. The parameters γ c (G) and c (G) are referred to as the connected domination number and upper connected
domination number of G, respectively. A CD-set of G of cardinality γ c (G) is called a γ c -set of G, while a minimal CD-set of cardinality c (G) is called a c -set of G.
Glossary of Common Terms
References 1. W. Ahrens, Mathematische Unterhaltungen und Spiele (Druck und Verlag von B. G. Teubner, Berlin, 1910), pp. 311–312 2. W.W.R. Ball, Mathematical Recreations and Essays, 4th edn.
(Macmillan, London, 1905) 3. C. Berge, The Theory of Graphs and its Applications (Methuen, London, 1962) 4. M. Bezzel, Schachfreund. Berliner Schachzeitung 3, 363 (1848) 5. A. Brandstädt, V.B. Le,
J.P. Spinrad, Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications (SIAM, Philadelphia, 1999) 6. E.J. Cockayne, S.T. Hedetniemi, Towards a theory of domination in graphs.
Networks 7, 247– 261 (1977) 7. C.F. de Jaenisch, Applications de l’Analyse Mathematique au Jeu des Echecs (Petrograd, Moscow, 1862) 8. H.E. Dudeney, The Canterbury Puzzles and Other Curious Problems
(E. P. Dutton and Company, New York, 1908) 9. R. Gera, T.W. Haynes, S.T. Hedetniemi, M.A. Henning, An annotated glossary of graph theory parameters with conjectures, in Graph Theory, Favorite
Conjectures and Open Problems, Volume 2, ed. by R. Gera, T.W. Haynes, S.T. Hedetniemi (Springer, Berlin, 2018), pp. 177– 281 10. T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination
in Graphs (Marcel Dekker, New York, 1998) 11. T.W. Haynes, S.T. Hedetniemi, P.J. Slater, eds., Domination in Graphs, Advanced Topics (Marcel Dekker, New York, 1998) 12. M.A. Henning, A. Yeo, Total
Domination in Graphs. Springer Monographs in Mathematics (Springer, New York, 2013). ISBN: 978-1-4614-6524-9 (Print) 978-1-4614-6525-6 (eBook) 13. D. König, Theorie der Endlichen und Unendlichen
Graphen. Akademische Verlagsgesellschaft M. B. H., Leipzig, 1936 (Chelsea, New York, 1950) 14. F. Nauck, Briefwechsel mit allen für alle. Illustrirte Zeitung 15, 182 (1850) 15. O. Ore, Theory of
Graphs. American Mathematical Society Colloquium Publications, vol. 38 (American Mathematical Society, Providence, 1962) 16. A.M. Yaglom, I.M. Yaglom, Challenging Mathematical Problems with
Elementary Solutions, Combinatorial Analysis and Probability Theory, vol. I (Holden-Day, San Francisco, 1964)
Part I
Related Parameters
Broadcast Domination in Graphs Michael A. Henning, Gary MacGillivray, and Feiran Yang
AMS Subject Classification: 05C65, 05C69
1 Introduction The concept of broadcast domination was birthed by combining the concepts of distance and domination in graphs and applying them to modeling the problem of positioning broadcasting
radio transmitters, where each transmitter may have a different effective radiated power. To formally define broadcast domination, we
The research of the author Michael A. Henning supported in part by the University of Johannesburg. The research of the author Gary MacGillivray supported by the Natural Sciences and Engineering
Research Council of Canada. M. A. Henning Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: [email protected] G. MacGillivray ()
Department of Mathematics and Statistics, University of Victoria, V8W 2Y2, P.O. Box 1700 STN CSC, Victoria, BC, Canada e-mail: [email protected] F. Yang Department of Mathematics and Applied
Mathematics, University of Johannesburg, Auckland Park, 2006 South Africa Department of Mathematics and Statistics, University of Victoria, V8W 2Y2, P.O. Box 1700 STN CSC, Victoria, BC, Canada
e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics
66, https://doi.org/10.1007/978-3-030-58892-2_2
M. A. Henning et al.
recall the fundamental concepts of distance and domination in graph theory. The distance between two vertices u and v in a graph G, denoted by dG (u, v), or simply d(u, v) if the graph G is clear
from context, is the length of a shortest (u, v)-path in G. The eccentricity eccG (v) of a vertex v in G is the maximum distance of a vertex from v in G. The maximum eccentricity among the vertices
of G is the diameter of G, denoted by diam(G), while the minimum eccentricity among the vertices of G is the radius of G, denoted by rad(G). A central vertex of G is a vertex whose eccentricity
equals rad(G). A tree is either central or bicentral, depending on whether it has one or two central vertices. A diametrical path in G is a shortest path whose length is equal to diam(G). We note
that the two vertices at the end of a diametrical path have maximum eccentricity in G. A dominating set in a graph G is a set S of vertices of G such that every vertex outside S is adjacent to at
least one vertex in S. The domination number of G, denoted by γ (G), is the minimum cardinality of a dominating set in G. A neighbor of a vertex v in G is a vertex adjacent to v. The open
neighborhood of a vertex v in G, denoted by NG (v), is the set of all neighbors of v in G, while the closed neighborhood of v is the set NG [v] = NG (v) ∪{v}. We denote the degree of a vertex v in G
by dG (v) = |NG (v)|. The minimum and maximum degrees among all vertices of G are denoted by δ(G) and (G), respectively. For an integer k ≥ 1, the closed k-neighborhood of v in G, denoted by Nk [v;
G], is the set of all vertices within distance k from v, that is, Nk [v;G] = {u : dG (u, v) ≤ k}. The open k-neighborhood of v, denoted by Nk (v;G), is the set of all vertices different from v and at
distance at most k from v in G, that is, Nk (v;G) = Nk [v;G] {v}. If the graph G is clear from context, we omit the subscript G. For example, we simply write N(v), N[v], Nk (v), and Nk [v] rather
than NG (v), NG [v], Nk (v;G), and Nk [v;G], respectively. When k = 1, the set Nk [v] = N[v] and the set Nk (v) = N(v). In what follows, for an integer k ≥ 1, we use the standard notation [k] = {1, .
. . , k} and [k]0 = [k] ∪{0} = {0, 1, . . . , k}. For a graph G = (V, E) with a vertex set V and an edge set E, a function f : V →{0, 1, 2, . . . , diam(G)} is called a broadcast on G. For each
vertex v in G, the value f (v) is called the strength (or the weight) of the broadcast from v. For each vertex u ∈ V , if there exists a vertex v in G (possibly, u = v) such that f (v) > 0 and d(u,
v) ≤ f (v), then f is called a dominating broadcast on G. A vertex v with f (v) > 0 can be thought of as the site from which the broadcast is transmitted with strength f (v), and such a vertex is
called an f-broadcast vertex or simply a broadcast vertex if the function f is clear from context. The ball of radius r around v is defined as Nr [v] = {u ∈ V : d(u, v) ≤ r}. Thus, the ball Nf (v)
[v] is the set of vertices that hear the broadcast from v. Vertices u with f (u) = 0 do not broadcast. For X ⊆ V , we define f (X) =
f (v).
The cost of the dominating broadcast f is the quantity f (V ), which is the sum of the strengths of the broadcasts over all vertices in G. The minimum cost of a dominating broadcast is the broadcast
domination number of G, denoted by γ b (G).
Broadcast Domination in Graphs 1
(a) f1
(b) f2
(c) f3
Fig. 1 Three broadcast dominating functions of a graph G
An optimal broadcast is a broadcast with cost equal to γ b (G). For the graph G shown in Figure 1, three broadcast dominating functions are illustrated in Figure 1(a), 1(b) and 1(c). The cost of f1 ,
f2 , and f3 is 4, 3, and 3, respectively. For this graph G, we have γ b (G) = 3 and both f2 and f3 are optimal broadcasts. Broadcast domination in graphs was first introduced and studied in 2001 by
Erwin [21, 22]. Erwin observed that if a dominating broadcast f satisfies f (v) ∈{0, 1} for all v ∈ V , then f is the characteristic function of a dominating set and hence has cost at most γ (G).
Furthermore, he observed that a broadcast f : V →{0, 1, . . . , diam(G)} that assigns the strength rad(G) to a central vertex of a connected graph G and the strength 0 to all remaining vertices of G
has cost f (V ) = rad(G). If G = K1 , then γ b (G) = 1 = γ (G), while rad(G) = 0. Hence, we assume that G=K1 and therefore has order at least 2. Thus, the broadcast domination number of a graph G is
at most its domination number and at most its radius. We state this formally as follows. Observation 1. ([21, 22]) If G is a connected graph of order at least 2, then γb (G) ≤ min{γ (G), rad(G)}.
Graphs for which the broadcast domination number is equal to the radius are called radial. In view of Observation 1, we can replace diam(G) by rad(G) in the definition of a dominating broadcast in a
graph G. Erwin [21, 22] showed that if the domination number or the radius of a graph is at most 3, then the broadcast domination number is determined. Proposition 2. ([21, 22]) If G is a connected
graph of order at least 2 and k = min{γ (G), rad(G)} where k ∈ [3], then γ b (G) = k. In 2006, Dunbar, Erwin, Haynes, Hedetniemi, and Hedetniemi [20] defined a key concept called efficient broadcast.
A dominating broadcast is efficient if no vertex hears a broadcast from two different vertices. If f is not an efficient dominating broadcast in a graph G = (V, E), then there exists a vertex v such
that d(v, x) ≤ f (x) and d(v, y) ≤ f (y), where x and y are broadcasting vertices in G. In this case, we can reassign the value 0 to both x and y, assign the value f (w) + f (x) + f (y) to a vertex w
that is within distance f (y) from x and also within distance f (x) from y, and leave the value of all other vertices unchanged under f. The cost of the new broadcast is equal
M. A. Henning et al.
to the cost of the original broadcast. This process can be repeated until an efficient broadcast is found. This yields the following result. Theorem 3. ([20]) Every graph G has an optimal dominating
broadcast that is efficient. As first observed by Herke [31], the broadcast domination number of a connected graph is equal to the minimum broadcast domination number among its spanning trees.
Observation 4. ([31]) If G is a connected graph, then γb (G) = min{γb (T ) | T is a spanning tree of G}.
2 The Dual of Broadcast Domination Graph theoretic minimization (respectively, maximization) problems expressed as linear programming problems have dual maximization (respectively, minimization)
problems. Much of the early work on linear programming duality problems for domination type parameters is done by Slater. A survey of these results can be found in the 1998 survey paper of Slater
[44]. The dual concept of coverings and packings is also well studied in graph theory. For a survey on the combinatorics underlying set packing and set covering problems, we refer the reader to the
2001 monograph by Cornuéjols [17]. In this section, we discuss the dual (in the sense of linear programming) of broadcast domination, namely multipacking. The term multipacking was first introduced
in the Master’s thesis of Teshima [47] in 2012. Here, broadcast domination was considered as a linear programming problem, and the linear programming dual was used to give the definition of a
multipacking. A multipacking is a set S ⊆ V in a graph G = (V, E) such that for every vertex v ∈ V and for every integer r ≥ 1, the ball of radius r around v contains at most r vertices of S, that
is, there are at most r vertices in S at distance at most r from v in G. We note that in this definition of a multipacking, we may restrict our attention to r ∈ [diam(G)]. By our earlier
observations, we can in fact restrict the integer r to belong to the set [rad(G)]. The multipacking number of G is the maximum cardinality of a multipacking of G and is denoted by mp(G). We define
next the multipacking number in terms of the dual of the linear programming problem for broadcast domination. Let G = (V, E) be a graph with V = {v1 , v2 , . . . , vn }. The definition of γ b (G)
leads to a 0–1 integer program, which we now describe. For each vertex vi and integer k ∈ [rad(G)], let xik be an indicator variable giving the truth value of the statement “the strength of the
broadcast f at vertex vi equals k,” that is, xik =
1 if f (vi ) = k 0 otherwise.
Broadcast Domination in Graphs
The formulation of the primal integer program for broadcast domination is given by Broadcast Domination γ b rad(G) n Minimize kxik , k=1 i=1
subject to xik ≥ 1 for all vertices vi and vj , d(vi ,vj )≤k
xik ∈{0, 1} for each vertex vi and integer k ∈ [rad(G)].
Multipacking Number mp(G) n Maximize yj , k=1
subject to yj ≤ k for all vertices vi and vj and integer k ∈ [rad(G)], d(vi ,vj )≤k
yk ∈{0, 1} for each k ∈ [n].
Since broadcast domination and multipacking are dual problems, we have the following observation. Observation 5. For every graph G, we have mp(G) ≤ γ b (G). The graph G shown in Figure 2 satisfies mp
(G) = 3, where the darkened vertices form a multipacking of maximum cardinality in G. As observed earlier, γ b (G) = 3, and so for this example, we have mp(G) = γ b (G). In 2014, Hartnell and
Mynhardt [26] provided the following lower bound on the multipacking number of a graph. Theorem 6. ([26]) If G is a connected graph, then mp(G) ≥ 13 (diam(G) + 1) . Proof.. Let P : v0 , v1 , . . . ,
vd be a diametrical path of G, where d = diam(G). Let Vi = {v ∈ V : d(v, v0 ) = i} for all i ∈ [d], and let M = {vi : i ≡ 0 (mod 3)}. We note that |M| = 13 (d + 1). By our choice of the set M, every
vertex v ∈ V (P) satisfies |Nr [v] ∩ M|≤ r for all integers r ≥ 1. We now consider an arbitrary vertex w ∈ V . We note that w ∈ Vj for some j ∈ [d]0 . Since vj ∈ Vj and M ⊆ V (P), we note that Nr [w]
∩ M ⊆ Nr [vj ] ∩ M, implying that |Nr [w] ∩ M|≤ r for all integers r ≥ 1. Since w ∈ V is arbitrary, this implies that the set M is a multipacking in G. Thus, mp(G) ≥ |M| = 13 (d + 1) = 13 (diam(G) +
1). 2
Fig. 2 A graph G with mp(G) = 3
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As an immediate consequence of Observation 5 and Theorem 6, we have the following lower bound on the broadcast domination number first observed by Erwin [21, 22]. Corollary 7. ([21, 22]) If G is a
connected graph, then γb (G) ≥ 13 (diam(G) + 1) . We note that if G is a path Pn on n ≥ 2 vertices, then γ (G) = 13 n = 13 (diam(G) + 1). Hence, by Observations 1 and 5 and Theorem 6, we have that
the lower bound of Theorem 6 is tight. Furthermore, we have the following result. Proposition 8. For every integer n ≥ 2, mp(Pn ) = γb (Pn ) = γ (Pn ) =
n 3
By Observation 4, for n ≥ 3, we have γ b (Cn ) = γ b (Pn ), and so by Proposition 8, γb (Cn ) = n3 . However, mp(Cn ) = n3 for all n ≥ 3. Thus, for cycles, we have the following result. Proposition
9. ([47]) For every integer n ≥ 3, mp(Cn ) = γ b (Cn ) if and only if n ≡ 0 (mod 3). For n (mod 3) ∈{1, 2}, we have mp(Cn ) = γ b (Cn ) − 1. By Theorem 6, if G is a connected graph, then 3mp(G) ≥diam
(G) + 1. By definition, diam(G) ≥rad(G). By Observation 1, rad(G) ≥ γ b (G). Hence, 3mp(G) ≥diam(G) + 1 ≥rad(G) + 1 ≥ γ b (G) + 1, or, equivalently, γ b (G) ≤ 3mp(G) − 1. Hence, as a consequence of
our earlier results, we have the following upper bound on the broadcast domination number in terms of its multipacking number. Corollary 10. ([26]) If G is a connected graph, then γ b (G) ≤ 3mp(G) −
1. If the multipacking number of a graph G is at least 2, then Hartnell and Mynhardt [26] improved the upper bound in Corollary 10 slightly. Theorem 11. ([26]) If G is a connected graph with mp(G) ≥
2, then γ b (G) ≤ 3mp(G) − 2. As a consequence of Corollary 10, we have the following upper bound on the ratio γ b (G)/mp(G). Corollary 12. ([26]) If G is a connected graph, then
γb (G) < 3. mp(G)
In 2012, Teshima [47] proved that the graph G shown in Figure 3 satisfies γ b (G) = 4 and mp(G) = 2. Assigning a strength 2 to each of the two vertices of degree 2 in G as illustrated in Figure 3,
and a strength of 0 to the remaining vertices of G produces an optimal broadcast of G. An example of a multipacking of maximum cardinality in G is given by the set of two darkened vertices of G
illustrated in Figure 3. This example serves to show the existence of a graph G for which the ratio γ b (G)/mp(G) = 2.
Broadcast Domination in Graphs
Fig. 3 A graph G with γ b (G) = 4 and mp(G) = 2
(a) G1
(b) G2
Fig. 4 Two graphs satisfying γ b (G) = 4 and mp(G) = 2
To date, no graph G has been found satisfying γ b (G)/mp(G) > 2. Beaudou, Brewster, and Foucaud [4] posed the following conjecture. Conjecture 1. ([4]) If G is a connected graph, then γ b (G) ≤ 2mp
(G). There are a few known examples of connected graphs G which achieve the conjectured bound, that is, γ b (G) = 2mp(G). For example, if G is a cycle C4 or C5 , then γ b (G) = 2 and mp(G) = 1, and
so γ b (G) = 2mp(G). As observed earlier, if G is the graph shown in Figure 3, then γ b (G) = 4 and mp(G) = 2, and so γ b (G) = 2mp(G). Two additional examples of graphs G with γ b (G) = 4 and mp(G)
= 2 are the graphs G = G1 and G = G2 shown in Figure 4(a) and 4(b), respectively. Graph G1 is attributed to C. R. Dougherty in [4, Figure 3(c)] as private communication, while graph G2 is given in
[4]. In 2014, Hartnell and Mynhardt [26] gave a construction of a graph Gk such that γ b (Gk ) −mp(Gk ) = k for any given integer k ≥ 1, showing that the difference γ b −mp can be arbitrarily large.
In order to explain their construction, let H be the graph obtained from three vertex-disjoint copies F1 , F2 , and F3 of K2,4 as follows. Let ui be a vertex of degree 2 in Fi for i ∈ [2], and let v1
and v2 be two vertices of degree 2 in F3 . Let H be obtained from the disjoint union of F1 , F2 , and F3 by joining vi to ui for i ∈ [2]. Let x be a vertex of degree 2 in F1 different from u1 , and
let y be a vertex of degree 2 in F2 different from u2 . The graph H is illustrated in Figure 5. Let M be a multipacking of maximum cardinality in H. Each induced subgraph Fi of H contains at that mp
(H) = |M|≤ 3. By Theorem most one vertex of M,implying 6, mp(H ) ≥ 13 (diam(H ) + 1) = 13 (8 + 1) = 3. Consequently, mp(H) = 3. An example of a multipacking of maximum cardinality in H is given by
the set
M. A. Henning et al. 4 x
Fig. 5 A graph H with γ b (H) = 4 and mp(H) = 3
of three darkened vertices of H illustrated in Figure 5. By Observation 5, we have γ b (H) ≥mp(H) = 3. If γ b (H) = 3, then since rad(H) = 4, every optimal broadcast in H must contain at least two
broadcast vertices (of positive strength), one of which therefore has strength 1 and the other strength 2. But then at least one of the vertices x and y hears no broadcast, a contradiction. Hence, γ
b (H) ≥ 4. Since rad(H) = 4 and γ (H) = 6, by Observation 1, we have γ b (H) ≤ 4. Consequently, γ b (H) = 4. We now return to the general construction given by Hartnell and Mynhardt [26]. For k = 1,
let Gk = H. For k ≥ 2, let H1 , H2 , . . . , Hk be k vertex-disjoint copies of the graph H, where xi and yi are the vertices in Hi named x and y in H. Let Gk be constructed from the disjoint union of
the graphs H1 , H2 , . . . , Hk by adding the edges yi xi+1 for i ∈ [k − 1]. As shown in [26], γ b (Gk ) = 4k and mp(Gk ) = 3k. This yields the following result. Theorem 13. ([26]) For every integer
k ≥ 1, there exists a connected graph Gk such that γ b (Gk ) = 4k and mp(Gk ) = 3k. Thus, the following hold in the graph Gk . (a) γ b (Gk ) −mp(Gk ) = k. (b) γb (Gk )/mp(Gk ) = 43 . Recall that in
Theorem 11, if G is a connected graph with mp(G) ≥ 2, then γ b (G) ≤ 3mp(G) − 2. Hartnell and Mynhardt [26] asked whether the factor 3 in this bound can be improved. In 2019, Beaudou, Brewster, and
Foucaud [4] answered their question in the affirmative, resulting in a significant improvement of this upper bound on the broadcast domination number in terms of its multipacking number. Theorem 14.
([4]) If G is a connected graph, then γ b (G) ≤ 2mp(G) + 3. Hartnell and Mynhardt [26] were the first to observe that Conjecture 1 is true when mp(G) ≤ 2. The conjecture is shown in [4] to hold for
all graphs with multipacking number at most 4. Theorem 15. ([4]) If G is a connected graph and mp(G) ≤ 4, then γ b (G) ≤ 2mp(G). By Observation 5, for every graph G, we have mp(G) ≤ γ b (G). In 2017,
Mynhardt and Teshima [47] proved that equality holds here for the class of trees, thereby extending a classic result due to Meir and Moon [37] that the domination number equals the 2-packing number
for trees. Theorem 16. ([47]) For every tree T, we have γ b (T) = mp(T). For any integer programming problem, a natural variation of the problem can be obtained by considering the LP relaxation.
Since broadcast domination
Broadcast Domination in Graphs
Fig. 6 A graph H with mpf (H) = 4 and mp(H) = 3
and multipacking can be regarded as integer programming problems, Brewster, Mynhardt, and Teshima [11] used this idea to study fractional broadcast domination and fractional multipacking. Here, the
broadcast strength of a vertex can be a fraction, and a vertex can be considered to be fractionally in a multipacking. For example, we can assign 1/3 strength to all vertices in C4 , for a total cost
of 4/3, resulting in a fractional dominating broadcast where each vertex hears a total strength at least one. On the other hand, we can pack 1/3 for each vertex in C4 and it will give a multipacking
of size 4/3. We denote the fractional broadcast domination number as γ b,f (G) and the fractional multipacking number as mpf (G). The duality theorem of linear programming yields the result below.
Theorem 17. ([11]) If G is a connected graph, then mp(G) ≤ mpf (G) = γb,f (G) ≤ γb (G). The difference mpf (G) −mp(G) can be arbitrarily large. The graph H shown in Figure 5 has fractional
multipacking number at least 4 since we can pack 1/3 on the degree 2 and 4 vertices with the exception of x and y, which are not packed. The resulting fractional multipacking is shown in Figure 6.
Thus, mpf (H) ≥ 4. As observed earlier, γ b (G) = 4, implying by Theorem 17 that mpf (H) ≤ 4. Consequently, mpf (H) = 4. Using the previous construction Gk given by Hartnell and Mynhardt [26], we can
readily deduce the following result. Theorem 18. For every integer k ≥ 1, there exists a connected graph Gk such that mpf (Gk ) = 4k and mp(Gk ) = 3k.
3 Broadcast Domination in Trees Broadcasts in trees have a special structure, which was exploited in the thesis by Herke [31] in 2007 and in the papers by Herke and Mynhardt [32] in 2009 and
Cockayne, Herke, and Mynhardt [16] in 2011. In order to determine the broadcast domination number of a tree, the above authors introduced the concept of a shadow tree of a tree, defined as follows.
M. A. Henning et al.
Suppose P : v0 v1 v2 . . . vd is a diametrical path in a tree T. The shadow tree is constructed in two steps. First, consider the forest F = T − E(P) obtained from T by deleting all edges on the path
P. For each vertex vk of P, let Qk be a longest path in F emanating from vk . Let Qk start at vk and end at the vertex bk (possibly, vk = bk ). We note, for example, that Q0 is the trivial path
consisting of the vertex v0 = b0 , and Qd is the trivial path consisting of the vertex vd = bk . For example, consider the tree T in Figure 7, where the vertices of the diametrical path P : v0 v1 v2
. . . vd and the vertices b1 , b4 , and b5 are labeled as shown. We note that in this example, vi = bi for i ∈{0, 2, 3, 6, 7}. In the first step of the construction of a shadow tree, we reduce the
tree T to the subtree, Treduced , of T induced by the vertices belonging to the set V (P ) ∪ (
V (Qk )).
For the tree T in Figure 7, the resulting reduced tree Treduced is shown in Figure 8. In the second step of the construction of a shadow tree, if d(vk , bk ) ≥ d(vk , bi ) for some k ∈ [d] and i ∈
[d] {k}, then we remove the vertices in V (Qi ) {vi } from the tree Treduced . We repeat this process until no such indices k and i exist. The resulting tree is a shadow tree of T, denoted by Tshadow
. The shadow of vertex bk is the set of vertices {v ∈ V (Tshadow ) : d(vk , bk ) ≥ d(vk , v)}. In our example, in the b5 b1
Fig. 7 A tree T
b5 b4
Fig. 8 A reduced tree Treduced of T
Broadcast Domination in Graphs
Fig. 9 A shadow tree Tshadow of T
reduced tree Treduced shown in Figure 8, we have d(v5 , b5 ) ≥ d(v5 , b4 ). According to the second step of our construction, we remove the vertex b4 from the tree Treduced . The resulting shadow
tree Tshadow of T is shown in Figure 9. Herke and Mynhardt [32] showed that the broadcast domination number of a tree equals the broadcast domination number of its shadow tree. Theorem 19. ([32]) For
a tree T and its shadow tree ST , γ b (T) = γ b (ST ). By Theorem 19, it therefore suffices for us to consider only the shadow tree ST of a tree T to determine the broadcast domination number of T.
Herke and Mynhardt [32] also introduced the important definitions of split-sets and split-edges. Let T be a tree with diametrical path P. A split-set is a set of edges on P whose removal splits T
into components such that for each component Ti has even positive diameter and Ti ∩ P is a diametrical path of Ti . A split-edge is an edge that is contained in some split-set. For example, in Figure
9, v2 v3 is a split-edge. On the other hand, the edge v3 v4 is not a split edge since its removal creates a subtree with diametrical path from b5 to v7 . In general, all the edges that have two ends
in some shadow (visually in Figure 9, the only edge that is not in some shadow is v2 v3 ) cannot be a split-edge. Herke and Mynhardt [32] showed that the broadcast domination number is a function of
the largest size of a split-set. Theorem 20. ([32]) If M is split-set with maximum cardinality m of a tree T, then γb (T ) =
1 (diam(T ) − m) . 2
Recall that by Observation 1, if G is a connected graph of order at least 2, then γ b (G) ≤rad(G). Graphs G satisfying γ b (G) = rad(G) are called radial graphs, which form an important class of
graphs with respect to broadcast domination. Several characterizations of radial graphs are given in the literature. A characterization of radial trees is given by Herke and Mynhardt [32]. Theorem
21. ([32]) A tree T is radial if and only if it has no non-empty split-set.
M. A. Henning et al.
As an application of shadow graphs, Herke and Mynhardt [32] gave an upper bound for the broadcast domination number of a tree in terms of its order. Theorem 22. ([32]) If T is a tree of order n, then
γb (T ) ≤ n3 . Proof.. We proceed by induction on the order n ≥ 1. The result is immediate for n ∈ [3]. This establishes the base case. Let n ≥ 4 and assume that every tree T of order n < n satisfies
γb (T ) ≤ 13 n . Let T be a tree of order n. Suppose that T has two adjacent vertices u1 and u2 of degree 2. Let vi be the neighbor of ui different from u3−i for i ∈ [2]. Thus, Q: v1 u1 u2 v2 is a
path in T. There exists an edge e ∈ E(Q) such that T − e has two components T1 and T2 of orders n1 and n2 , respectively, where n1 ≡ 0 (mod 3). Thus, n1 = 3t for some t ≥ 1. Applying the inductive
hypothesis to T1 and T2 , we have γ b (T1 ) ≤ t and γb (T2 ) ≤ 13 n2 = 13 (n − 3t). Hence,
1 1 (n − 3t) = n . γb (T ) ≤ γb (T1 ) + γb (T2 ) ≤ t + 3 3
Hence, we may assume that T does not have two adjacent vertices of degree 2, otherwise the desired result follows. Let rad(T) = k, and let P : v1 v2 . . . vd be a diametrical path in a tree T, and so
diam(T) = d − 1. By Observation 1, γ b (T) ≤rad(T) = k. Recall that a tree is central if it contains exactly one central vertex (whose eccentricity equals the radius of the tree), while a tree is
bicentral if it has two central vertices. Suppose, firstly, that T is central. In this case, d = 2k + 1. By our assumption that T has no adjacent vertices of degree 2, the pigeonhole principle shows
that at least 12 (2k −1) = k −1 of the vertices in V (P) {v1 , vd } are adjacent to vertices not on P, implying that n ≥ (2k + 1) + (k − 1) = 3k. Therefore, in this case, γb (T ) ≤ k ≤ 13 n. Suppose,
secondly, that T is bicentral. In this case, d = 2k. Analogously, as before, at least 12 (2k−2) = k−1 of the vertices in V (P) {v1 , vd } are adjacent to vertices not on P, implying that n ≥ 2k + (k
− 1) = 3k − 1. Therefore, in this case, γb (T ) ≤ k ≤ 13 n. This completes the proof by induction. 2 By Observation 4, Theorem 22 gives an upper bound on the broadcast domination number of a graph.
Corollary 23. ([32]) If G is a connected graph of order n, then γb (G) ≤ n3 . This bound of Corollary 23 is tight for paths and cycles.
4 Broadcast Domination in Graph Products In this section, we present selected results on broadcast domination in graph products. By a graph product G ⊗ H on graphs G and H, we mean the graph that has
vertex set the Cartesian product of the vertex sets of G and H, that is,
Broadcast Domination in Graphs
V (G ⊗ H ) = V (G) × V (H ) = {(g, h) | g ∈ V (G) and h ∈ V (H )}, and has an edge set that is determined entirely by the adjacency relations of G and H. Exactly how it is determined depends on what
kind of graph product we are considering. In this section, we consider four such graph products, namely the Cartesian product (2), the direct product (×), the strong product () of graphs, and the
lexicographic product (•). Two vertices (g1 , h1 ) and (g2 , h2 ) in the Cartesian product G2H of graphs G and H are adjacent if either g1 = g2 and h1 h2 is an edge in H or h1 = h2 and g1 g2 is an
edge in G. Two vertices (g1 , h1 ) and (g2 , h2 ) in the direct product graph G × H of graphs G and H are adjacent if g1 g2 ∈ E(G) and h1 h2 ∈ E(H). Two vertices (g1 , h1 ) and (g2 , h2 ) in the
strong direct product G H of G and H are adjacent if and only if u = v and u v ∈ E(H) or u = v and uv ∈ E(G) or uv ∈ E(G) and u v ∈ E(H). Two vertices (g1 , h1 ) and (g2 , h2 ) in the lexicographic
product G•H of G and H are adjacent if and only if either g1 g2 ∈ E(G) or g1 = g2 and h1 h2 ∈ E(H). In 2009, Braser and Spacaman [10] studied broadcast domination in the Cartesian product, the direct
product, and the strong product of graphs and established the following upper bounds on the broadcast domination number in the respective product graphs. Theorem 24. ([10]) For all graphs G and H, γb
(G2H ) ≤
3 (γb (G) + γb (H )). 2
Theorem 25. ([10]) For all graphs G and H, γb (G H ) ≤
3 max{γb (G), γb (H )}. 2
Theorem 26. ([10]) For all graphs G and H, γb (G × H ) ≤
if rad(G) = rad(H ) 3 max{γb (G), γb (H )} 3 min{γb (G), γb (H )} + 1 otherwise.
Dunbar, Erwin, Haynes, Hedetniemi, and Hedetniemi [20] presented results on broadcast domination in m × n grid graphs Gn,m , or equivalently in the Cartesian product Pm 2Pn of paths Pm and Pn . They
showed that it suffices to have one broadcast vertex in the center of the grid, broadcasting with strength rad(Gn,m ) = m2 + n2 . This is illustrated in Figure 10(a) in the case of a 4 × 4 grid.
Theorem 27. ([20]) For integers m ≥ 1 and n ≥ 1, γb (Pm 2Pn ) = rad(Pm 2Pn ) =
m 2
n 2
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(a) γb (P4 P4) = 4
(b) γb (C4 P4) = 4
(c) γb (C4 C4) = 3
Fig. 10 Broadcast domination in P4 2P4 , C4 2P4 , and C4 2C4
We remark that this result was also established by Braser and Spacaman [10]. In 2019, Beaudou and Brewster [3] later extended the result in m × n grids to multipacking. Theorem 28. ([3]) For m ≥ 4
and n ≥ 4, mp(Pn 2Pm ) = γb (Pn 2Pm ), with the exception of P4 2P6 , where mp(P4 2P6 ) = 4 and γb (P4 2P6 ) = 5. In 2015, Koh and Soh [33] determined the broadcast domination number of the Cartesian
product of a cycle and a path. We illustrate this in Figure 10(b) in the case of the Cartesian product C4 2P4 . Theorem 29. ([33]) For integers m ≥ 3 and n ≥ 2, γb (Cm 2Pn ) =
if n = 2 and m is even, 2 m2 + n2 otherwise.
Braser and Spacaman [10] gave results on broadcast domination in the Cartesian products Cm 2Cn of cycles Cm and Cn , also called the torus in the literature. Theorem 30. ([10]) For m ≥ 3 and n ≥ 3,
γb (Cm 2Cn ) =
rad(Cm 2Cn ) − 1 if both m and n are even, otherwise. rad(Cm 2Cn )
Using a differen approach to that used in [10], Soh and Koh [46] determined the broadcast domination number of the torus Cm 2Cn . We illustrate this in Figure 10(c) in the case of the torus C4 2C4 .
We remark that the broadcast domination number of the grid Pm 2Pn and the torus Cm 2Cn is the same, except when both m and n are even. Theorem 31. ([46]) For m ≥ 3 and n ≥ 3, we have γb (Cm 2Cn ) =
m+n 2 − 1.
Broadcast Domination in Graphs
In 2014, Soh and Koh [45] studied broadcast domination in the strong product, the direct product, and the lexicographic product of two paths. Theorem 32. ([45]) For integers m ≥ n ≥ 1,
m 1 γb (Pm Pn ) = m− , 2 max{p, 3} where p = 2
n−1 2
+ 1.
The broadcast domination number of the strong product P4 P4 equals 2 and is illustrated in Figure 11(a). Theorem 33. ([45]) For integers m ≥ n ≥ 1,
γb (Pm × Pn ) =
⎧ m ⎪ ⎪ ⎪ ⎪ ⎨ n · m+1 n+1
if n = 1, if n ≥ 2, both m and n are odd with
⎪ ⎪
⎪ ⎪ ⎩2 · 1 m − 2
m n+1
m+1 n+1
an integer, otherwise.
The broadcast domination number of the direct product P4 × P4 equals 2 and is illustrated in Figure 11(b). Theorem 34. ([45]) The broadcast domination number of the lexicographic product of Pm and Pn
has ⎧ m n ⎪ ⎨ max{ 3 , 3 } if m = 1 or n ∈ {1, 2, 3}, γb (Pm • Pn ) = ⎪ ⎩ max{ 2m , 2} if m ≥ 2 and n ≥ 4. 5 The broadcast domination number of the lexicographic product P4 •P4 equals 2 and is
illustrated in Figure 11(c), where the darkened vertex has a strength of 2.
(a) γb (P4 P4) = 2
(b) γb (P4 × P4) = 4
Fig. 11 Broadcast domination in P4 P4 , P4 × P4 , and P4 •P4
(c) γb (P4 • P4) = 2
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5 Irredundant Broadcasts The upper domination number (G) of a graph G is the maximum cardinality of a minimal dominating set of G. The concept of the upper broadcast domination number was first
defined by Erwin [21] and also studied, for example, in [1, 20, 38]. A dominating broadcast is minimal if reducing the strength of any vertex that is broadcasting results in a broadcast that is no
longer dominating. Thus, a dominating broadcast f on a graph G is a minimal dominating broadcast if no broadcast g < f is dominating. The maximum cost of a minimal dominating broadcast is the upper
broadcast domination number of G, denoted as b (G). Thus, for a graph G = (V, E), b (G) = max{f (V ) | f is a minimal dominating broadcast of G}. Ahmadi, Fricke, Schroder, Hedetniemi, and Laskar [1]
point out that if you broadcast with strength diam(G) from a vertex v with ecc(v) = diam(G), then you have a minimal dominating broadcast. Hence, we have the following lower bound on the upper
broadcast domination number. Theorem 35. ([1]) Every graph G satisfies diam(G) ≤ Γ b (G). A set X ⊆ V (G) is irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex
in X. We note that a maximal irredundant set is not necessarily a dominating set. The concept of an irredundance broadcast was first introduced in 2015 by Ahmadi et al. [1]. Essential for their
definition of an irredundant broadcast, they use an important property given by Erwin [21] that makes a dominating broadcast a minimal dominating broadcast. Thereafter, they define an irredundant
broadcast f to be maximal irredundant if no broadcast g > f is irredundant and observe that any minimal dominating broadcast is maximal irredundant. The lower and upper broadcast irredundance numbers
of G are given by irb (G) = min{f (V ) | f is a maximal irredundant broadcast of G} and IRb (G) = max{f (V ) | f is an irredundant broadcast of G}, respectively. We note that if the strength of any
vertex in an optimal dominating broadcast is reduced, this will decrease the number of vertices that hears the broadcast (as all vertices in G can no longer hear the broadcast), so every optimal
dominating broadcast is maximal irredundant. As a result, we have that irb (G) ≤ γ b (G). More generally, by the properties established by Erwin [21], and by the above definitions, we have the
following inequality chain.
Broadcast Domination in Graphs
Theorem 36. ([1]) Every graph G satisfies the inequality chain given by irb (G) ≤ γb (G) ≤ γ (G) ≤ (G) ≤ b (G) ≤ IRb (G). Conditions under which an irredundant broadcast is maximal irredundant were
determined by Mynhardt and Roux [38]. Their main result is that the ratio γ b /irb is bounded. Theorem 37. ([38]) For every graph G, we have γb (G) ≤ 54 irb (G). However, Mynhardt and Roux [38] give
constructions illustrating that the ratio IRb / b is unbounded for general graphs. In 2017, Bouchemakh and Fergani [8] continued the study of the upper broadcast number and established the following
upper bound for the upper broadcast domination number. Theorem 38. ([8]) If G is a connected graph of order n, then Γ b (G) ≤ n − δ(G) and the bound is sharp on paths, stars, and complete graphs.
Bouchemakh and Fergani also studied the upper broadcast number on grids. Recall that the Cartesian product of graphs G and H is denoted by G2H . Theorem 39. ([8]) For integers m ≥ n ≥ 2, we have b
(Pm 2Pn ) = m(n − 1).
6 Independent Domination Broadcasts In 2006, Dunbar, Erwin, Haynes, Hedetniemi, and Hedetniemi [20] defined the concept of an independent broadcast. A broadcast f on a connected graph G is an
independent broadcast if every pair of vertices u and v for which f (u) > 0 and f (v) > 0, we have d(u, v) > max{f (u), f (v)}. Equivalently, an independent broadcast on G is a broadcast f of G such
that for every vertex x of G, f (v) > 0 implies that f (u) = 0 for every vertex u of G within distance at most f (v) from v. Thus, if f is an independent broadcast, then no broadcast vertex can hear
a broadcast from any other broadcast vertex. As observed in [20], an independent broadcast need not be a dominating broadcast. The broadcast independence number α b (G) of G is the maximum cost of an
independent broadcast of G. The lower broadcast independence number ib (G) of G equals the minimum cost of a maximal independent broadcast of G. Thus, ib (G) = min{f (V ) | f is a maximal independent
broadcast of G}, and αb (G) = max{f (V ) | f is an independent broadcast of G}. An independent broadcast of G of cardinality α b (G) is called an α b -broadcast of G. In 2014, Bouchmakh and Zemir [9]
studied broadcast independence on grids
M. A. Henning et al. 3
(a) αb (G5,5) = 15
(b) αb (G5,6) = 16
Fig. 12 α b -Broadcasts of G5,5 and G5,6
and gave bounds for the broadcast independence number on 2 × n, 3 × n, and 4 × n grids. Recall that the m × n grid graph is denoted by Gn,m , and so Gn,m = Pm 2Pn . Theorem 40. ([9]) The following
hold. (a) (b) (c) (d)
For n ≥ 2, α b (G2,n ) = 2(n − 1). For n ≥ 3, α b (G3,n ) = 2n. For n ≥ 4, α b (G4,n ) = 2(n + 1). α b (G5,5 ) = 15 and α b (G5,6 ) = 16.
An α b -broadcast of G5,5 and G5,6 is illustrated in Figure 12(a) and 12(b), respectively, where the broadcast vertices are darkened and their strengths are given. Theorem 41. ([9]) For integers n ≥
m ≥ 5 where (m, n)∈{(5, 5), (5, 6)}, αb (Gm,n ) =
mn 2
Later, Ahmane, Bouchmakh, and Sopena studied the broadcast independence number for caterpillars [2]. In 2019, Bessy and Rautenbach [5] studied the relationship between the broadcast independence
number and the independence number, where the independence number α(G) of G is the maximum cardinality of an independent set in G. Their main result is that the broadcast independence number and the
independence number are within a constant factor from each other. Theorem 42. ([5]) For every connected graph G, we have α(G) ≤ α b (G) ≤ 4α(G). Bessy and Rautenbach [5] also characterize all
extremal graphs satisfying equality in the bound of Theorem 42. Imposing a girth condition on the graph, Bessy and Rautenbach [6] improve the upper bound in Theorem 42. Theorem 43. ([5]) Let G be a
connected graph of girth g and minimum degree δ. If g ≥ 6 and δ ≥ 3 or g ≥ 4 and δ ≥ 5, then α b (G) ≤ 2α(G).
Broadcast Domination in Graphs
Theorem 44. ([5]) For every positive integer k, there is a connected graph G of girth at least k and minimum degree at least k such that 1 αb (G) ≤ 2 1 − α(G). k The results of Theorems 43 and 44
imply that lower bounds on the girth and the minimum degree of a connected graph G can lower the fraction α b (G)/α(G) from 4 to below 2, but not any further.
7 k-Broadcast Domination Another variation of broadcast domination is the k-broadcast domination, which was first studied in 2018 by Henning, MacGillivray, and Yang [29] (also see [49]). In a
k-broadcast, instead of requiring every vertex to hear at least one broadcast, this time we restrict every vertex to hear at least k broadcasts. The following motivation is given in [29]. Consider a
large city partitioned into many neighborhoods, each of which has a radio tower to transmit emergency information. It is desirable to have some redundancy in the system so that everyone can hear a
broadcast even if some of the towers are not functioning. The goal is to design a broadcast protocol with the property that every neighborhood receives some number k of broadcasts, and which does not
require every tower to be used. Formally, let f : V →{0, 1, . . . , diam(G)} be a broadcast on a graph G = (V, E), and let Vf+ be the set of all vertices in G with positive strength under f, that is,
Vf+ = {v ∈ V | f (v) > 0}. If, for each vertex u ∈ V , there exist k different vertices v1 , v2 , . . . , vk ∈ Vf+ such that for i ∈ [k] we have d(u, vi ) ≤ f (vi ), then f is called a dominating
k-broadcast. The cost of a dominating k-broadcast is the quantity f (V ). The minimum cost of a dominating k-broadcast is the k-broadcast domination number of G and is denoted by γbk (G). When k = 1,
we note that the 1-broadcast domination number is the broadcast domination number, γ b (G), of G. The integer programming formulation of k-broadcast domination is a bit more complicated than for
1-broadcast domination. Here, we need to introduce the adjacency matrix and the ball matrix. Adopting the notation in [29], the -adjacency matrix A is the n × n incidence matrix, where the rows
correspond to vertices and the columns correspond to closed -neighborhoods. The (i, j)th entry of A is 1 if the vertex vi is contained in the closed -neighborhood of vj , otherwise it is 0. Clearly,
the closed neighborhood adjacency matrix is just A1 . We define the ball matrix to be A∗ = A A2 . . . Ar ,
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where r is the radius of the graph. The columns of A∗ are the characteristic vectors of closed -neighborhoods of vertices. For i ∈ [n] and ∈ [r], let xi be a Boolean variable representing the truth
value of the statement “there is a broadcast from vertex vi of strength .” Let x = [x11 , x21 , . . . , xn1 , x12 , x22 , . . . , xn2 , . . . , x1r , x2r , . . . , xnr ]T . We have A∗ x ≥ [k, k, . .
. , k]T if and only if every vertex hears at least k broadcasts. We must also add constraints to guarantee that each vertex broadcasts at most once. For each vertex vi , the constraint xi1 + xi2 + ·
· · + xir ≤ 1, or equivalently − xi1 − xi2 −· · · − xir ≥−1, guarantees that vi broadcasts with at most one strength, and therefore at most once. The 0–1 integer program Bk (G) for finding γbk (G) is
described below: Minimize
rad(G) n
xi ,
=1 i=1
subject to A∗ x ≥ [k, k, . . . , k]T , xi1 + xi2 + · · · + xir ≤ 1 for each i ∈ [n], xi ∈{0, 1} for vi ∈ V and ∈ [r]. Recall that the definition of multipacking arose from the dual of the linear
programming relaxation of the broadcast domination integer program, and then consider it as an integer program. Analogously, we shall use the dual of the linear programming relaxation of Bk (G) to
formulate the definition of k-multipacking. Considering the dual of the linear programming relaxation of Bk (G) as an integer program leads to the following integer program, Mk (G): Maximize
n j =1
subject to
k · cj − rj ,
− ri ≤ for each ∈ [r] and each i ∈ [n],
d(vi ,vj )≤
cj , rj ∈ N for each vertex vj . There are two values, cj and rj , associated with each vertex vj . The variable cj indicates how many times a vertex vj is chosen in the k-multipacking (vertices can
be chosen more than once). The variable rj is the relaxation value on vertex vj which allows the multipacking restriction on that specific vertex to be relaxed by rj . Thus, a k-multipacking is a
pair (c, r), where c : V → N and r : V → N such that for every vertex vi ∈ V , we have d(vi ,vj )≤
c(vj ) ≤ + r(vi )
Broadcast Domination in Graphs
for each ∈ [r]. The value of a k-multipacking (c, r) is the quantity
(k · c(v) − r(v)).
The largest value of a k-multipacking of G is the k-multipacking number of G, denoted by mpk (G). As remarked in [29], multipacking is the same as 1multipacking. To see this, suppose we take a
maximum multipacking of a graph G and consider whether it is an optimal solution to M1 (G). If we want to increase the value of some variable cj , we must also increase some ri from 0 by at least the
same amount, otherwise a constraint is violated at some vertex. Conversely, given a maximum 1-multipacking, if we want to decrease some positive ri , then some cj must be decreased by at least the
same amount, otherwise a constraint will be violated. As a result, mp1 (G) = mp(G). Thus, the k-multipacking problem is a generalization of multipacking. The fractional k-broadcast domination number
of G, γbk ,f (G), is the optimum solution of the linear programming relaxation of Bk (G), and the fractional kmultipacking number of G, mpk,f (G), is the optimum solution of the linear programming
relaxation of Mk (G). By the strong duality theorem of linear programming, we have the following result. Theorem 45. ([29]) If G is a connected graph, then mpk (G) ≤ mpk,f (G) = γbk ,f (G) ≤ γbk (G).
As observed in [29], by the definition of k-multipacking, we can always let ci be 1 for all the vertices included in a maximum multipacking and ri be 0 for all vertices, yielding k ·mp(G) as a lower
bound on the k-multipacking number. Observation 46. ([29]) For a graph G, mpk (G) ≥ k ·mp(G). As an example of a dominating 2-broadcast and a 2-multipacking, consider the tree shown in Figure 13(a).
A dominating 2-broadcast is obtained by broadcasting with strength 1 from v2 and strength 2 from v1 , as illustrated in Figure 13(b). This gives γb2 (G) ≤ 3. On the other hand, we can assign the
function values of (c(vi ), r(vi )) as (1, 0), (0, 1), (1, 0), and (0, 0) at the vertices v1 , v2 , v3 , and v4 , respectively. This gives a 2-multipacking with value 3, thus mp2 (G) ≥ 3. By the
duality theorem of linear programming, we have 3 ≤ mp2 (G) ≤ γb2 (G) ≤ 3, implying that γb2 (G) = mp2 (G) = 3. The 2-broadcast domination number can differ from twice the broadcast domination number
by an arbitrarily large constant difference as shown in [29]. Theorem 47. ([29]) For any integer t, there exists a connected graph G with γb2 (G) ≤ 2 · γb (G) − t and there exists a graph H with γb2
(H ) ≥ 2 · γb (H ) − t. The following upper bound on the 2-broadcast domination number of a tree in terms of its order is given in [29].
M. A. Henning et al. v4
v2 (a) T
(0, 0)
(b) γb2(T) = 3
(1, 0)
(0, 1)
(1, 0)
(c) γb2(T) = 3
Fig. 13 A tree T with γb2 (T ) = mp2 (T ) = 3
Theorem 48. ([29]) If T is a tree of order n ≥ 2, then γb2 (T ) ≤ 4n 5 . The proof of Theorem 48 is by induction on the order n ≥ 2. The tree is carefully split into two subtrees, and various cases
are considered in terms of different properties of the resulting subtrees. Since a dominating 2-broadcast in a spanning tree of a connected graph G is also a dominating 2-broadcast in G, as an
immediate consequence of Theorem 48, we have the following result. Corollary 49. ([29]) If G is a connected graph of order n ≥ 2, then γb2 (G) ≤ 4n 5 . The authors in [29] believe that the bound in
Theorem 48 could possibly be improved using a more detailed analysis and pose the following conjecture. Conjecture 2. ([29]) If T is a tree of order n ≥ 2, then γb2 (T ) ≤ 13 (2n + 4). If T is a path
Pn where n ≡ 1 (mod 3), then γb2 (T ) = (2n + 4)/3. Thus, if Conjecture 2 is true, then the bound is tight.
8 Limited Broadcast Domination In this section, we consider a limited version of the broadcast function. For a graph G = (V, E) and an integer k ≥ 1, a function f : V →{0, 1, . . . , k} is called a
k-limited dominating broadcast, abbreviated kLD-broadcast, in G if for each vertex u ∈ V , there exists a vertex v in G such that f (v) > 0 and d(u, v) ≤ f (v). The minimum cost of a kLD-broadcast is
the k-limited broadcast domination number of G, denoted by γ b,k (G). A kLD-broadcast of cost γ b,k (G) is called a γ b,k -broadcast of G; that is, a γ b,k -broadcast of G is a minimum kLD-broadcast
of G. The 1-limited broadcast domination number, γ b,1 (G), of G is precisely the domination number, γ (G). For k ≥ 1, the function that assigns the weight 1 to the vertices of a minimum dominating
set of G (of cardinality γ (G)) and the weight 0 to the remaining vertices of G is a kLD-broadcast of cost γ (G), implying that γ b,k (G) ≤ γ (G). By definition, every kLD-broadcast is a dominating
broadcast, and so γ b (G) ≤ γ b,k (G). We state this formally as follows. Observation 50. For an integer k ≥ 1 and for every graph G,
Broadcast Domination in Graphs
(a) γb (T) = 3
(b) γb,2 (T) = 5
(c) γ(T) = 6
Fig. 14 A tree T with γ (T) = γ b,2 (T) = γ (T) = 3
Fig. 15 The tree T9 with gluing vertex v
γb (G) ≤ γb,k (G) ≤ γ (G)
γb,k (G) ≤ γb,k+1 (G).
The inequalities in the inequality chain of Observation 50 can be strict. For example, consider the tree T shown in Figure 14. In the case when k = 2, we have γ b (T) = γ b,3 (T) = 3, γ b,2 (T) = 5,
and γ (T) = γ b,1 (T) = 6. An optimal broadcast with cost equal to γ b (G) = 3 is shown in Figure 14(a). A 2-limited dominating broadcast of cost γ b,2 (T) = 5 is shown in Figure 14(b), while a
dominating set of T of cardinality γ (T) = 6 is shown in Figure 14(c). We note that the optimal broadcast in Figure 14(b) is not efficient. More generally, the result of Theorem 3, that every graph
has an optimal dominating broadcast that is efficient, does not apply to k-limited broadcasting for k ≥ 2. This above example illustrates that, even for a tree, an optimal 2-limited dominating
broadcast may not be efficient. However, every graph has efficient k-limited broadcasts for some values of k, e.g., k = rad(G) and k = diam(G). This raises the question: what is the smallest value of
k for which G has an efficient k-limited broadcast? Limited broadcast domination in graphs was first studied in 2013 by Rad and Khosvravi [42], where some fundamental properties were introduced. The
first major results for limited broadcast domination were given in 2018 by Cáceres, Hernando, Mora, Pelayo, and Puertas [13, 14]. Theorem 51. ([13, 14]) For k ≥ 2, if G is a connected graph, then
γb,k (G) = min{γb,k (T ) | T is a spanning tree of G}. In order to establish a tight upper bound for the 2-limited broadcast domination number of a general graph, Cáceres et al. [13] construct a
family of trees F as follows. Let T9 be the tree shown in Figure 15. We call the central vertex of degree 2 (that is not a support vertex) the gluing vertex of T9 . A tree T belongs to the family T
if T is obtained from k ≥ 1 vertex-disjoint copies of the tree T9 shown in Figure 15 by adding k − 1 edges between the gluing vertices.
M. A. Henning et al. 1
Fig. 16 A tree T in the family T
An example of a tree T in the family T is shown in Figure 16. Furthermore, an example of a 2-limited dominating broadcast of cost γ b,2 (T) = 12 for the tree T is illustrated in Figure 16. We are now
in a position to state the result of Cáceres et al. [13]. Theorem 52. ([13]) If T is a tree of order n, then γb,2 (G) ≤ 49 n , with equality if and only if T ∈ F ∪ {P1 , P2 , P4 }. As a consequence
of Theorems 51 and 52, we have the following upper bound on the 2-limited broadcast domination number of a graph. Theorem 53. ([13]) If G is a connected graph of order n, then γb,2 (G) ≤ 49 n .
Cáceres et al. [13] showed that the upper bound in Theorem 53 can be improved if the graph G contains a dominating path, that is, a path P such that every vertex not on P has a neighbor on P. In this
case, we note that the graph G has a caterpillar as a spanning tree, where a caterpillar is a tree in which the removal of all leaves yields a path. Theorem 54. ([13]) If G is a graph of order n that
contains a dominating path, then γb,2 (G) ≤ 25 n . In a subsequent paper, Cáceres et al. [14] generalized the upper bound on the 2limited broadcast domination number of a tree given in Theorem 52 to
the k-limited broadcast domination number for all k ≥ 2. Theorem 55. ([14]) If T is a tree of order n and k ≥ 2 is an integer such that k < rad(T), then γb,k (T ) ≤
k+2 ·n , 3(k + 1)
and this bound is tight. As a consequence of Theorems 51 and 55, we obtain the following bound on the k-limited broadcast domination number of a graph. Theorem 56. ([13]) For k ≥ 2, if G is a
connected graph of order n, then γb,k (G) ≤ k+2 3(k+1) · n .
Broadcast Domination in Graphs
Recently, Henning, MacGillivray, and Yang [30] studied 2-limited broadcast domination in subcubic graphs. A subcubic graph is a graph whose maximum degree is at most 3, while a cubic graph (also
called a 3-regular graph in the literature) is a graph in which every vertex has degree 3. The following conjecture is posed in [30]. Conjecture 3. ([30]) If G is a cubic graph of order n, then γb,2
(G) ≤ 13 n. Conjecture 3 is shown in [30] to be true if the cubic graph G is (C4 , C6 )-free, where a (C4 , C6 )-free graph is a graph that does not contain a 4-cycle or a 6-cycle as an induced
subgraph. Theorem 57. ([30]) If G is a cubic graph of order n that is (C4 , C6 )-free, then γb,2 (G) ≤ 13 n.
9 Algorithmic and Complexity Results We consider in this section the problem of finding the broadcast domination number of an arbitrary graph. We state the decision problem formally as follows:
BROADCAST DOMINATION Input: A graph G, and an integer k ≥ 1. Question: Is γ b (G) ≤ k?
The most interesting feature about dominating broadcasts is that the broadcast domination number γ b (G) can be computed in polynomial time for any graph, as shown by Heggernes and Lokshtanov [27] in
2006. This is quite counter-intuitive since computing the domination number of a graph is in general NP-hard. Theorem 58. ([27]) The broadcast domination number of a graph of order n can be computed
in O(n6 ) time, implying that Broadcast Domination is solvable in polynomial time. To find an optimal dominating broadcast, Heggernes and Lokshtanov first considered a ball graph of the original
graph. The ball graph of a dominating broadcast is a graph whose vertices are the broadcast neighborhoods of the original graph where two vertices of the ball graph are adjacent if the two broadcast
neighborhoods contain a pair of adjacent vertices in the original graph. By Theorem 3, every graph G has an optimal efficient dominating broadcast, implying that there exists an optimal efficient
broadcast whose ball is either a path or a cycle. The idea is to assume that for each vertex v ∈ V , the broadcast neighborhood of v is an end point
M. A. Henning et al.
of a ball graph which is a path. This finds all possible optimal dominating broadcasts which are paths. Next, the case when the ball graph is a cycle is considered. A broadcast neighborhood from the
original graph is first removed, giving a path ball graph for the remaining subgraph. The running time of this process when the ball graph is a path is O(n4 ), and when the ball graph is a cycle, the
running time is O(n6 ). Heggernes and Sæther [28] later conjectured that BROADCAST DOMINATION can be solved in O(n5 ) time in general. In the literature, several algorithms have been given to find
the broadcast number and the multipacking number for trees. In 2009, Dabney, Dean, and Hedetniemi [19] (also see [18]) gave a linear algorithm to find an optimal dominating broadcast for trees. The
linearity of the algorithm is based on a complex data structure. Later, Brewster, MacGillivray, and Yang [12] (see also [49]) gave a simpler greedy algorithm which makes use of shadow trees,
split-edges, and split-sets. Theorem 59. ([18, 19]) The broadcast domination number of a tree of order n can be computed in O(n) time, implying that Broadcast Domination is solvable in linear time
for trees. We consider next the following decision problem:
MULTIPACKING Input: A graph G, and an integer k ≥ 1. Question: Is mp(G) ≥ k?
In 2014, Mynhardt and Teshima [39] (also in [47]) showed that the multipacking number of a tree of order n can be computed in linear time. Brewster, MacGillivray, and Yang [12] (also see [49]) gave a
simpler algorithm for finding an optimal multipacking on trees. Theorem 60. ([39, 47]) The multipacking number of a tree of order n can be computed in O(n) time, implying that Multipacking is
solvable in linear time for trees. A block graph is a graph in which every block is a complete graph. In particular, every tree is a block graph. Heggernes and Sæther [28] showed that BROADCAST
DOMINATION can be solved efficiently on block graphs. As block graphs form a superclass of trees, their result extends the result given by Theorem 59. Due to the tree-like structure of a block graph,
their algorithm is efficient and elegant. Theorem 61. ([28]) The broadcast domination number of a block graph of order n can be computed in O(n + m) time.
Broadcast Domination in Graphs
A graph G is an interval graph if there exists a one-to-one correspondence between its vertex set and a family of closed intervals in the real line, such that two vertices are adjacent if and only if
their corresponding intervals intersect. Blair, Heggernes, Horton, and Manne [7] studied algorithmic and complexity results for broadcast domination in interval graphs, series-parallel graphs, and
trees. Employing a dynamic programming method, they found optimal broadcasts for interval graphs and for series-parallel graphs. Theorem 62. ([7]) The broadcast domination number of an interval graph
of order n can be computed in O(n3 ) time. Theorem 63. ([7]) The broadcast domination number of a series-parallel graph of order n and radius r can be computed in O(nr4 ) time. The complexity result
for broadcast domination in interval graphs given in Theorem 62 was subsequently improved by Chang and Peng [15]. Although their method is similar to that employed in [7], they use a better data
structure resulting to improve the running time. Theorem 64. ([15]) The broadcast domination number of an interval graph of order n and size m can be computed in O(n + m) time. A chordal graph is a
graph in which every cycle of length at least 4 has a chord, where a chord of a cycle is an edge that is not part of the cycle but joins two vertices of the cycle. Equivalently, a chordal graph is a
graph in which every induced cycle contains exactly three vertices. Chordal graphs can also be defined in terms of a perfect elimination ordering. A perfect elimination ordering in a graph G is an
ordering of the vertices of the graph such that, for each vertex v, the set of neighbors of v that occur after v in the order form a clique. A graph is chordal if and only if it has a perfect
elimination ordering. Heggernes and Sæther [28] gave the following complexity result to solve BROADCAST DOMINATION in chordal graphs. Theorem 65. ([28]) The broadcast domination number of a chordal
graph of order n can be computed in O(n4 ) time. Furthermore, Heggernes and Sæther [28] conjectured that BROADCAST DOMIcan be solved in O(n2 ) time for chordal graphs of order n. Although partial
results have been obtained, their conjecture has yet to be fully settled. However, pleasing progress for the important subclass of chordal graphs, called strongly chordal graphs, has been made. For k
≥ 3, a graph G is called a k-trampoline (also called a k-sun in the literature) if it contains a k-clique with vertex set {v1 , v2 , . . . , vk } and, for each pair {vi , vi+1 }, there is a vertex wi
of degree 2 adjacent only to vi and vi+1 in G for all i ∈ [k], where addition is taken modulo k. Thus, a k-trampoline has order 2k. A 3-trampoline is shown in Fig. 17. A graph G is a strongly chordal
graph if it is chordal and does not contain a k-trampoline as an induced subgraph, for any k. Brewster, MacGillivray, and Yang [12] (see also [49]) showed that for strongly chordal graphs, BROADCAST
DOMINATION can be solved in O(n3 ) time. This NATION
M. A. Henning et al.
Fig. 17 A 3-trampoline
algorithm is different from previous algorithms in that it uses integer programming to find an optimal broadcast. As shown in Section 2, BROADCAST DOMINATION can be defined as a linear programming
problem. However, if the solution to the linear program is fractional, it is not a solution of BROADCAST DOMINATION. In general, although the class of chordal graphs does not necessarily have an
integer solution to the linear program, the subclass of strongly chordal graphs always has integer solutions. A matrix is totally balanced if it does not contain any cycle of length at least 3. It
was proved in [12] that the constraint matrix of a strongly chordal graph is totally balanced. A matrix is called -free if it does not contain
11 = 10
as a submatrix. Lubiw [34, 35] proved that a totally balanced matrix can have a -free ordering, and Farber [23] showed that the linear programming problem associated with -free matrices always has an
integer solution and it can be solved greedily. Combining all the results above, Brewster et al. [12] provided an efficient algorithm for the class of strongly chordal graphs. Theorem 66. ([12, 49])
The broadcast domination number of a strongly chordal graph of order n can be computed in O(n3 ) time. Brewster et al. [12] noted that every strongly chordal graph G satisfies γ b (G) = mp(G) by the
duality theorem of linear programming. Corollary 67. ([12, 49]) If G is strongly chordal graph, then γ b (G) = mp(G). We consider next the following decision problem:
k-LIMITED BROADCAST DOMINATION Input: A graph G, integers k ≥ 1 and ≥ 1. Question: Is γ b,k (G) ≤ ?
Cáceres et al. [13, 14] used a reduction from 3-SAT to prove that for k ≥ 2, kLIMITED BROADCAST DOMINATION is NP-complete. Theorem 68. ([13, 14]) For k ≥ 2, k-Limited Broadcast Domination is
NP-complete for general graphs.
Broadcast Domination in Graphs
Cáceres et al. [13] considered trees and proved the following result. Theorem 69. ([13]) 2-Limited Broadcast Domination can be solved in linear time for trees.
10 Concluding Comments In this chapter, we have surveyed selected results on the broadcast domination in graphs. Other results on broadcast domination can be found, for example, in [24, 25, 36, 40,
41, 43, 48]. We close with a small list of conjectures and open problems. We repeat our earlier three conjectures. Conjecture 1 ([4]). If G is a connected graph, then γ b (G) ≤ 2mp(G). Conjecture 2
([29]). If T is a tree of order n ≥ 2, then γb2 (T ) ≤ 13 (2n + 4). Conjecture 3 ([30]). If G is a cubic graph of order n, then γb,2 (G) ≤ 13 n. We present next a list of open problems, taken from
the “Stephen Hedetniemi treasure chest of intriguing ideas and open research questions.” Problem 1. Recall that by Theorem 35, every graph G satisfies diam(G) ≤ b (G). Is diam(G) < b (G) possible?
Problem 2. What can you say about γ b,3 (T) for trees, as compared with γ b,2 (T)? Problem 3. What is the smallest value of k for which a graph G has an efficient k-limited broadcast? Problem 4. In
normal domination, every vertex has broadcast strength of 1. In klimited broadcasting, a vertex can be assigned any integer value between 0 and k, that is, if f is a k-limited dominating broadcast,
then f : V →{0, 1, . . . , k}. What if the only values allowed are 0 and k, that is, only one type of broadcast vertex is made, one of strength k, and so f : V →{0, k}? Problem 5. Does a broadcast
domination chain analogous to that presented in Theorem 36 exist for k-limited broadcasting? Problem 6. It would seem worthwhile to define a type of dominating broadcast f in a graph G in which for
every vertex v in V , there exists a distinct broadcast vertex w in V such that d(v, w) ≤ f (w). This would enable a broadcast vertex v to compare its broadcast with that being given by another
broadcast vertex. This is the total version of broadcast domination, called broadcast total domination. Problem 7. Let us define a new concept called connected dominating broadcasting. Given a set B
of broadcast vertices, we construct a corresponding broadcast network N = (B, C), whose vertices are the broadcast vertices B, and two broadcast vertices are connected by an edge uv in C if d(u, v) ≤
min{f (u), f (v)}, that is u and v
M. A. Henning et al.
can hear each others’ broadcasts. What we want is that this broadcast network N is connected. The problem with normal broadcast domination is that the broadcast stations are all independent and
totally disconnected. We want connected broadcast domination. This is a much more realistic model. Problem 8. Even more interesting than connected or network broadcasting is the directed model,
whereby a broadcast vertex v can hear a broadcast from a broadcast vertex u, but not necessarily conversely. In this case, you need a central, or originating, broadcast vertex v∗ from which there is
a directed path to all other broadcast vertices. Such a vertex v∗ then originates a broadcast to all broadcast vertices along these directed edges, which in turn broadcast the originating message to
all remaining non-broadcast vertices. That is, between any two broadcast vertices u and v, if d(u, v) ≤ f (v), then there is an arc from v to u, meaning that u can hear a broadcast from v. But it is
possible that v cannot hear a broadcast from u. In this way, you can define the directed broadcast network existing among the broadcast vertices, and this is a directed graph. You want this broadcast
network to be connected in the further sense that there is a central vertex from which a given broadcast message can be relayed to all broadcast vertices over the arcs in the network, which in turn
can then broadcast this message to all non-broadcast vertices, i.e., the listeners. Problem 9. Consider, as an example, the 7 × 7 grid graph. Place the value 4 in the center vertex, square (4, 4).
Place the value 1 at the four vertices of degree 4 at distance 2 from a vertex of degree 2, as illustrated in Fig. 18(a). You have a dominating broadcast with a connected broadcast network that is a
directed K1,4 . If you decrease the broadcast strength of the center vertex to 3, then you have a less expensive 3-limited dominating broadcast, but now the broadcast network is totally disconnected.
The two broadcasts are illustrated in Figure 18(a) and 18(b), respectively.
(a) A connected broadcast of cost 8
Fig. 18 Two broadcasts in the 7 × 7 grid graph
(b) A disconnected broadcast of cost 7
Broadcast Domination in Graphs
References 1. D. Ahmadi, G.H. Fricke, C. Schroder, S.T. Hedetniemi, R.C. Laskar, Broadcast irredundance in graphs. Congr. Numer. 224, 17–31 (2015) 2. M. Ahmane, I. Bouchmakh, E. Sopena, On the
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Alliances and Related Domination Parameters Teresa W. Haynes and Stephen T. Hedetniemi
1 Introduction An alliance is generally thought of as a treaty or formal agreement between two or more parties, made in order to unite for a common cause. In 2002, P. Kristiansen, S. M. Hedetniemi,
and S. T. Hedetniemi [69] introduced several types of alliances in graphs to model such agreements. The study of alliances in graphs has become a popular area of research with around 100 papers
published since its inception in 2002. In fact, three survey papers have been published on the topic [39, 73, 99]. Since these recent overviews are readily available, the purpose of this chapter is
not to give a comprehensive survey of alliances in graphs. Instead our goal is to provide selected results along with sample proof techniques used in studying alliances. Furthermore, since an
alliance need not be a dominating set and this book is on domination in graphs, the main focus of this chapter will be on alliances that are also dominating sets. In particular, we select a specific
dominating alliance (a global defensive alliance) to serve as an illustration of work in the field. Also, we present a brief overview of some precursors of alliances and two recently defined related
parameters, namely cost effective sets and distribution sets. For algorithms and complexity of alliances, we refer the reader to Chapter 17 of this volume.
T. W. Haynes () Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN, USA Department of Mathematics and Applied Mathematics, University of Johannesburg,
Johannesburg, South Africa e-mail: [email protected] S. T. Hedetniemi School of Computing, Clemson University, Clemson, SC, USA e-mail: [email protected] © The Author(s), under exclusive license to
Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://doi.org/10.1007/978-3-030-58892-2_3
T. W. Haynes and S. T. Hedetniemi
We begin with some terminology and a discussion of precursors of alliances in Section 2. Definitions, preliminary results, and examples of alliances are given in Section 3. In Section 4, we present a
survey of core results on a type of dominating alliance, namely, a global defensive alliance. The related concepts of cost effective sets and distribution sets along with open problems and ideas for
future work are discussed in Section 5. We will use the following terminology throughout the chapter. Given a vertex set S ⊆ V , we let S = V \ S and we use the notation dS (v) = |N(v) ∩ S| for the
number of neighbors of v that are in S and dS (v) = |N(v) ∩ S| for the number of neighbors of v in S. Thus, d(v) = dS (v) + dS (v). The boundary of a set S is the set ∂(S) = N (S) ∩ S. Let the
generalized corona G ◦ H be the graph obtained by adding a copy of H for each vertex v of G and joining v to every vertex of H.
2 Preliminary Definitions and Background This section contains a discussion of parameters that can be considered as precursors to alliances. We present several types of sets, all of which are defined
in terms of conditions on the degrees of vertices with respect to sets S, ∂(S), and S, such as dS (v), dS (v), and d(v), either for vertices v ∈ S or vertices v ∈ S. Each of these types of sets
naturally arise in different real-world contexts. We present these as they occurred chronologically in the literature.
2.1 Unfriendly Sets and Satisfactory Partitions Problems of partitioning the vertex set of a graph with constraints on the degrees of vertices in the sets can be traced to a problem of unfriendly
partitions of graphs introduced by Borodin and Kostochka [9] in 1977. Definition 2.1 A partition {S, S} of the vertex set V of a graph G is unfriendly if for every vertex v ∈ S, dS (v) ≥ dS (v) and
for every vertex u ∈ S, dS (u) ≥ dS (u). Stated equivalently, a partition of the vertex set of a graph G into two sets is called unfriendly if every vertex v ∈ V has at least as many neighbors in the
opposite set as it has in its own set. In 1990, Aharoni, Milner, and Prikry [1] settled, in the affirmative, a conjecture by Cowan and Emerson that every graph has an unfriendly partition (see also
Shelah and Milner [85]). Theorem 2.2 ([1]) Every nontrivial graph has an unfriendly partition. Proof. Among all partitions of the vertex set V of G into two nonempty sets, select {S, S} to be one
that maximizes the number of edges having one end in S and the
Alliances and Related Domination Parameters
other in S. We claim that {S, S} is an unfriendly partition. If not, then there must be at least one vertex having more neighbors in its own set than in the other set. Moving this vertex to the other
set would therefore increase the number of edges between the two sets, a contradiction. It can also be shown that if you start with any arbitrary partition of V into two nonempty sets and repeatedly
find a vertex v in either set, which has more neighbors in the set containing it than neighbors in the other set, and then move v to the other set, this process will repeatedly increase the number of
edges between the two sets. Therefore, after a finite number of such moves, this process will terminate with an unfriendly bipartition. Corollary 2.3 Every graph G without isolated vertices has a
partition into two dominating sets. Proof Let {S, S} be an unfriendly partition of the vertex set of G. We show that both S and S are dominating sets of G. It suffices to consider only set S as the
same argument used for S applies to S. If S is not a dominating set, then there exists a vertex, say x ∈ S having no neighbors in S. But in this case, x has more neighbors (at least one) in S than it
has in S (none), contradicting that {S, S} is an unfriendly partition. It follows that both S and S are dominating sets of G. Corollary 2.3 is reminiscent of the following well-known theorem and
corollary of Ore [72]. Theorem 2.4 (Ore [72]) If G is a graph having no isolated vertices, then the complement S of any minimal dominating set S is a dominating set. Corollary 2.5 Every graph G
without isolated vertices has a partition into two dominating sets. Although Corollary 2.3 and Corollary 2.5 are the same, the reasoning behind them makes a difference. With Corollary 2.5, we know
that at least one of the two dominating sets is a minimal dominating set, and furthermore, it can be a minimum dominating set of cardinality γ (G). We do not have such an assurance in Corollary 2.3.
Our discussion of unfriendly partitions gives rise to the following definitions. Definition 2.6 A dominating set S is unfriendly if for every vertex w ∈ S, dS (w) ≥ dS (w), that is, w has at least as
many neighbors in S as it has in S. An unfriendly dominating set is very unfriendly if this inequality is strict, that is, dS (w) > dS (w). Definition 2.7 A dominating set S is friendly if for every
vertex w ∈ S, dS (w) ≤ dS (w), that is, w has at least as many neighbors in S as it has in S. A friendly dominating set is very friendly if this inequality is strict, that is, dS (w) < dS (w). This
leads to the observation: every isolate-free graph G has a partition into an independent dominating set S and an unfriendly dominating set S, since the
T. W. Haynes and S. T. Hedetniemi
complement S of every maximal independent set S is necessarily a (very) unfriendly dominating set. Notice that, by definition, the vertex set V is vacuously a friendly dominating set, so every graph
has a friendly dominating set. We will revisit friendly and unfriendly dominating sets in Section 5. In some sense dual to an unfriendly partition, a satisfactory partition is a partition of the
vertex set of a graph into two sets such that each vertex has at least as many neighbors in the set containing it as it has in the opposite set. We note that this “friendly” version is equivalent to
a partition of the vertex set into two strong defensive alliances. Satisfactory partitions have been studied in [43–45] and [81]. However, unlike unfriendly partitions, not every graph has a
satisfactory partition. For example, complete graphs of odd order and complete bipartite graphs Kr,s , when r or s is odd, do not have satisfactory partitions. In fact, it is an NP-complete problem
to decide if an arbitrary graph has a satisfactory partition [4].
2.2 (σ , ρ)-sets In 1994, Telle [92, 93] introduced the idea of (σ , ρ)-sets S, in which σ is a nonnegative integer condition that must hold on the number of neighbors a vertex in S must have in S,
and ρ is a nonnegative integer condition that must hold on the number of neighbors a vertex in S must have in S. This was generalized by Haynes, Hedetniemi, and Slater in 1998 [50] as follows. There
are four possible values under consideration, namely, dS (v) for v ∈ S, dS (v) for v ∈ S, dS (v) for v ∈ S, and dS (v) for v ∈ S. Table 1 illustrates how different domination parameters are defined
using combinations of these four values. A blank
Table 1 Degree Conditions S is a D-set an independent set an ID-set a TD-set a PD-set a RD-set a k-dominating set a D-set and S is a D-set a [1, k]-dominating set an odd D-set an open odd D-set an
efficient D-set a 1-dependent D-set
v ∈ S, dS (v)
v ∈ S, dS (v)
=0 =0 ≥1
≥1 even odd =0 ≤1
v ∈ S, dS (v) ≥1 ≥1 ≥1 =1 ≥1 ≥k ≥1 ≥ 1 and ≤ k odd odd =1 ≥1
v ∈ S, dS (v)
Alliances and Related Domination Parameters
entry in the table implies that this condition is not relevant to the definition. Let Dset, TD-set, ID-set, PD-set, and RD-set denote dominating set, total dominating set, independent dominating set,
perfect dominating set, and restrained dominating set, respectively.
2.3 Signed Domination In 1995, Dunbar, Hedetniemi, Henning, and Slater [30] introduced the concept of signed domination in graphs as follows. Definition 2.8 A function f : V →{−1, 1} is called a
signed dominating function if for every vertex v ∈ V , f (N[v]) = Σ u ∈ N[v] f (u) ≥ 1. In effect, a signed dominating function defines a partition of the vertex set of G into two sets S and S, where
S = {v : f (v) = 1} and S = {u : f (u) = −1}, such that for every vertex v ∈ V , dS (v) > dS (v). We note that in a signed dominating function the set S is a very unfriendly dominating set of the set
S, or equivalently, S is a global offensive alliance of G, which we will define in the next section.
2.4 Minus Domination In 1999, Dunbar, Hedetniemi, Henning, and McRae [31] introduced a variation on signed domination called minus domination, as follows. Definition 2.9 A function f : V →{−1, 0, 1}
is a minus dominating function if for every vertex v ∈ V , f (N[v]) = Σ u ∈ N[v] f (u) ≥ 1. In effect, a minus dominating function defines a partition {S−1 , S0 , S1 } of the vertex set V , where S−1
= {v : f (v) = −1}, S0 = {v : f (v) = 0}, and S1 = {u : f (u) = 1}, such that for every vertex v ∈ V , dS 1 (v) > dS −1 (v).
2.5 Strong and Weak Dominating Sets In 1996, Sampathkumar and Pushpa Latha [80] focused on the degrees of the vertices in a dominating set S and how they related to the degrees of vertices in S.
Definition 2.10 A dominating set S is said to be strong if for every vertex v ∈ S, there exists a vertex u ∈ S ∩ N(v) such that d(u) ≥ d(v). Similarly, a dominating set S is said to be weak if for
every vertex v ∈ S, there exists a vertex u ∈ S ∩ N(v) such that d(u) ≤ d(v).
T. W. Haynes and S. T. Hedetniemi
2.6 α-Dominating Sets In 2000, Dunbar, Hoffman, Laskar, and Markus [32] introduced the concept of αdomination, where α is a number 0 < α ≤ 1. Definition 2.11 A set S of vertices in a graph G is an
α-dominating set if for every S (v) vertex v ∈ S, dd(v) ≥ α, where 0 < α ≤ 1. Note that when α ≥ 1/2, the vertices in S corresponding to an α-dominating set S satisfy the unfriendly condition of
Aharoni et al. that every vertex in S has at least as many neighbors in S as it has in S. However, no unfriendly condition is required for the vertices in an α-dominating set S.
2.7 Communities In 2000, Flake, Lawrence, and Giles [41] introduced the concept of a community in a graph as follows. Definition 2.12 A community is a vertex subset C ⊆ V of a graph G, such that for
all vertices v ∈ C, v has at least as many edges connecting to vertices in C as it does to vertices in C. This definition can be rephrased as follows. A community is a set S of vertices having the
property that for every vertex v ∈ S, dS (v) ≥ dS (v).
3 Alliances In 2002, concepts almost the same as communities were introduced by Kristiansen, Hedetniemi, and Hedetniemi [69], but in a completely different context, that of alliances in networks
rather than communities. In 2004, the authors followed the proceedings [69] with a more detailed introduction to alliances in [57]. An alliance is generally thought of as a treaty or formal agreement
between two or more parties or nations, made in order to unite for a common cause or for mutual support. For example, defensive alliances are formed during times of war, where the allies agree to
join forces if one or more of them are attacked, and offensive alliances can be formed in times of peace, where allies might have to join forces in order to keep peace. In addition to alliances for
national defense, applications of alliances are widespread in nature from social and business associations to political and scientific groupings. As mentioned in the introduction, the study of
alliances in graphs has become a popular area of research with around 100 papers published since its introduction in 2002.
Alliances and Related Domination Parameters
The popularity of alliances is further evidenced by three recent survey papers on the topic [39, 73, 99]. The first survey by Fernau and Rodríguez-Velázquez [39] in 2014 focuses mainly on defensive
alliances. Yero and Rodríguez-Velázquez [99] wrote a second survey in 2017. In it, they note that graph parameters and types of alliances have been studied under many different names, and they
provide a new general and unifying framework for a wide variety of alliances. In 2018, Ouazine, Slimani, and Tari [73] published the third survey on alliances. This survey gives another
generalization of alliances and presents results for both defensive and offensive alliances. In this section, we present definitions, examples, and preliminary results on alliances. Recall that for a
set S, the boundary of S, denoted ∂(S), is the set of vertices in S that have a neighbor in S. Definition 3.1 A nonempty set of vertices S of a graph G is a defensive alliance if for every v ∈ S, |N
[v] ∩ S| ≥ |N(v) ∩ S|. The minimum cardinality of a defensive alliance of G is the defensive alliance number of G, denoted by a(G). This can be stated equivalently as follows. A defensive alliance is
a nonempty set S of vertices having the property that for every vertex v ∈ S, dS (v) ≥ dS (v) − 1. Conceptually, each vertex in S is in alliance with its neighbors in S for defense against possible
attacks from neighbors in ∂(S). For each vertex v ∈ S, an attack at v by the vertices in ∂(S) that are adjacent to v can result in no worse than a draw (assuming strength in numbers). Thus, each
vertex in S can be successfully defended against attacks from its neighbors in ∂(S). By changing focus from vertices in S to vertices in ∂(S), Hedetniemi et al. [57] defined the following. Definition
3.2 A nonempty set of vertices S of a graph G is an offensive alliance if for every vertex v ∈ ∂(S), |N(v) ∩ S| ≥ |N[v] ∩ S|. The minimum cardinality of an offensive alliance of G is the offensive
alliance number of G, denoted by ao (G). Equivalently, an offensive alliance is a set S of vertices having the property that for every vertex v ∈ ∂(S), dS (v) ≥ dS (v) + 1. In terms of application of
an offensive alliance S, it is reasonable to think that each vertex in S is in alliance with its neighbors in S against its neighbors in ∂(S). For the set S as a whole, since an attack by an
offensive alliance S on the vertices of ∂(S) can result in no worse than a “tie,” the vertices in S can “successfully” attack any single vertex in ∂(S). For examples, we note that the offensive
alliance and defensive alliance numbers are equal for a complete graph, that is, a(Kn ) = ao (Kn ) = n2 . Note also that any vertex of degree 0 or 1 is a defensive alliance. It is shown in [57] that
a(G) = 1 if and only if G has a vertex of degree 0 or 1, and it is shown in [37] that ao (G) = 1 if and only if G is a star. The alliance numbers for paths and cycles follow. Proposition 3.3 For
paths Pn and cycles Cn with n ≥ 3, 1. ([57]) a(Pn ) = 1 and a(Cn ) = 2, 2. ([37]) ao (Pn ) = n2 and ao (Cn ) = n2 .
T. W. Haynes and S. T. Hedetniemi
Paths and cycles provide examples where the offensive alliance number can be larger than the defensive alliance number. To see that these two numbers are incomparable, consider the complete bipartite
graphs Kr,s , where 2 ≤ r ≤ s. Then r s r+1 a(Kr,s ) = 2 + 2 , while ao (Kr,s ) = 2 . Thus, the defensive alliance number is larger than the offensive alliance number for Kr,s when r ≥ 4. We mention
upper bounds on these two alliance numbers before defining more alliance numbers. For two sets of vertices A and B, we define an edge having one end in A and the other in B, an AB-edge. Theorem 3.4
([42]) If G is a connected graph of order n ≥ 2, then a(G) ≤ n2 . Proof The result is trivial if G has a vertex of degree at most 1. Among all balanced bipartitions {A, B} of V , where |A|−|B|≤ 1,
let π = {A, B} be one that minimizes the number of AB-edges. Without loss of generality, we can assume that |A| = n2 and B = n2 . If A or B is a defensive alliance, then the result holds. Hence,
assume that neither A nor B is a defensive alliance. Thus, there exist vertices a ∈ A and b ∈ B such that |N[a] ∩ A| < |N(a) ∩ B| and |N[b] ∩ B| < |N(b) ∩ A|. But then swapping a and b will produce a
balanced bipartition with fewer AB-edges, contradicting our choice of π . As we have seen, Theorem 3.4 is sharp for complete graphs. Theorem 3.5 ([37]) If G is a graph of order n ≥ 2, then ao (G) ≤
2n 3 .
Proof Since the result is trivial if G has an isolated vertex, we may assume that the minimum degree of G is at least 1. Color the vertices of V with three colors, say Red, Green and Blue, so that
the number of monochromatic edges is minimum. Let v be a vertex colored red. Then v has at least as many green (respectively, blue) neighbors as it has red ones, else v could be recolored green
(respectively, blue), decreasing the number of monochromatic edges. Thus, the union of any two color classes is an offensive alliance, implying that ao ≤ 2n 3 . The authors of [37] note that the
bound of Theorem 3.5 is tight for K3 , K2,2,2 , and the generalized corona K3 ◦ K2 . A defensive (respectively, offensive) alliance S is called strong if the inequality is strict. We state the
definitions formally as follows. Definition 3.6 A nonempty set of vertices S of a graph G is a 1. strong defensive alliance if for every v ∈ S, |N[v]∩S| > |N(v)∩S|. The minimum cardinality of a
strong defensive alliance of G is the strong defensive alliance number of G, denoted by a(G). ˆ 2. strong offensive alliance if for every vertex v ∈ ∂(S), |N(v) ∩ S| > |N[v] ∩ S|. The minimum
cardinality of a strong offensive alliance of G is the strong offensive alliance number of G, denoted by aˆ o (G). In 2009, Shafique and Dutton [84] published a paper that had an interesting
connection to the Aharoni, Milner, and Prikry theorem (Theorem 2.2) that every graph has an unfriendly partition. A partition {S, S} of a vertex set of a graph G
Alliances and Related Domination Parameters
is called an alliance-free partition, if neither S nor S contains a strong defensive alliance as a subset. Shafique and Dutton [84] prove that a connected graph G has an alliance-free partition
exactly when G has a block that is neither an odd clique nor an odd cycle. For more on alliance numbers, the reader is referred to [2, 3, 5, 8, 20, 37, 59, 63, 65–67, 70]. In 2009, Brigham, Dutton,
Haynes, and Hedetniemi [16] studied alliances that are both defensive and offensive. Definition 3.7 A nonempty set of vertices S of a graph G is a powerful alliance if for every v ∈ N[S], |N[v] ∩ S|≥
|N[v] S|. The minimum cardinality of a powerful alliance of G is the powerful alliance number of G, denoted by ap (G). This can be stated equivalently as follows. A powerful alliance is a set S of
vertices having the property that for every vertex v ∈ S, dS (v) ≥ dS (v) − 1 and for every vertexv ∈ ∂(S), dS (v) ≥ dS (v) + 1. As examples, for the complete graph Kn , ap (Kn ) = n2 , and the
values of the powerful alliance number of paths and cycles are given in the next result. Proposition 3.8 ([16]) For paths Pn and cycles Cn with n ≥ 3, ap (Pn ) =
2n 3
and ap (Cn ) =
2n . 3
Powerful alliances are also studied in [14, 15, 46]. An alliance S of any type (defensive, offensive, or powerful) is called global if S is in addition a dominating set. We state the definitions
formally as follows. Definition 3.9 Any alliance S is a global alliance if S is a dominating set. 1. The global defensive alliance number γ a (G) (respectively, global strong defensive alliance
number γaˆ (G)) is the minimum cardinality of a global defensive alliance (respectively, global strong defensive alliance) of G. 2. The global offensive alliance number γ o (G) (respectively, global
strong offensive alliance number γoˆ (G)) is the minimum cardinality of a global offensive alliance (respectively, global strong offensive alliance) of G. 3. The global powerful alliance number γap
(G) is the minimum cardinality of a global powerful alliance of G. The next result follows directly from the definitions. Proposition 3.10 For any graph G, γ (G) γa (G) γaˆ (G), γ (G) γo (G) γoˆ (G),
and γ (G) ≤ γap (G).
T. W. Haynes and S. T. Hedetniemi
As we mentioned in the introduction, we will survey the results on global defensive alliances as a sampling of global alliances. For more on global offensive alliances, see [10, 12, 23, 26, 29, 48,
62, 74, 89, 98]. Shafique and Dutton [82, 83] generalized alliances to k-alliances for an integer k. Formally, a nonempty set S of vertices of a graph G is a defensive k-alliance (respectively, an
offensive k-alliance) if every vertex of S (respectively, the boundary of S) has at least k more neighbors in S than it has in S. Note that for k = −1, a defensive k-alliance is the standard
defensive alliance and for k = 0 it is a strong defensive alliance. Similarly, the case for k = 1 (respectively, k = 2) in a k-offensive alliance corresponds to the normal offensive alliance
(respectively, strong offensive alliance). A set S ⊆ V is a powerful k-alliance if it is both a defensive k-alliance and an offensive (k + 2)-alliance. Much of the research on alliances has been on
these generalized k-alliances. See [6, 18, 21, 24, 25, 38, 40, 60, 77, 78, 86, 87, 90, 91, 94–97, 100], for example. We conclude this section by mentioning that Haynes and Lachniet [49] defined the
alliance partition number of a graph as follows. A partition of the vertex set of G into defensive alliances is called an alliance partition. The alliance partition number ψ a (G) is the maximum
order of any alliance partition of G. For example, the alliance partition number of grid graphs Gr,c = Pr 2Pc (the Cartesian product of path graphs on r and c vertices) was determined in [49]. For
examples, see Figures 1 and 2, where the alliances in the partition are circled with dashed blue lines. Theorem 3.11 ([49]) For the grid graph Gr,c ,
ψa (Gr,c ) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
c+1 2
if 1 = r ≤ c if 2 = r ≤ c
c if 3 = r ≤ c and c is odd ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c+1 if 3 = r ≤ c and c is even ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎪ ⎪ c−2 ⎩ r−2 + r + c − 2 if 4 ≤ r ≤ c. 2 2
We note that the alliance partition number is also studied in [35] and studied under a different name, the quorum coloring number, in [58]. As defined by Hedetniemi, Hedetniemi, Laskar, and Mulder
[58] in 2013, the quorum coloring number is the maximum order k of a vertex partition π = {V1 , V2 , . . . , Vk } such that for every vertex v ∈ Vi , |N[v] ∩ Vi |≥|N[v]|/2, that is, at least half of
the vertices in the closed neighborhood of every vertex v have the same color as v. They determined the quorum number, and hence the alliance partition number, of a hypercube Qn as follows.
Alliances and Related Domination Parameters
Fig. 1 ψ a (G10,10 ) = 34 n Theorem 3.12 ([58]) For the hypercube Qn , ψa (Qn ) = 2 2 .
Sahbi and Chellali [79] showed that the decision problem associated with the quorum coloring number (alliance partition number) is NP-complete. Partitioning the vertex set of a graph into alliances
for other types of alliances is also studied in [84, 90, 96, 100], for example.
4 Global Defensive Alliances Global defensive alliances were defined in [57, 69] and first studied in [51] and [52]. As examples, we determine the global defensive alliance and global strong
defensive alliance numbers of the Petersen graph P. Proposition 4.1 For the Petersen graph P, γ a (P) = 4, while γaˆ (P ) = 5.
T. W. Haynes and S. T. Hedetniemi
Fig. 2 ψ a (G10,11 ) = 35 Fig. 3 Petersen Graph P, γ a (P) = 4
Proof Figure 3 demonstrates a global defensive alliance (the set of darkened vertices) of the Petersen graph P, while Figure 4 illustrates a global strong defensive alliance of P. Hence, γ a (P) ≤ 4
and γaˆ (P ) ≤ 5. First, to see that γ a (P) ≥ 4, let S be a γ a -set of P. Since P is 3-regular and for every vertex v ∈ S, dS (v) + 1 ≥ dS (v), it follows that there are no isolated vertices in P
[S]. That is, every vertex in S must have a neighbor in S. Furthermore, no set of three vertices having this property dominates P, implying that γ a (P) ≥ 4, and so γ a (P) = 4.
Alliances and Related Domination Parameters
Fig. 4 Petersen Graph P, γaˆ (P ) = 5
Table 2 Global defensive alliance numbers for families of graphs Family of graphs G Complete graphs Kn Complete bipartite graphs Kr,s , 2 ≤ r ≤ s Cycles Cn Paths Pn , n ≥ 3, n≡2 (mod 4) Paths Pn , n
≥ 3, n ≡ 2 (mod 4) Double Star S(r, s), 1 ≤ r ≤ s
γ a (G)
γaˆ (G)
n+1 2
r 2
n n n
r−1 2
n+1 2
n 4
n n 4
s−1 2
−1 4 +2
s 2
n n s
n 4
n 4
We note that γa (P ) = 4 ≤ γaˆ (P ). To see that γaˆ (P ) ≥ 5, suppose to the contrary that γaˆ (P ) = 4 and let D be a γaˆ -set of P. Since P has no 4-cycle as a subgraph, at least one of the
vertices, say x, in D has dD (x) = 1 and so dD (x) + 1 = dD (x) = 2, contradicting that D is a global strong defensive alliance. Hence, γaˆ (P ) ≥ 5, and so γaˆ (P ) = 5. Values of the global
defensive alliance and global strong defensive alliance numbers for several families of graphs are given [52]. We summarize these results in Table 2. For the remainder of this section, we focus on
bounds on the global (strong) defensive alliance numbers for general graphs and trees. Bounds for several other families of graphs have been studied in [19, 33, 34, 47, 61, 68, 71, 76, 88, 91, 101].
4.1 Bounds for General Graphs From our previous discussion, γ (G) γa (G) γaˆ (G) ≤ n for any graph G of order n. Bullington, Eroh, and Winters [17] showed that for positive integers a, √ b, and c,
where 2 ≤ b and c ≤ 12 (ab + 2b − a b/a, there exists a graph G with γ (G) = a, γ a (G) = b, and γaˆ (G) = c. For examples showing that equality and strictness can occur in each of the inequalities,
we consider a family of caterpillars. A caterpillar is a tree that reduces
T. W. Haynes and S. T. Hedetniemi
Fig. 5 Caterpillars G5,1 , G5,2 , and G5,3
to a nonempty path, called the spine, upon the removal of all its leaves. For k ≥ 2, let Gk,i denote the caterpillar that has the path Pk as its spine and each vertex on the spine is adjacent to i
leaves. In other words, Gk,i is the generalized corona Pk ◦ K i . We note that γ (Gk,1 ) = γa (Gk,1 ) = γaˆ (Gk,1 ) = k and that the vertices of the spine form a γ -set, a γ a -set, and a γaˆ -set.
Furthermore, γ (Gk,2 ) = γa (Gk,2 ) = k < k + 2 = γaˆ (Gk,2 ), while γ (Gk,3 ) = k < k + 2 = γa (Gk,3 ) < 2k = γaˆ (Gk,3 ). See Figure 5 for caterpillars G5,1 , G5,2 , and G5,3 , for examples, where
the vertices on the spine form a γ -set and the darkened vertices form a γaˆ -set. The vertices on the spine also form a γ a -set of Gk,1 and Gk,2 , while for Gk,3 , the spine vertices along with two
leaves (one adjacent to each end of the spine) form a γ a -set. It is observed in [52] that a graph G has γ a (G) = n if and only if G = K n . In fact, Haynes, Hedetniemi, and Henning [52] prove the
following sharp upper bounds. Theorem 4.2 ([52]) Let G be a graph with order n and minimum degree δ = δ(G). 1. γa (G) ≤ n − 2δ , and 2. γaˆ (G) ≤ n − 2δ . Complete graphs achieve both the bounds of
Theorem 4.2. As we have seen, the domination number is a lower bound on the global defensive alliance number and hence for the global strong defensive alliance number. It was established in [52] that
the total domination number γ t (G) is also a lower bound on the global (strong) defensive alliance number for graphs G with given minimum degree. Theorem 4.3 ([52]) Let G be a graph with minimum
degree δ(G). 1. If δ(G) ≥ 1, then γt (G) ≤ γaˆ (G). 2. If δ(G) ≥ 2, then γ t (G) ≤ γ a (G). These bounds are sharp as can be seen with cycles, that is, for cycles Cn with n ≥ 3, γt (Cn ) = γa (Cn ) =
γaˆ (Cn ). The caterpillar Gk,3 (as previously defined) shows that strictness in each bound can occur as γ (Gk,3 ) = γt (Gk,3 ) = k < k+2 = γa (Gk,3 ) < 2k = γaˆ (Gk,3 ), for k ≥ 3. The graph G in
Figure 6 gives another example of strictness in the inequalities, where γ (G) = γ t (G). To see this, we note that the support vertices of G along with a single vertex from the triangle form a γ set
of G, while the support vertices along with their nonleaf neighbors form a γ t -set of G. Further, the leaves of G along with the three vertices of the triangle form a γ a -set of G, and the set of
darkened vertices in Figure 6 is a γaˆ -set of G. Favaron [36] considered relationships between the global (strong) defensive alliance numbers and the independent domination number. We give here the
Alliances and Related Domination Parameters
Fig. 6 Graph G with γ (G) = 7, γ a (G) = 10, γ t (G) = 12, and γaˆ (G) = 13
and the proof to the first part of the theorem but omit the second proof as it can be proven similarly. Let F1 be the family of graphs G obtained from the complete graph Kk by attaching k leaves
adjacent to each vertex of the Kk , that is, G = Kk ◦ K k . Let F2 be the family of graphs G obtained from the complete graph Kk by attaching k − 1 leaves adjacent to each vertex of the Kk , that is,
G = Kk ◦ K k−1 . Theorem 4.4 ([36]) For every graph G, 1. i(G) ≤ (γ a (G))2 − γ a (G) + 1 with equality if and only if G ∈ F1 . 2. i(G) ≤ (γaˆ (G))2 − 2γaˆ (G) + 2 with equality if and only if G ∈ F2
. Proof Let S be a γ a -set of G, A be a maximal independent set of G[S], and B a maximal independent set of G[S \ N(A)]. Then A ∪ B is a maximal independent set of G, and so i(G) ≤|A| + |B|. Since S
is a defensive alliance, for each v ∈ S, dS (v) ≤ dS (v) + 1. Furthermore, since S is a dominating set, |B| ≤ |S \ N (A)| ≤
dS (v) ≤
(dS (v) + 1) ≤ |S| − |A| +
dS (v).
(1) Thus, i(G) ≤ |A| + |B| ≤ |S| +
dS (v).
Since every vertex v ∈ S has at most |S|− 1 neighbors in S, i(G) ≤|S| + (|S|−|A|) (|S|− 1) with |A|≥ 1. Hence, i(G) ≤|S|2 −|A|(|S|− 1) ≤|S|2 −|S| + 1 = (γ a (G))2 − γ a (G) + 1. If i(G) = (γ a (G))2
− γ a (G) + 1, then |A| = 1 and dS (v) = |S|− 1 for every v ∈ S A, that is, S is a clique and A consists of a vertex a ∈ S. Moreover, for any a ∈ S, equality in (1) gives |B| = |S \ N(A)|. Thus, S \
N(A) is independent, and |S \ N (A)| = v∈S\{a} dS (v) = v∈S\{a} (dS (v) + 1). This implies that NS (v) is independent and dS (v) = dS (v) + 1 for all v ∈ S. Moreover, NS (v) ∩ NS (u) = ∅ for all u, v
∈ S. It follows that G ∈ F1 .
T. W. Haynes and S. T. Hedetniemi
For the converse, let G be a graph in F1 , that is, G = Kk ◦ K k for k ≥ 1. Note that the vertices of the clique Kk form a minimum dominating set of G and a global alliance of G, so γ a (G) = k. Let
v be a vertex of the clique Kk in G, L(G) be the set of leaves of G, and L(v) the set of leaves adjacent to v in G. Then the set (L(G) L(v)) ∪{v} is a minimum independent dominating set of G. Hence,
i(G) = |(L(G) L(v)) ∪{v}| = k2 − k + 1 = (γ a (G))2 − γ a (G) + 1. A similar argument establishes Part (2) of the theorem. Sharp lower bounds on the global defensive alliance and global strong
defensive alliance numbers in terms of the order of a graph are given in [52]. Theorem 4.5 ([52]) If G is a graph of order n, then √
1. γa (G) ≥ √4n+1−1 , and 2 2. γaˆ (G) ≥ n. Proof Let G be a graph of order n, and let S be a γ a -set of G with = γ a (G) = k.
|S| neighbors in Since S is a defensive alliance, each vertex v ∈ S has at least d(v)
2 S. Hence, k = |S| ≥ |{v}| + d(v) . Moreover, dS (v) ≤ d(v) ≤ k. Since S 2 2 2 is a dominating set, |S| = √n − |S| = n − k ≤ v∈S dS (v) ≤ k . Equivalently, , proving part (1). k2 + k − n ≥ 0, and so
k ≥ 4n+1−1 2 Taking into account that strict inequality must hold for a global strong defensive alliance, a similar argument gives the bound of (2).
Families of generalized coronas achieve sharpness for the bounds of Theorem 4.5. In particular, Kk ◦ K k for k ≥ 3 has order n = k2 + k and γa (Kk ◦ K k ) = k = √ √ 4n+1−1 , while Kk ◦ K k−1 has
order n = k2 and γaˆ (Kk ◦ K k−1 ) = k = n. 2 2n Haynes, Hedetniemi, and Henning [52] also proved the lower bounds of +3 2n and +2 on the global defensive alliance number and the global strong
defensive alliance number, respectively, of bipartite graphs with order n and maximum degree . Rodríguez-Velázquez and Sigarreta [75] showed the bipartite condition was not necessary for the bound on
the global defensive alliance number to hold and improved the bound of Theorem 4.5 for the global strong defensive alliance number as follows. Theorem 4.6 ([75]) If G is a graph of order n with
maximum degree Δ = Δ(G), then 1. γa (G) ≥ 2. γaˆ (G) ≥
2n +3 , and n /2+1 .
Proof Let G be a graph of order n and maximum = (G). Let S be a
degree γ a -set of G. Now each vertex v ∈ S has at least d(v) neighbors in S and has at 2 most d(v) neighbors in S. That is, for each v ∈ S, dS (v) ≤ dS (v) + 1, and so 2
Alliances and Related Domination Parameters
dS (v) ≤
dS (v) + |S|.
Moreover, since S is a dominating set, n − |S| ≤
dS (v).
(dS (v) + dS (v)) ≤ d(v) ≤ |S|.
By Equations 3 and 4, we have 2n − 3|S| ≤
2n Thus, |S| = γa (G) ≥ +3 . Next let S be a γaˆ -set of G. Then,
dS (v) ≤ By Equations 4 and 6, γaˆ (G) ≥
d(v) . 2
n /2+1 .
Recall that Kr,s denotes the complete bipartite graph with partite sets of cardinality r and s. To see that the bounds of Theorem 4.6 are sharp, we again consider generalized coronas. It is noted in
[52] that Kk,k ◦ K k+1 for k ≥ 1 has = 2k + 1 and n = 2k + 2k(k + 1) = 2k2 + 4k. Since the 2k vertices of the Kk,k form a global 2n defensive alliance, +3 = 2k ≤ γa (Kk,k ◦ K k+1 ) ≤ 2k, and so the
bound is sharp. Moreover, as noted in [75] and by Theorem 4.1, the Petersen graph P has order n n = 10, (P) = 3, and γaˆ (P ) = 4 = /2+1 . Rodríguez-Velázquez and Sigarreta [75] also determined lower
bounds on global defensive alliance numbers of a graph in terms of its spectral radius λ (the largest eigenvalue of the adjacency matrix of the graph). Theorem 4.7 ([75]) If G is a graph of order n
with spectral radius λ, then 1. γa (G) ≥ 2. γaˆ (G) ≥
n λ+2 , n λ+1 .
4.2 Bounds for Trees We present bounds on the global (strong) defensive alliance numbers of trees. In particular, upper bounds are detailed in Section 4.2.1 and lower bounds in Section 4.2.2.
T. W. Haynes and S. T. Hedetniemi
Upper Bounds
The upper bound of Theorem 4.2 on the global defensive alliance number for general graphs was improved for trees in [52]. The authors also characterized trees attaining the new bound. To present the
new bound and characterization, we introduce some additional notation and define two families of trees. For a vertex v in a rooted tree T, let C(v) and D(v) denote the sets of children and
descendants, respectively, of v, and let D[v] = D(v) ∪{v}. Let T1 be the family of all trees defined as follows: Let T ∈{P5 , K1,4 } or let T be the tree obtained from tK1,4 (the disjoint union of t
≥ 2 copies of the star K1,4 ) by adding t − 1 edges between leaves of these copies of K1,4 in such a way that the center of each K1,4 is adjacent to exactly three leaves in T. Theorem 4.8 ([52]) If T
is a tree of order n ≥ 4, then γa (T ) ≤ and only if T ∈ T1 .
3n 5 ,
with equality if
Proof We proceed by induction on the order n ≥ 4 of T. If n = 4, then either T is the path P4 or the star K1,3 , and so γ a (T) = 2 < 3n/5. Suppose, then, that for all trees T of order n , where 4 ≤
n < n, γ a (T ) ≤ 3n /5, with equality if and onlyif T ∈ T1 . + 1 ≤ 3n Let T be a tree of order n ≥ 4. If T is a star, then γa (K1,n−1 ) = n−1 2 5 with equality if and only n = 5, that is, if and
only if T = K1,4 ∈ T1 . If T is a double star, then from the results in Table 2, γ a (T) < 3n/5. If T is the path P5 , then γ a (T) = 3 = 3n/5 and T ∈ T1 . Hence, we may assume that diam(T) ≥ 4 and
that T = P5 . Among all support vertices of T of eccentricity diam(T) − 1, let v be one of minimum degree. Root T at a vertex r, where r is at distance diam(T) − 1 from v. Let u denote the parent of
v and x the parent of u. Let T be the tree of order n obtained from T by deleting v and its children, that is, T = T − D[v]. Since diam(T) ≥ 4 and T = P5 , it follows from our choice of v that n ≥ 4.
Applying the inductive hypothesis to T , γ a (T ) ≤ 3n /5. Let S be a γ a -set of T . Let |C(v)| = v , and so n = n + v + 1. If u ∈ S , then adding v and v2−1 children of v to S forms a global
defensive alliance of T, and so γ a (T) ≤|S | + (v + 1)/2 ≤ 3(n − v − 1)/5 + (v + 1)/2 < 3n/5. Hence, we may assume that u∈S , else we have the desired result. If d(v) = 2, then adding the child of v
to S produces a global defensive alliance of T, and so γ a (T) ≤|S | + 1 ≤ 3(n − 2)/5 + 1 < 3n/5. Hence, we many assume that v ≥ 2. We consider two possibilities based on the d(u). Assume first that
d(u) ≥ 3. If u has a child v different from v that is a support vertex, then, by our choice of v, |C(v )|≥ v ≥ 2. But then we can always choose S to contain u and v , contradicting our assumption
that u∈S . Hence, every child of u different from v must be a leaf. If u is adjacent to more than one leaf, then again we can choose u ∈ S , a contradiction. Hence, d(u) = 3 and the child y (say) of
u different from v is a leaf. Since u∈S , it follows that y ∈ S . Deleting y from S and adding u, v,
Alliances and Related Domination Parameters
and (v − 1)/2 children of v to S yields a global defensive alliance of T, and so γ a (T) ≤|S |− 1 + 2 + (v − 1)/2 = |S | + (v + 1)/2 < 3n/5. Second assume that d(u) = 2. If diam(T) = 4, then x is
adjacent to r, that is, x is a support vertex. By our choice of v, d(x) ≥ d(v) ≥ 3. Thus, T − D[u] is a star K1,k with x as its center, where k ≥ v ≥ 2. The set {x, u, v} together with (v − 1)/2
leaves adjacent to v and (k − 1)/2 leaves adjacent to x is a global defensive alliance of T, and so γ a (T) ≤ 3 + (v − 1)/2 + (k − 1)/2 = (k + v + 4)/2 = (n + 1)/2. Since n ≥ 7, γ a (T) < 3n/5. Thus,
we may assume that diam(T) ≥ 5. Let T∗ = T − D[u] have order n∗ . Since diam(T) ≥ 5, it follows from our choice of v that n∗ ≥ 4. Applying the inductive hypothesis to T∗ , γ a (T∗ ) ≤ 3n∗ /5 with
equality if and only if T ∗ ∈ T1 .
Let S∗ be a γ a -set of T∗ . Adding u and v along with v2−1 children of v to S∗ gives a global defensive alliance of T. Hence, if v = 2, then γ a (T) ≤|S∗ | + 2 = 3(n − 4)/5 + 2 < 3n/5; while if v ≥
3, then γ a (T) ≤|S∗ | + (v + 3) /2 ≤ 3(n − v − 2)/5 + (v + 3)/2 ≤ 3n/5. Furthermore, if γ a (T) = 3n/5, then v = 3 and γ a (T∗ ) = |S∗ | = 3n∗ /5. By the inductive hypothesis, T ∗ ∈ T1 . Moreover,
by our choice of v, the support vertex adjacent to r has at least three leaf neighbors in T∗ . Hence, T∗ is not the path P5 . If T∗ = K1,4 , then T ∈ T1 . So we may assume that T∗ = K1,4 . That is,
T∗ is a tree obtained from t ≥ 2 copies of K1,4 by adding t − 1 edges as described in the definition of the family T1 . We note that a γ a -set of T∗ can be chosen to consist of the center, the
nonleaf neighbor of the center, and one leaf neighbor from each of the t copies of K1,4 . Suppose x is a central vertex of one of the copies of K1,4 in T∗ . Now S∗ contains at least one child of x
that is a leaf in T∗ . Deleting this child of x from S∗ and adding u, v, and one child of v forms a global defensive alliance of T. Thus, γ a (T) ≤|S∗ | + 3 − 1 = |S∗ | + 2 = 3(n − 5)/5 + 2 < 3n/5, a
contradiction. Hence, x must be a leaf of one of the copies of K1,4 in T∗ . Let z be the center of the K1,4 containing x in T∗ , and let N(z) = {z1 , z2 , z3 , x}. Now x may or may not be a leaf in
T∗ . If x is a leaf in T∗ , then in T, z is adjacent to exactly two leaves, z1 and z2 , say. Now let D∗ be a γ a -set of T∗ that contains all the central vertices of the t copies of K1,4 in T∗ ,
exactly one leaf adjacent to each central vertex and all the leaves of K1,4 that are incident to the t − 1 added edges when constructing T∗ . In particular, z, z3 ∈ D∗ . We may assume that x ∈ D∗ .
Let D = (D∗ −{x, z, z3 }) ∪{z1 , z2 , u, v, w}, where w is any child of v. Therefore, D is a global defensive alliance of T of cardinality γ a (T∗ ) + 2 < 3n/5, a contradiction. Hence, x is not a
leaf in T∗ , that is, x is adjacent to a vertex (a leaf) in a copy of K1,4 in T∗ . Thus, z1 , z2 , and z3 are leaves in T∗ , and it follows that T ∈ T1 . A parallel result for the global strong
defensive alliance number is also given [52]. Since its proof is similar to the one for Theorem 4.8, we omit it and only state the theorem here. Let T2 be the family of trees described as follows:
Let T be the tree obtained from the disjoint union t ≥ 1 copies of the star K1,3 by adding t − 1 edges between leaves of these copies of K1,3 in such a way that the center of each K1,3 is adjacent to
at least one leaf in T. Let T2 be the family of all such trees T. Theorem 4.9 ([52]) If T is a tree of order n ≥ 3, then γaˆ (T ) ≤ and only if T ∈ T2 .
3n 4 ,
with equality if
T. W. Haynes and S. T. Hedetniemi
The global defensive alliance number can be much larger than the vertex independence number for general graphs. For example, the complete graph Kn has
= γa (Kn ). However, the following result shows that for trees α(Kn ) = 1 n+1 2 T, γ a (T) is bounded above by α(T). To prove the theorem, we will use the following observation. Observation 4.10
([22]) If T is a tree obtained from a tree T by attaching a star K1,p , for p ≥ 1, with center u by adding edge uv for some vertex v of T , then α(T) = α(T ) + p. Theorem 4.11 ([22]) For any tree T,
γa (T ) α(T ), and this bound is sharp. Proof We proceed by induction on the order n of T. It is straightforward to check the result for trees of order n = 1 and n = 2. Let T be a tree of order n 3
assume that γa (T ) α(T ) for every tree T of order n < n. If T is a star, then γa (T ) = n/2 α(T ) = n − 1, and the result holds. Assume that T is not a star, and let v be a support vertex of T with
exactly one nonleaf neighbor, say w. Let T be the tree obtained from T by removing v and all its
leaf neighbors. Since T is not a star, T has order at least two. Let S be a γ a -set of T . We consider two cases based on the number v of leaves adjacent to v in T.
Assume first that v ≥ 2. If w
∈ S ,then S can be extended to a global defensive v −1 leaves adjacent to v. If w∈S , then adding v alliance of T by adding v and 2 and v2−1 leaves from N(v) gives a global defensive alliance of T.
In either case,
γa (T ) γa (T ) + v2−1 + 1. By Observation 4.10, α(T) = α(T ) + v . Applying
the inductive hypothesis to T , we obtain γa (T ) γa (T ) + v2−1 + 1 α(T ) + v −1 v −1 + 1 α(T ) − + 1, and therefore γa (T ) α(T ). + v 2 2 Next assume that v = 1. Let v be the leaf neighbor of v.
Then S can be extended
to a global defensive alliance of T by adding v if w ∈ S or adding v if w∈S . Thus, γa (T ) γa (T )+1. By Observation 4.10, α(T) = α(T ) + 1. Applying the inductive
hypothesis to T , we obtain γa (T ) γa (T ) + 1 α(T ) + 1 = α(T ). That this bound is sharp may be seen by considering the tree Hk , formed from an odd path P2k+1 , for k ≥ 0, labelled 1, 2, . . . ,
2k + 1, where for each odd labelled vertex v of the path, a new P5 is added by identifying its center vertex with v. Then γ a (Hk ) = α(Hk ) = 3(k + 1). For example, see H3 in Figure 7.
Fig. 7 The tree H3
Alliances and Related Domination Parameters
Since α(T ) (n + − 1)/2 for every nontrivial tree T with leaves [7], the next corollary is an improvement on the bound of Theorem 4.8 for n/5. Corollary 4.12 ([22]) For every nontrivial tree T with
leaves, γa (T ) (n + − 1)/2. The following bounds on the global strong defensive alliance number are also found in [22]. Theorem 4.13 ([22]) If T is a tree of order n ≥ 3 with s support vertices,
then 1. γaˆ (T ) ≤ 3α(T2)−1 , and 2. γaˆ (T ) ≤ α(T ) + s − 1. 4.2.2
Lower Bounds
From our previous discussion, γ (T ) ≤ γa (T ) ≤ γaˆ (T ) for all trees T. Trees T having γ (T ) = γaˆ (T ) are characterized in [53]. The following sharp lower bounds for trees are determined in
[52]. Theorem 4.14 ([52]) If T is a tree of order n, then 1. γ a (T) ≥ (n + 2)/4, and 2. γaˆ (T ) ≥ (n + 2)/3. Proof For part (1), let S be a γ a -set of T and γ a (T) = k. Further, let F = T[S], and
so V (F) = S. Since F is a forest, v ∈ S dS (v) = 2|E(F)|≤ 2(|S|− 1) = 2(k − 1). Since S is a global defensive alliance, dS (v) ≤ dS (v) + 1 for all v ∈ S. Combining these inequalities with the fact
that S is a dominating set, it follows that n − k = |S| ≤
dS (v) ≤
(dS (v) + 1) ≤ 2(k − 1) + k = 3k − 2.
Hence, k ≥ (n + 2)/4. For part (2), let S be a γaˆ -set of T and γaˆ (T ) = k. Let F = T[S]. As in part (1), v ∈ S dS (v) = 2|E(F)|≤ 2(|S|− 1) = 2(k − 1). Since S is a global strong defensive
alliance, dS (v) ≤ dS (v) for every v ∈ S. Since S is a dominating set, n − k = |S| ≤
and so k ≥ (n + 2)/3.
dS (v) ≤
dS (v) ≤ 2(k − 1),
We note that the tree T of order n obtained from a tree F of order n by adding dF (v) + 1 leaves adjacent to each vertex v of F has γ a (T) = n = (n + 2)/4, attaining the bound of Theorem 4.14(1).
Further, the tree T of order n obtained from a tree F of order n by adding dF (v) leaves adjacent to each vertex v of F has γaˆ (T ) = n = (n + 2)/3, attaining the bound of Theorem 4.14(2).
T. W. Haynes and S. T. Hedetniemi
Fig. 8 Tree T1
Fig. 9 Tree T2
Rodríguez-Velázquez and Sigarreta [76] generalized the lower bounds of Theorems 4.14 as follows. Theorem 4.15 ([76]) Let T be a tree of order n. 1. If S is a γ a -set of T and T[S] has c components,
then γ a (T) ≥ (n + 2c)/4. 2. If S is a γaˆ -set of T and T[S] has c components, then γaˆ (T ) ≥ (n + 2c)/3. Note that if T[S] is connected, then the bounds of Theorem 4.15 reduce to the bounds of
Theorem 4.14. Bouzefrane, Chellali, and Haynes [13] also improved the lower bounds given in Theorem 4.14 as follows. Theorem 4.16 ([13]) Let T be a tree of order n ≥ 2 with leaves and s support
vertices. Then, 1. γa (T ) ≥ 2. γaˆ (T ) ≥
3n−−s+4 , 8 3n−−s+4 . 6
We note that constructive characterizations are given in [13] for the trees attaining each of the bounds of Theorem 4.16. For examples of trees achieving the bounds, consider the trees T1 and T2 in
Figures 8 and 9. The tree T1 shown in Figure 8 has order n = 18, = 13, and s = 5. The set of support vertices of T1 is a γ a -set of T1 , and so γ a (T1 ) = 5 = (3n − − s + 4)/8. Also, the tree T2
shown in Figure 9 has order n = 11, = 4, and s = 3. The set of darkened vertices form a γaˆ -set of T2 , so γaˆ (T2 ) = 5 = (3n − − s + 4)/6. In [36], Favaron considered relationships between the
global (strong) defensive alliance numbers and the independent domination number of trees. We state the result but omit the proofs as they are similar to the proof of Theorem 4.4 for general graphs.
Let H1 be the family of trees T obtained from a tree H by attaching dH (u) + 1 leaves to each vertex u of H. Let H be the family of trees H such that for every
Alliances and Related Domination Parameters
maximal independent set I of H, the number of components of the forest of H − I is at most |n(H)|/2. Further, let H2 be the family of trees T obtained from a tree H ∈ H by attaching dH (u) leaves to
each vertex u of H. Theorem 4.17 ([36]) For every tree T of order n ≥ 2, 1. i(T) ≤ 2γ a (T) − 1 with equality if and only if T ∈ H1 . 2. i(T ) ≤ 3γaˆ (T )/2 − 1 with equality if and only if T ∈ H2 .
In concluding this section, we note that relationships between global defensive alliance and global offensive alliance numbers are given in [11, 102].
5 Related Parameters and Future Work We conclude this chapter with ideas for future research involving alliances. We begin by presenting two related concepts, namely cost effective sets and
distribution sets, that were defined subsequent to the introduction of alliances. As they are relatively unstudied compared to alliances, we include them here. Additional avenues for future research
are discussed in Section 5.1.3.
5.1 Cost Effective and Distribution Sets Cost effective and distribution sets depend on degrees of vertices in the sets S and S and are similar to alliances.
Cost Effective Sets
In 2012 Haynes, Hedetniemi, Hedetniemi, McCoy, and Vasylieva [54] introduced cost effective sets and cost effective domination in graphs. This was a reworking of the 1990 concept of unfriendly
partitions of Aharoni, Milner, and Prikry [1]. Cost effective sets were proposed to model applications, where services are provided to clients. Consider the client–server model of human
relationships, in which we let a set S represent a collection of servers, providing services to the vertices in ∂(S) over the edges between S and ∂(S). We say that a server, a vertex u ∈ S, is cost
effective if it serves at least as many clients as other servers, that is, if dS (v) ≥ dS (v). Definition 5.1 A subset S of vertices of a graph G is cost effective if for every vertex v ∈ S, dS (v) ≥
dS (v). The cost effective number CE(G) equals the maximum cardinality of a cost effective set in G, and the lower cost effective number ce(G) equals the minimum cardinality of a maximal cost
effective set in G.
T. W. Haynes and S. T. Hedetniemi
Notice that the property of being a cost effective set is hereditary, that is, every subset of a cost effective set is cost effective. Notice also that every independent set is a cost effective set.
Proposition 5.2 For any graph G, α(G) ≤ CE(G). Since a set can be maximal independent but not maximal cost effective, no inequality exists between ce(G) and i(G). If the inequality is strict, that
is, if dS (v) > dS (v) for a vertex v ∈ S, then v is said to be very cost effective. Definition 5.3 A subset S of vertices in a graph G is very cost effective if every vertex of S is very cost
effective. The very cost effective number V CE(G) equals the maximum cardinality of a very cost effective set in G, and the lower very cost effective number vce(G) equals the minimum cardinality of a
maximal very cost effective set in G. Cost effective domination numbers are defined as expected. The following results are given in [54]. Observation 5.4 ([54]) For a connected graph G of order n ≥
2, γce (G) ≤
n 2
Observation 5.5 ([54]) Every independent dominating set S in an isolate-free graph G is a very cost effective dominating set. Corollary 5.6 ([54]) For any isolate-free graph G, γ (G) ≤ γce (G) ≤ γvce
(G) ≤ i(G) ≤ α(G) ≤ vce (G) ≤ ce (G) ≤ (G). For more on cost effective sets, the reader is referred to [27, 55, 56, 64].
Distribution Centers
In 2018 Desormeaux, Haynes, Hedetniemi, and Moore [28] defined a distribution center in a graph to model a supply and demand situation. In business, a distribution center for products is a structure
or a group of units used to store goods that are to be distributed to retailers, to wholesalers, or directly to consumers. Distribution centers are usually thought of as being demand driven.
Definition 5.7 A nonempty set S of vertices in a graph G is a distribution center if for each vertex v ∈ ∂(S), there exists a vertex u ∈ S such that u ∈ N(v) and |N[u] ∩ S| ≥ |N [v] ∩ S|. The minimum
cardinality of a distribution center of a graph G is the distribution center number dc(G).
Alliances and Related Domination Parameters
Equivalently, a nonempty set S of vertices in a graph G is a distribution center if for each vertex v ∈ ∂(S), there exists a vertex u ∈ S ∩ N(v) with dS (u) ≥ dS (v). For such vertices, we say that u
supplies the demand of v, or equivalently, v is supplied by u. One perspective of a distribution center S is to think of a vertex v ∈ ∂(S) and its neighbors in S as needing some amount of resource
units, one unit per vertex, while each vertex in S is able to supply one unit of the resource. Thus, a vertex in ∂(S) makes a demand on the distribution center S and is supplied by one of its
neighbors in S. Vertex v asks a vertex u ∈ S ∩ N(v) to deliver dS (v) + 1 units for itself and its neighbors in S. This is possible only if the vertex u can receive from itself and its neighbors in S
at least this demand, that is, dS (u) ≥ dS (v). Hence, such a set S models a distribution center that is capable of providing two-day delivery to any vertex (customer) in ∂(S): on day 1, each
neighbor of u ∈ S ships one unit of resource to u, and then, on day 2, vertex u ships dS (v) + 1 units of resource to its neighbor v ∈ ∂(S). Notice the contrast between a distribution center and an
offensive alliance. With an offensive alliance S, the neighbors of a vertex v ∈ S that are in S can provide one unit of resource in one day. Thus, the total demand of dS (v) + 1 by vertex v can be
met by the vertices in S, each sending one unit of resource to v. In this way, an offensive alliance is like a one-day distribution center. One can think of a distribution center as a type of an
alliance between the vertices of S to service the vertices in ∂(S). Although distribution centers and offensive alliances are similar concepts, the corresponding parameters can easily be shown to be
incomparable. To see this, note that for cycles Cn with n ≥ 5, ao (Cn ) = n2 > 2 = dc(Cn ). On the contrary, for the complete bipartite graph < r = dc(Kr,s ). Kr,s with 1 ≤ r ≤ s, ao (Kr,s ) = r+1 2
As with alliances, a distribution center that is also a dominating set is called a global distribution center. Definition 5.8 A set S of vertices of a graph G is a global distribution center if for
each vertex v ∈ S, there exists a vertex u ∈ S ∩ N(v) such that dS (u) ≥ dS (v). The global distribution center number γ dc (G) is the minimum cardinality of a global distribution center of G.
Clearly, every global distribution center is a dominating set. Moreover, every graph G has a distribution center and a global distribution center since the set V (G) is trivially both of these.
Observation 5.9 ([28]) For any graph G of order n, γ (G) ≤ γ dc (G) and dc(G) ≤ γ dc (G) ≤ n. We conclude our discussion of distribution centers by illustrating the distribution and global
distribution numbers of the Petersen graph P in Figures 10 and 11, respectively, where the darkened vertices represent the appropriate sets. Notice in this example that both the set of darkened
vertices and the set of undarkened vertices are global distribution centers. In fact, in any prism, that is,
T. W. Haynes and S. T. Hedetniemi
Fig. 10 Petersen Graph P, dc(P) = 4
Fig. 11 Petersen Graph P, γdc (P ) = 5
a Cartesian product of the form G2K2 , the set of vertices in each copy of G in G2K2 is a global distribution center. As another illustration, it can be seen that the set of vertices in any two (or
more) consecutive rows (or columns) of a grid graph of the form Gm,n = Pm 2Pn forms a distribution center in a grid graph.
Future Research
Much work remains to be done on cost effective sets and distribution sets in graphs, since they were only introduced in 2012 and 2017, respectively. We note that Corollary 2.3 raises two optimization
questions: Problem 5.10 Over all possible unfriendly partitions {S, S} of a graph G, what is the smallest and largest cardinality of sets S and S, or, equivalently, what is the largest difference |S|
− |S|? Problem 5.11 Over all possible unfriendly partitions {S, S} of a graph G, what is the smallest and largest cardinality of the set of edges between S and S? The definitions given in this
chapter suggest a broader avenue for future research. For example, our discussion of unfriendly partitions suggests the definition of friendly and unfriendly sets as follows. A set S is friendly if
for every vertex v ∈ ∂(S), dS (v) ≤ dS (v), and is very friendly if this inequality is strict, that is, dS (v) < dS (v). If we reverse this inequality, we get the following:
Alliances and Related Domination Parameters Table 3 Degree Conditions for every v ∈ S for every v ∈ ∂(S)
73 dS (v) ≤ dS (v) cost effective friendly
dS (v) ≥ dS (v) internally strong unfriendly
A set S is unfriendly if for every vertex v ∈ ∂(S), dS (v) ≥ dS (v), and is very unfriendly if this inequality is strict, that is, dS (v) > dS (v). If we change focus from every vertex v ∈ S to every
vertex u ∈ S, we get the following: A set S is cost effective if for every vertex u ∈ S, dS (u) ≤ dS (u), and is very cost effective if this inequality is strict, that is, dS (u) < dS (u). If we
reverse this inequality, we get the following: A set S is internally strong if for every vertex u ∈ S, dS (u) ≥ dS (u), and is very internally strong if this inequality is strict, that is, dS (u) >
dS (u). We conclude by summarizing these concepts in Table 3. As usual, if ∂(S) = S, then the relevant table entries represent dominating sets. Thus, three new types of dominating sets are defined in
Table 3.
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Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph Wayne Goddard and Michael A. Henning
AMS Subject Classification: 05C69, 05C65
1 Introduction In this chapter, we survey some results concerning the fractional domatic, fractional idomatic, and fractional total domatic numbers of a graph. First, we recall the fundamental
concepts of a dominating set, an independent dominating set, and a total dominating set. A dominating set of a graph G is a set S of vertices of G such that every vertex not in S has a neighbor in S.
The domination number of G, denoted γ (G), is the minimum cardinality of a dominating set. An independent dominating set of G is a set that is both a dominating set and an independent set. The
independent domination number, denoted i(G), is the
The research of the author Michael A. Henning was supported in part by the University of Johannesburg. W. Goddard () School of Computing and School of Mathematical and Statistical Sciences, Clemson
University, Clemson, SC 29634, USA Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park 2006, Johannesburg, South Africa e-mail: [email protected] M. A.
Henning Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature
Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://doi.org/10.1007/978-3-030-58892-2_4
W. Goddard and M. A. Henning
minimum cardinality of an independent dominating set of G. An independent set of vertices in a graph G is a dominating set of G if and only if it is a maximal independent set. Thus, i(G) is
equivalently the minimum cardinality of a maximal independent set of vertices in G. A survey on independent domination in graphs can be found in [9]. A total dominating set, abbreviated TD-set, of a
graph G with no isolated vertex is a set S of vertices such that every vertex in G is adjacent to a vertex in S. The total domination number, denoted by γ t (G), is the minimum cardinality of a
TD-set of G. For a recent book on total domination in graphs, we refer the reader to [14]. The parameters studied in this chapter represent the fractional relaxation of the count of the maximum
number of disjoint dominating, independent dominating, and total dominating sets. We discuss these in the next three sections. In Section 5, we discuss a common framework in hypergraphs and in
Section 6 some generalizations.
2 The Fractional Domatic Number In this section, we survey results on the fractional domatic number of a graph. First, we recall the concept of the domatic number of a graph. The maximum number of
vertex-disjoint dominating sets in a graph G is called the domatic number of G. While this is often denoted by d(G), we will use here the notation dom(G). The domatic number was introduced in 1975 by
Cockayne and Hedetniemi [4] and has since been the subject of a large number of publications; a rough estimate says that it occurs in more than 200 papers to date. Much of the early work on the
domatic number of a graph and its variants was due to Zelinka. As he remarked in [31], the word “domatic” was created from the words “dominating” and “chromatic” since, although it is defined using
the concept of domination in graphs, it is somewhat analogous to the chromatic number of a graph, where we partition the vertex set into classes having certain properties (in this case, each class is
a dominating set). We consider here a fractional analog of the parameter dom(G). For a family F of subsets of V (G), let m(F) denote the maximum number of times a vertex appears in F (equivalently,
the maximum degree of the hypergraph with F as the hyperedges). The fractional domatic number of a graph G, denoted FDOM(G), is defined as FDOM(G) = max
|F| , m(F)
where the maximum is taken over all families F of dominating sets of G. (For a discussion of why FDOM can be viewed as a fractional analog and why maximum can be used instead of supremum in the above
formula, see Section 5.) The fractional domatic number seems to have been formally introduced in 2006 by Suomela [26], although the concept was studied in 2000 by Fujita, Yamashita, and Kameda [8].
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
We start with the following immediate lower and upper bounds on the fractional domatic number of a graph. Theorem 1 For every graph G, we have dom(G) ≤ FDOM(G) ≤
n(G) . γ (G)
Proof. To prove the lower bound, consider a family F that consists of a maximum number of vertex-disjoint dominating sets of G. In this case, |F| = dom(G) and m(F) = 1, implying that FDOM(G) ≥ |F|/m
(F) = dom(G). To prove the upper bound, let F be a family of dominating sets of G. Each set in the family F has size at least γ (G). Thus, γ (G) · |F| ≤
|F | ≤ n(G) · m(F),
F ∈F
or, equivalently, |F|/m(F) ≤ n(G)/γ (G), whence the result.
We note that equality occurs in the upper bound of Theorem 1 for all complete graphs. We show next that equality also occurs in the upper bound for all cycles. We use the standard notation [k] = {1,
. . . , k}. Proposition 2 For n ≥ 3, we have FDOM(Cn ) = n/γ (Cn ). Proof. Let G be the cycle v1 v2 . . . vn v1 . Let S be an arbitrary minimum dominating set of G; so |S| = γ (G). Let S = {j | j ∈
[n] and vj ∈ S}. For i ∈ [n], let Si = {vi+j | j ∈ S}, where addition is taken modulo n. Each set Si is a minimum dominating set of G. Let F = {S1 , S2 , . . . , Sn }. We note that each vertex of G
appears in exactly γ (G) of these sets, implying that m(F) = γ (G) and therefore that |F| n = . m(F) γ (G) Hence, FDOM(G) ≥ n/γ (G). The desired result now follows from the upper bound of Theorem 1.
2 The above proposition immediately generalizes to any circulant. Recall that γ (Cn ) = n/3. Hence, as a consequence of Proposition 2, the fractional domatic number of a cycle is determined as
follows: ⎧ ⎪ ⎨ 3 if n ≡ 0 (mod 3) 3n if n ≡ 1 (mod 3) FDOM(Cn ) = n+2 ⎪ ⎩ 3n if n ≡ 2 (mod 3). n+1
W. Goddard and M. A. Henning
It is immediate that a graph G of minimum degree δ has dom(G) ≤ δ + 1. The same upper bound holds for the fractional domatic number: Theorem 3 If a graph G of order n has minimum degree δ, then n ≤
FDOM(G) ≤ δ + 1. n−δ Proof. Let v be a vertex of minimum degree in G, and so dG (v) = δ. Let F be a family of dominating sets of G. Each set in the family F must contain the vertex v or a neighbor of
v in order to dominate v. By the pigeonhole principle, at least one vertex in the closed neighborhood of v appears in at least |F|/(δ + 1) sets, and so m(F) ≥ |F|/(δ + 1), or, equivalently, |F|/m(F)
≤ δ + 1. This is true for every family F of dominating sets of G, implying that FDOM(G) ≤ δ + 1. To prove the lower bound, consider the collection F of all subsets F of the vertex set of exactly n −
δ elements. Since every vertex belongs to F or has a neighbor in F, each such subset F is a dominating set !of H; that is, F is a family of dominating n−1 " sets of G. Furthermore, every vertex is in
n−δ−1 sets of F, and so !
n " n−δ n−1 " n−δ−1
|F| =! m(F)
n . n−δ
Hence, FDOM(G) ≥ n/(n − δ).
Thus, for example, by Theorem 3, if a graph G contains an isolated vertex, then dom(G) = FDOM(G) = 1. However, if G has no isolated vertex and S is a maximal independent set in G, then both S and V
(G) S are dominating sets, implying that dom(G) ≥ 2. Thus, FDOM(G) ≥ 2 if and only if G contains no isolated vertex. It remains an open problem to characterize the graphs G achieving equality in the
upper bound of Theorem 3. In the special case when the graph is a regular graph, Fujita, Yamashita, and Kameda proved in [8] the following result. Theorem 4 ([8]) If G is a δ-regular graph of order
n, then FDOM(G) = δ + 1 if and only dom(G) = δ + 1. We note that the parameter FDOM is monotonic, in that it cannot decrease on the addition of edges. We next examine the fractional domatic number of
the disjoint union or join of graphs. Let G and H be two graphs. The join of G and H, written G ⊕ H, is the graph obtained from the disjoint union of G and H by joining each vertex of G to every
vertex of H, while the union of G and H, written G + H, is the graph consisting of the disjoint union of G and H. As a consequence of Lemma 32 later, we get the following result. Theorem 5 For graphs
G and H, it holds that FDOM(G + H ) = min(FDOM(G), FDOM(H )).
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
A universal vertex of a graph G is one adjacent to all other vertices in G. Lemma 6 For graphs G and H without a universal vertex with orders n1 and n2 , respectively, the following hold: (a) FDOM(G
⊕ H) = n1 if n1 = n2 . (b) min(n1 , n2 ) < FDOM(G ⊕ H ) ≤ (n1 + n2 )/2 otherwise. Proof. The upper bound follows from Theorem 1, since the join of G and H has domination number 2. For the lower bound
in (a), let F consist of n1 disjoint pairs, each pair containing one vertex of G and one vertex of H. For the lower bound in (b), assume n1 < n2 . Then, let F consist of (i) all pairs of one vertex
in G and one vertex in H and (ii) the set V (H). Then, |F| = n1 n2 + 1 and m(F) = n2 , whence the result. 2 Fujita, Yamashita, and Kameda proved the following surprising and beautiful result in [8].
Theorem 7 ([8]) If G is a cubic graph, then G has a family F of five dominating sets of G such that m(F) ≤ 2. Furthermore, such a family F can be constructed in polynomial time. As an immediate
consequence of Theorem 7, we have the following lower bound on the fractional domatic number of a cubic graph. Theorem 8 If G is a cubic graph, then FDOM(G) ≥ 52 . As remarked by Fujita et al. [8],
it is not true that if G is a cubic graph, then G has a family F of six dominating sets of G such that every vertex is in at most two of these. We note that if a cubic graph had such a property, then
this would imply that FDOM(G) ≥ 6/2 = 3. However, that in turn would imply that γ (G) ≤ n(G)/3, but Kostochka and Stodolsky [19] showed there are cubic graphs G, where γ (G) = 8n(G)/23 + o(1). Abbas,
Egerstedt, Liu, Thomas, and Whalen [1] generalized Theorem 7 to a larger class of graphs, namely the class of K1,6 -free graphs; that is, graphs with no induced subgraph isomorphic to K1,6 . Their
study was motivated by a problem encountered both in the multiagent robotics and in the mobile sensor networks domains. As remarked in [1], the generalization to K1,6 -free graphs is of interest in
multiagent robotics, because the class of K1,6 -free graphs includes the class of unit disk graphs, where for every vertex there is a disk of radius 1 centered at the vertex representing its
transmission or interaction range (see [23]). In order to state their result, we recall a 1989 result due to McCuaig and Shepherd [22]. Let B be the family of seven graphs shown in Figure 1. McCuaig
and Shepherd [22] showed that if G is a connected graph of minimum degree at least 2 and G is not one of graphs in the family B, then the domination number of G is at most two-fifths its order.
W. Goddard and M. A. Henning
Fig. 1 The family B of seven exceptional graphs
We are now in a position to state the result of Abbas et al. [1]: Theorem 9 ([1]) If G is a K1,6 -free connected graph with minimum degree at least 2 and G ∈ / B ∪ {K2,3 }, then G has a family F of
five dominating sets of G such that m(F) ≤ 2, implying that FDOM(G) ≥ 52 . We remark that the proof of Theorem 9 given by Abbas et al. [1] is algorithmic and gives a polynomial-time algorithm to find
such a family F. We conclude this section with a comment about planar graphs. By Theorem 3, the maximum fractional domatic number of a planar graph is at most 6, since the minimum degree is at most
5. As observed in [12], an example of a planar graph with fractional domatic number equal to 6 is the icosahedron; this has six disjoint independent dominating sets of size 2, each consisting of a
vertex and the unique vertex at distance 3 from it.
3 The Fractional Idomatic Number In this section, we survey results on the fractional idomatic number of a graph. First, we recall the concept of the idomatic number of a graph. The maximum number of
vertex disjoint independent dominating sets in a graph G is called the idomatic number of G denoted by idom(G). This terminology was introduced by Zelinka [29], but the parameter was originally
defined by Cockayne and Hedetniemi [5]. In this section, we consider a fractional version of the idomatic number of a graph. The fractional idomatic number of graph G, denoted FIDOM(G), is defined as
FIDOM(G) = max
|F| , m(F)
where the maximum is taken over all families F of independent dominating sets of G and where, as before, m(F) is the maximum number of times an element appears in F. (We note that this parameter
should be defined as the supremum, but as explained in Section 5, one can show that the supremum is always achieved.) The results in this section are mainly due to the authors [12]. Using analogous
proofs to those presented in Theorems 1 and 3, one can establish the following immediate lower and upper bounds on the fractional idomatic number of a graph. Theorem 10 ([12]) The following hold in a
graph G with minimum degree δ.
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
(a) idom(G) ≤FIDOM(G) ≤ n(G)/i(G). (b) FDOM(G) ≤ δ + 1. Examples of equality in the upper bound of Theorem 10(a) are the cycles of length a multiple of 3. We note that FIDOM(G) = 1 only if the graph
G has an isolate. As shown, for example, by Favaron [7], the independent domination number of a graph of order n can be as much as n −o(n), even with prescribed minimum degree. For such a graph G, we
have that idom(G) = 1 and that FIDOM(G) = 1 + o(1). Even restricted to special classes of graphs, one can still see this behavior. Here is one such result. Recall that a graph is claw-free if it does
not contain K1,3 as an induced subgraph. Proposition 11 ([12]) There exist connected claw-free graphs G with arbitrarily large minimum degree for which FIDOM(G) = 1 + o(1). Proof. Let G be the
claw-free graph G constructed as follows. For a and d positive integers, let G be obtained from a complete graph H of order ad as follows. Let X1 , . . . , Xa be a partition of V (H) into sets each
of size d. For each resulting set Xi , we add a vertex xi of degree d adjacent to every vertex of Xi for i ∈ [a]. The resultant split graph G is claw-free and has minimum degree d. Letting X = {x1 ,
. . . , xa }, we note that every independent dominating set of G contains at least a − 1 vertices of X, implying that FIDOM(G) ≤
a . a−1
Furthermore, for all i ∈ [a] if Fi = (X \ {xi }) ∪ {xi }, where xi is an arbitrary vertex in Xi , then F = {F1 , . . . , Fa } is a family of independent dominating sets of G satisfying m(F) = a − 1,
implying that FIDOM(G) ≥
a |F| = . m(F) a−1
Consequently, FIDOM(G) = a/(a − 1). The desired result now follows by taking a and d arbitrarily large. 2 We next present a lower bound on the fractional idiomatic number in terms of dynamic
colorings. An r-dynamic coloring, also called r-hued coloring in the literature, of a graph G is a proper coloring of the vertices of G such that every vertex v has at least min(dG (v), r) colors in
its neighborhood, where dG (v) is the degree of the vertex v in G. For more details on r-dynamic colorings, we refer the reader to Jahanbekam et al. [17]. Theorem 12 ([12]) If G is a graph with
minimum degree at least r that has an r-dynamic coloring using k colors, then FIDOM(G) ≥
k , k−r
W. Goddard and M. A. Henning
and therefore i(G) ≤ (k − r)n/k. As a consequence of Theorem 12, we have the following lower bound on the fractional idiomatic number in terms of the chromatic number. Theorem 13 ([12]) If G is an
isolate-free graph with chromatic number k, then FIDOM(G) ≥
k . k−1
By Theorem 10(b), the maximum fractional idomatic number of a planar graph is at most 6, since the minimum degree is at most 5. As noted earlier, an example of a planar graph with fractional idomatic
number equal to 6 is the icosahedron; this has six disjoint independent dominating sets of size 2, each consisting of a vertex and the unique vertex at distance 3 from it. We next consider lower
bounds on the fractional idomatic number of a planar graph. As a consequence of Theorem 13 and a result by MacGillivray and Seyffarth [20] that provides an upper bound for the independent domination
number in terms of the chromatic number, we obtain the following result. Theorem 14 ([12]) The following hold in a planar connected graph G of order n. (a) If n ≥ 2, then FIDOM(G) ≥ 43 . (b) If n ≥
10, then i(G) ≤ 34 n − 2. Recall that the corona G ◦ P1 of a graph G, also denoted cor(G) in the literature, is the graph obtained from G by adding a pendant edge to each vertex of G. As remarked in
[12], the two bounds in Theorem 14 are sharp because of the corona K4 ◦ P1 of K4 illustrated in Figure 2. If one considers planar graphs of minimum degree at least 2, then the lower bound on the
fractional idiomatic number in Theorem 14(a) can be improved. The key is a result due to Kim, Lee, and Park [18], who showed that every connected planar graph has a 2-dynamic coloring using at most
four colors, except for C5 . Therefore, by Theorem 12 (with r = 2 and k = 4), we have that every connected planar graph G, except possibly for C5 , satisfies FIDOM(G) ≥ 2. However, since the 5-cycle
has FIDOM(C5 ) = 52 , the 5-cycle is no exception to the lower bound FIDOM(G) ≥ 2. We therefore have the following lower bound on the fractional idiomatic number of a planar graph with minimum degree
at least 2. Theorem 15 ([12]) If G is a planar graph with δ(G) ≥ 2, then FIDOM(G) ≥ 2. Fig. 2 A planar graph G with FIDOM(G) = 43
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
As an immediate consequence of Theorem 10(a) and Theorem 15, we have the following upper bound on the independent domination number of a planar graph with minimum degree at least 2. Corollary 16
([12]) If G is a planar graph with δ(G) ≥ 2, then i(G) ≤ 12 n(G). The following construction, given in [12], shows that there exists an infinite family of planar graphs G with minimum degree two that
satisfy FIDOM(G) = 2. For s ≥ 2, let Hs be the graph obtained from a 4-cycle v1 v2 v3 v4 v1 by adding s new vertices whose neighbors are the pair {vi , vi+2 } for each i ∈ [2]. When s = 4, for
example, the resulting graph Hs is illustrated in Figure 3. As observed in [12], it holds that i(Hs ) = 12 n(Hs ) and FIDOM(Hs ) = 2. Thus, the bounds of Theorem 15 and Corollary 16 are tight. As
remarked in [12], there are numerous families of planar graphs G with minimum degree two that satisfy FIDOM(G) = 2 and i(G) < 12 n(G). It remains, however, an open problem to characterize the graphs
achieving equality in the bounds of Theorem 15 and Corollary 16. It is also noted in [12] that there are numerous families of planar graphs G with minimum degree two that do not have two disjoint
independent dominating sets, and therefore such graphs G satisfy idom(G) = 1 and FIDOM(G) ≥ 2. It remains an open problem to determine a best possible lower bound on the fractional idiomatic number
of a planar graph with minimum degree 3. In this case, we believe that the upper bound of Theorem 15 can be improved from 2 to 52 . We pose this formally as a conjecture. Conjecture 1 If G is a
planar graph with δ(G) ≥ 3, then FIDOM(G) ≥ 52 . We note that if G = C5 2 K2 is the 5-prism illustrated in Figure 4, then i(G) = 4 = 25 n and FIDOM(G) = 52 . Thus, if Conjecture 1 is true, the bound
is best possible. Fig. 3 The planar graph H4 with i(H4 ) = 12 n(H4 ) and FIDOM(H4 ) = 2
Fig. 4 The 5-prism C5 2 K2
W. Goddard and M. A. Henning
The fractional idiomatic number of a maximal outerplanar graph is easy to compute. If G is a maximal outerplanar graph, then G is 3-colorable and every color class is an independent dominating set,
implying that idom(G) ≥ 3. Furthermore, since G has minimum degree 2, Theorem 10(b) implies that FIDOM(G) ≤ 3. Consequently, idom(G) = FIDOM(G) = 3 for a maximal outerplanar graph G, as observed in
[12]. If G is a general outerplanar graph, then G is 3-chromatic and, by Theorem 13, we therefore have FIDOM(G) ≥ 32 . This, in turn, implies by Theorem 10(a) that i(G) ≤ 23 n(G). If G is obtained
from K3 by attaching k ≥ 1 pendant edges to each vertex of the triangle, then the resulting graph G has order n(G) = 3(k + 1) and satisfies i(G) = 2k + 1. Thus, for k sufficiently large, the bounds i
(G) ≤ 23 n(G) and FIDOM(G) ≥ 32 are asymptotically sharp. If, however, we impose a minimum degree condition, then we can improve the lower bound of 32 on the fractional idiomatic number to a lower
bound of 2, as shown by the following result in [12]. Theorem 17 ([12]) If G is an outerplanar graph with minimum degree at least 2, then FIDOM(G) ≥idom(G) ≥ 2. Let G and H be two graphs. As above,
the join of G and H is written as G ⊕ H, while their disjoint union is written as G + H. The lexicographic product of G and H, written G[H], is the graph with vertex set V (G) × V (H), where two
vertices (g1 , h1 ) and (g2 , h2 ) are adjacent in G[H] if and only if g1 g2 ∈ E(G) or g1 = g2 and h1 h2 ∈ E(H). For these graph operations, the fractional idiomatic number behaves as follows.
Theorem 18 ([12]) For graphs G and H, the following hold: (a) FIDOM(G + H ) = min(FIDOM(G), FIDOM(H )). (b) FIDOM(G ⊕ H) = FIDOM(G) + FIDOM(H). (c) FIDOM(G[H]) = FIDOM(G) ×FIDOM(H). The following
result is shown in [11]. Theorem 19 ([11]) If G is a graph with minimum degree at least 2 and maximum degree at most 3, then idom(G) ≥ 2. As an immediate consequence of Theorem 19, every cubic graph
G satisfies idom(G) ≥ 2, a result attributed to Berge. We note that there are many cubic graphs G satisfying idom(G) = 2. We pose the following question. Question 1 Is it true that if G is a
connected cubic graph different from K3,3 , then FIDOM(G) ≥ 52 ? As observed earlier, if G = C5 2 K2 is the 5-prism shown in Figure 4, then FIDOM(G) = 52 . Hence, if Question 1 is true, then the
lower bound value 5/2 for the fractional idiomatic number would be best possible.
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
4 The Fractional Total Domatic Number In this section, we survey results on the fractional total domatic number of a graph. First, we recall the concept of the total domatic number of a graph. The
total domatic number of a graph G, denoted by tdom(G) and first defined by Cockayne, Dawes, and Hedetniemi [6], is the maximum number of total dominating sets into which the vertex set of G can be
partitioned. The parameter tdom(G) is equivalent to the maximum number of colors in a (not necessarily proper) coloring of the vertices of a graph, where every color appears in every open
neighborhood. Chen, Kim, Tait, and Verstraete [3] called this the coupon coloring problem. This parameter is now well studied. We refer the reader to Chapter 13 in the book [14] on total domination
in graphs for a survey of results on the total domatic number and to [10] for a recent paper on this topic. In this section, we consider a fractional version of the total domatic number of a graph.
The fractional total domatic number of a graph G, denoted FTD(G), is defined as FTD(G) = max
|F| , m(F)
where the maximum is taken over all families F of total dominating sets of G and where, as before, m(F) is the maximum number of times an element appears in F. (As before, the parameter should be
defined as the supremum, but as shown in Section 5, the supremum is always achieved.) The following trivial lower and upper bounds on the fractional total domatic number of a graph are established in
[10]. Theorem 20 ([10]) If G is an isolate-free graph, then tdom(G) ≤ FTD(G) ≤
n(G) . γt (G)
Theorem 21 ([10]) If a graph G of order n has minimum degree δ ≥ 1, then n ≤ FTD(G) ≤ δ. n−δ+1 Thus, for example, by Theorems 20 and 21, we get the following observation. Proposition 22 ([10]) The
following hold in a graph G with minimum degree δ. (a) If δ ≥ 1, then FTD(G) = 1. (b) If δ ≥ 2, then FTD(G) > 1. We note that there are graphs G with arbitrarily large minimum degree with FTD(G) < 1
+ ε for any given ε > 0. Indeed, these are the graphs that Zelinka [30] provided as examples that have tdom(G) = 1 and arbitrarily large minimum degree.
W. Goddard and M. A. Henning
A lower bound on the fractional total domatic number of a claw-free graph with minimum degree at least 2 is determined in [10]. Theorem 23 ([10]) If G is a claw-free graph with δ ≥ 2, then FTD(G) ≥
32 . The lower bound of Theorem 23 is in a sense best possible in that the graphs K3 and C6 have fractional total domatic number exactly 3/2. However, asymptotically the bound should be improvable.
As remarked in [10], perhaps it is true that if G is a connected, claw-free graph with δ ≥ 2, then FTD(G) ≥ 2 −o(1). Furthermore, if G is a connected, claw-free graph with δ ≥ 3, then maybe this
guarantees that FTD(G) ≥ 2. As shown in [10], there are arbitrarily large connected K1,4 -free graphs with fractional total domatic number exactly 3/2. We note that the union of two disjoint
dominating sets is a total dominating set. Thus, it is immediate that tdom(G) ≥dom(G)/2. But, in the case that the ordinary domatic number is odd, one can say slightly more: Theorem 24 ([10]) If G is
an isolate-free graph, then FTD(G) ≥ 12 dom(G). Proof. Let dom(G) = k, and let D1 , . . . , Dk be k disjoint dominating sets in the graph G. The family F = { Di ∪ D j | 1 ≤ i < j ≤ k } !" is a family
of k2 total dominating sets of G. Since every vertex of G appears in at most k − 1 sets in the family F, we note that m(F) = k − 1. Thus, !k " |F| k FTD(G) ≥ = 2 = . m(F) k−1 2 The desired result now
follows recalling that k = dom(G).
Equality in Theorem 24 occurs, for example, in complete graphs. We consider now planar graphs. A triangulated disc is a (simple) planar graph all of whose faces are triangles, except possibly for the
outer face. Matheson and Tarjan [21] showed that if G is a triangulated disc, then dom(G) ≥ 3. Hence, as an immediate consequence of Theorem 24 and the Matheson–Tarjan result, we have the following
lower bound on the fractional total domatic number of a triangulated disc. Theorem 25 ([10]) If G is a triangulated disc, then FTD(G) ≥ 32 . As remarked in [10], the lower bound of Theorem 25 is
tight as may be seen by considering the triangulated disc G illustrated in Figure 5, where the shaded area consists of any maximal planar graph (or, equivalently, triangulation). Let S be the set of
three vertices on the outer face of G that have degree at least 4. If F is an arbitrary family of total dominating sets of G, then each set in the family F contains at least two vertices of S. By
averaging, there is a vertex in S that belongs to at least 2|F|/3 sets in F, implying that m(F) ≥ 2|F|/3, or, equivalently, |F|/m(F) ≤ 3/2.
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
Fig. 5 A triangulated disc G with FTD(G) = 32
any triangulation
Since this is true for every family F of total dominating sets of G, this implies that FTD(G) ≤ 3/2. Consequently, by Theorem 25, FTD(G) = 3/2. We next consider triangulations, where by triangulation
we mean a simple graph embedded in some orientable surface such that every region is a triangle. The following key lemma establishes an upper bound on the fractional total domatic number of a graph G
in terms of its average degree, which we denote by dav (G). Lemma 26 ([10]) If G is a triangulation of order at least 4, then FTD(G) ≤ dav (G) − 1. If G is a planar triangulation of order n, then dav
(G) = 6 − 12 n . Thus, as an immediate consequence of Lemma 26, we have the following upper bounds on the total domatic and fractional total domatic numbers of a planar graph. Theorem 27 ([10]) The
following hold in a planar graph G. (a) tdom(G) ≤ 4. (b) FTD(G) ≤ 5 −
12 n.
As remarked in [10] there are planar graphs G with tdom(G) = 4. For example, if G is obtained from a truncated tetrahedron and adding a vertex inside each hexagonal face that is joined to all
vertices on the boundary, then G is a planar graph G of order 16 satisfying tdom(G) = 4. Illustrated in Figure 6 (which corrects a figure in [10]) is a spanning subgraph thereof that still has four
disjoint total dominating sets: the vertices labeled i form a total dominating set for each i ∈ [4]. As shown in [10], there are planar graphs G for which FTD(G) > 4. We note that the result of Lemma
26 applies on all surfaces. In particular, since the average degree of a toroidal graph is at most 6, this yields the following upper bounds on the total domatic and fractional total domatic numbers
of a toroidal graph. Theorem 28 If G is a toroidal graph, then tdom(G) ≤FTD(G) ≤ 5. There are toroidal graphs G satisfying tdom(G) = FTD(G) = 5. The example provided in [10] is illustrated in Figure
7, where the top and bottom dotted lines
W. Goddard and M. A. Henning
Fig. 6 A planar graph G with tdom(G) = 4
Fig. 7 A toroidal graph G with tdom(G) = 5
should be identified and similarly with the left and right dotted lines; the vertices labeled i form a total dominating set of G for each i ∈ [5]. It remains generally an open problem to determine
good lower bounds for triangulations. Since every planar triangulation is a triangulated disc, Theorem 25 implies that every planar triangulation G satisfies FTD(G) ≥ 32 . However, we believe this
lower bound can be improved. In this regard, the following conjectures are posed in [10]. Conjecture 2 ([10]) If G is a planar triangulation of order at least 4, then tdom(G) ≥ 2. Using the Four
Color Theorem, it is shown in [10] that Conjecture 2 is true if every vertex has odd degree or if the dual of G is Hamiltonian. By characterizing the maximal outerplanar graphs H that have tdom(H) <
2, Nagy [24] showed that the conjecture is true if G is Hamiltonian. A stronger version of the above is the following conjecture. Conjecture 3 ([10]) Every planar triangulation with at least four
vertices has a proper 4-coloring (C1 , C2 , C3 , C4 ) such that C1 ∪ C2 and C3 ∪ C4 are total dominating sets.
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
Fig. 8 The Heawood graph
As with the parameter FDOM, the parameter FTD is monotonic. The fractional total domatic number of the disjoint union of graphs behaves as expected. Theorem 29 ([10]) For all graphs G and H, it holds
that FTD(G + H ) = min{FTD(G), FTD(H )}. We conclude this section with some comments on regular graphs. It remains a long-standing open problem to characterize those cubic graphs that do not have two
disjoint total dominating sets; that is, the 3-regular graphs with tdom(G) = 1. It is well known that the Heawood graph, shown in Figure 8, is the smallest example of such a graph G without two
disjoint total dominating sets. Nevertheless, a natural question is whether the fractional total domatic number of a cubic graph is always at least 2. This question was answered in [16]. Theorem 30
([16]) If G is a connected cubic graph, then FTD(G) ≥ 2. Recall that Fujita, Yamashita, and Kameda proved in Theorem 7 the beautiful result that every connected cubic graph G has a family F of five
dominating sets such that every vertex is in at most two of these. The following strengthening of Theorem 30 is conjectured in [10]. Conjecture 4 ([10]) If G is a connected cubic graph, then G has a
family of four total dominating sets of G such that every vertex is in at most two of these. For regular graphs with higher minimum degree, we have the following lower bound on the fractional total
domatic number of a regular graph. Theorem 31 ([16]) For all k ≥ 3, if G is a k-regular graph, then FTD(G) >
k . 1 + ln(k)
5 Fractional Definitions and Hypergraphs The three parameters explored in this chapter can be defined in terms of hypergraphs. The fractional matching number of a hypergraph H, denoted ν ∗ (H), is
defined by the linear program
W. Goddard and M. A. Henning
w(e) such that ∀e ∈ E : w(e) ≥ 0 and ∀v ∈ V (H ) :
w(e) ≤ 1.
e∈E(H )
The matching number of H, denoted ν(H), is the maximum number of disjoint hyperedges. Clearly, ν ∗ (H) ≥ ν(H). Also, by linear programming duality, the fractional matching number equals the
fractional transversal/cover number. As before, for a multiset F of E(H), we define m(F) as the maximum number of times a vertex of H appears in F. Since the linear program has a rational solution, ν
∗ (H ) = sup
|F| |F| = max , m(F) m(F)
where the maximum and the supremum are over all such multisets F. See Chapter 1 of [25] for a fuller discussion. Consider a family D of subsets of the vertex set of a graph G. This set can naturally
be thought of as a hypergraph HD . Then, the matching number of HD is the maximum number of disjoint members of D. If we let D be the set of dominating sets, we get the domatic number and fractional
domatic number, discussed in Section 2. If we let D be the set of independent dominating sets, we get the idomatic number and fractional idomatic number, discussed in Section 3. If we let D be the
set of total dominating sets, we get the total domatic number and fractional total domatic number, discussed in Section 4. The results about the behavior of the three fractional parameters under
disjoint union are a special case of a result in hypergraphs. Given two disjoint hypergraphs H1 and H2 , we define their direct sum as the hypergraph with vertex set V (H1 ) ∪ V (H2 ) and edge set
{e1 ∪ e2 e1 ∈ E(H1 ), e2 ∈ E(H2 )}. Bujtás and Tuza [2] showed that the matching number of the direct sum of two hypergraphs equals the smaller of the two matching numbers. We note that the analogous
result is true for the fractional matching number too. Lemma 32 If hypergraph H is the direct sum of hypergraphs H1 and H2 , then ν ∗ (H ) = min(ν ∗ (H1 ), ν ∗ (H2 )). Proof. For ∈{1, 2}, let hi be
an optimal weighting of E(Hi ). Let Y max(ν ∗ (H1 ), ν ∗ (H2 )). Then define the weighting h of the direct sum H by
h(e1 ∪ e2 ) = h1 (e1 )h2 (e2 )/Y. For each vertex v ∈ V (H1 ), we have ev
h(e) =
e1 v
h1 (e1 )
h2 (e2 )/Y ≤
h1 (e1 ) ≤ 1.
e1 v
Similarly, the constraint is satisfied for v ∈ V (H2 ). And the total weight of h is ν ∗ (H1 )ν ∗ (H2 )/Y = min(ν ∗ (H1 ), ν ∗ (H2 )). It follows that
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
ν ∗ (H ) ≥ min(ν ∗ (H1 ), ν ∗ (H2 )). Conversely, let h be the optimal weighting for the direct sum H. Then, define the weighting h1 on hypergraph H1 by h1 (e1 ) =
It follows readily that for each vertex v ∈ V (H1 ), we have e1 v h1 (e1 ) ≤ 1. That is, the weighting h1 represents a fractional matching of H1 . Thus, ν ∗ (H1 ) ≥ ν ∗ (H). Analogously, ν ∗ (H2 ) ≥
ν ∗ (H). That is, ν ∗ (H ) ≤ min(ν ∗ (H1 ), ν ∗ (H2 )). The two inequalities combined give the desired result. 2 As a consequence the results on disjoint union follow for the fractional domatic,
independent domatic, and total domatic numbers given earlier.
6 More on Hypergraphs While the previous section described a general conversion from a particular domatic number of a graph to the matching number of an associated hypergraph, another hypergraph
provides a more general setting for the fractional domatic and fractional total domatic number. Recall that a subset T of vertices is a transversal (also called vertex cover or hitting set) in a
hypergraph H if T has a nonempty intersection with every edge of H. The transversal number τ (H) of H is the minimum size of a transversal in H. We denote by disjτ (H) the disjoint transversal number
of a hypergraph H, which is the maximum number of disjoint transversals in H. Analogous to the fractional total domatic number, one can define the fractional disjoint transversal number. The
fractional disjoint transversal number of H, denoted FDT(H), is defined as FDT(G) = max
|F| , m(F)
where the maximum is taken over all families F of transversals of H. Analogous to earlier results, we have the following bounds on the fractional disjoint transversal number. Theorem 33 ([10]) For
every isolate-free hypergraph H of order n, disjτ (H ) ≤ FDT(H ) ≤
n . τ (H )
For k ≥ 2, if H is the complete k-uniform hypergraph of order n, then τ (H) = n − k + 1, and so by Theorem 33, FDT(H) ≤ n/(n − k + 1). To prove that
W. Goddard and M. A. Henning
Fig. 9 The Fano plane F7
FDT(H) ≥ n/(n − k + 1), we consider the collection F ! of all" n − k + 1 element n transversals of H. subsets of V (H). The resulting family F is a family of n−k+1 !n−1" Each vertex is in n−k sets of
F, and so ! n " n |F| FDT(H ) ≥ = n−k+1 . !n−1" = m(F) n−k+1 n−k
Consequently, FDT(H) = n/(n − k + 1). Thus, the upper bound of Theorem 33 is achieved, for example, by the complete k-uniform hypergraph of order n. As another example, if F7 is the Fano plane,
illustrated in Figure 9, then τ (F7 ) = 3, and so by Theorem 33, FDT(F7 ) ≤ 7/3. If we take F to be the family consisting of the seven edges of F7 , then F is a family of transversals of F7 . Each
vertex belongs to exactly three sets of F, and so FDT(F7 ) ≥ |F|/m(F) = 7/3. Consequently, FDT(F7 ) = 7/3. Thus, the Fano plane achieves equality in the upper bound of Theorem 33. We note however
that disjτ (F7 ) = 1. Analogous to Lemma 32, we have the following result on the fractional disjoint transversal number of the disjoint union of hypergraphs. Theorem 34 ([10]) If H is the disjoint
union of isolate-free hypergraphs H1 and H2 , then FDT(H ) = min{FDT(H1 ), FDT(H2 )}. We describe next the interplay between the fractional (total) domatic number and the fractional disjoint
transversal number. The open neighborhood hypergraph, abbreviated ONH, of a graph G is the hypergraph ON (G) whose vertex set is V (G) and whose hyperedges are the open neighborhoods of vertices in
G. Thus, if H = ON (G), then V (H) = V (G) and E(H) = { NG (x)x ∈ V (G)}. As first observed by Thomassé and Yeo [27], a total dominating set in G is a transversal in ON (G) and conversely. Thus, the
transversal number of ON (G) is precisely the total domination number γ t (G). Similarly, the closed neighborhood hypergraph, abbreviated CNH, of a graph G is the hypergraph CN (G) whose vertex set
is V (G) and whose hyperedges are the closed neighborhoods of vertices in G. Again, a dominating set in G is a transversal in CN (G) and conversely. We state this connection formally as follows.
Proposition 35 ([10]) For every graph G, FDOM(G) = FDT(CN (G)) and for every isolate-free graph G, FTD(G) = FDT(ON (G)).
Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
As an example, if G is the Heawood graph, illustrated in Figure 8, then the ONH consists of two disjoint copies of the Fano plane F7 . Therefore, by Theorem 34 and Proposition 35, we have FTD(G) =
FDT(ON (G)) = FDT(F7 ∪ F7 ) = FDT(F7 ) = 7/3. In contrast, the Heawood graph does not have two disjoint total dominating sets; that is, tdom(G) = 1. Nevertheless, there is a general fractional lower
bound. Theorem 36 ([16]) If H is a 3-regular 3-uniform hypergraph, then FDT(H) ≥ 2, and this bound is tight. As regards tightness, it is shown in [13] that there are infinitely many (connected)
3-regular 3-uniform hypergraphs H satisfying τ (H ) = 12 n(H ), implying by Theorem 33 that each of these hypergraphs satisfies FDT(H) ≤ 2. The lower bound in Theorem 36 shows that FDT(H) ≥ 2.
Consequently, FDT(H) = 2 for these hypergraphs. As a consequence of a result due to Thomassen [28] and a relationship given in [15] between the total domatic number of a k-regular graph and
2-colorings of k-uniform k-regular hypergraphs, we have that if H is a 4-regular 4-uniform hypergraph, then FDT(H) ≥disjτ (H) ≥ 2. However, it is conjectured in [16] that this lower can be improved
when H is a 4-regular 4-uniform hypergraph. Conjecture 5 ([16]) If H is a 4-regular 4-uniform hypergraph, then FDT(H ) ≥
7 3.
We note that if Conjecture 5 is true, then it implies that every 4-regular graph G satisfies FTD(G) ≥ 73 . Using probabilistic arguments, the following lower bound on the fractional disjoint
transversal number of a k-regular k-uniform hypergraph was established in [16]. Theorem 37 ([16]) For all k ≥ 3, if H is a k-regular k-uniform hypergraph, then FDT(H ) >
k . 1 + ln(k)
Furthermore, this bound is essentially best possible as there exist k-regular kk uniform hypergraphs Hk with FDT(Hk ) ≤ ln(k) (1 + o(1)).
7 Conclusion In this chapter, we survey results on the fractional relaxation of the count of the maximum number of disjoint dominating, independent dominating, and total dominating sets in a graph.
We discuss a common framework in hypergraphs and show that the fractional domatic and fractional total domatic numbers of a graph can be placed in a more general hypergraph setting. We present the
main results known to date on the fractional domatic parameters and list several outstanding open problems and conjectures that have yet to be settled.
W. Goddard and M. A. Henning
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Dominator and Total Dominator Colorings in Graphs Michael A. Henning
AMS Subject Classification: 05C65, 05C69
1 Introduction A dominating set of a graph G is a set S ⊆ V (G) such that every vertex in V (G) S is adjacent to at least one vertex in S. The domination number of G, denoted by γ (G), is the minimum
cardinality of a dominating set of G. A dominating set of G of cardinality γ (G) is called a γ -set of G. A total dominating set, abbreviated a TD-set, of a graph G with no isolated vertex is a set S
⊆ V (G) such that every vertex in V (G) is adjacent to at least one vertex in S. The total domination number of G, denoted by γ t (G), is the minimum cardinality of a TD-set of G. A TD-set of G of
cardinality γ t (G) is called a γ t -set of G. Total domination is now well studied in graph theory. The literature on the subject of total domination in graphs has been surveyed and detailed in the
book [19]. A proper vertex coloring of a graph G is an assignment of colors (elements of some set) to the vertices of G, one color to each vertex, so that adjacent vertices are assigned distinct
colors. A proper vertex coloring whose colors are taken from a set of k colors, usually the set [k] = {1, 2, . . . , k}, is called a proper k-coloring. In a given coloring of G, a color class of the
coloring is a set consisting of all those vertices assigned the same color. The vertex chromatic number of G, denoted χ (G), is the smallest positive integer k for which G has a proper k-coloring. A
χ -coloring of G is a proper k-coloring of G that uses χ (G) colors. In what follows, we simply
M. A. Henning () Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: [email protected] © The Author(s), under exclusive license to
Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://doi.org/10.1007/978-3-030-58892-2_5
M. A. Henning
call a proper vertex coloring a coloring, and we refer to the vertex chromatic number as the chromatic number. In this chapter, we combine the concept of domination (total domination) in graphs with
the concept of colorings in graphs and study dominator colorings (respectively, total dominator colorings) of a graph. In Section 3, we formally define dominator colorings in graphs, and in Section
4, we formally define the analogous concept of total dominator colorings in graphs. In these sections, we present selected results on the so-called dominator chromatic number and total dominator
chromatic number of a graph.
2 Graph Theory Notation For completeness, we include some graph theory terminology that we will use in this chapter. A vertex of degree 1 is called a leaf, and its unique neighbor is called a support
vertex. Two vertices v and w are neighbors in a graph G if they are adjacent, that is, if vw ∈ E(G). The open neighborhood of a vertex v in G is the set of neighbors of v, denoted NG (v), whereas the
closed neighborhood of v is NG [v] = NG (v) ∪{v}. The open neighborhood of a set S ⊆ V (G) is the set of all neighbors of vertices in S, denoted NG (S), whereas the closed neighborhood of S is NG [S]
= NG (S) ∪ S. The S-private neighborhood of a vertex v ∈ S is defined by pnG (v, S) = {w ∈ V (G) : NG [w] ∩ S = {v}}. Thus, pnG (v, S) = NG [v] NG [S {v}]. We note that if v ∈pnG (v, S), then the
vertex v is isolated in the subgraph G[S]. A vertex outside the set S that belongs to the set pnG (v, S) is called an S-external private neighbor of v. If the graph G is clear from the context, we
omit the subscript G in the above definitions. For example, we write N[v] and N[S] rather than NG [v] and NG [S], respectively. We denote a complete graph on n vertices by Kn , and we denote a path
and cycle on n vertices by Pn and Cn , respectively. We denote a complete bipartite graph with partite sets of cardinality m and n by Km,n . A star is a graph K1,n for some n ≥ 1. A double star is a
tree with exactly two (adjacent) non-leaf vertices. If one of these vertices is adjacent to 1 leaves and the other to 2 leaves, then we denote the double star by S(1 , 2 ). By a nontrivial graph, we
mean a graph of order at least two. The corona cor(G) of a graph G, also denoted G ◦ K1 in the literature, is the graph obtained from G by attaching a leaf v to every vertex v of G. The 2-corona G ◦
P2 of G is the graph of order 3|V (G)| obtained from G by attaching a path of length 2 to each vertex of G so that the resulting paths are vertex-disjoint. Given a graph F, a graph G is F-free if it
does not contain any induced subgraph isomorphic to F. If G is K1,3 -free, then G is said to be claw-free. A graph is chordal if it contains no induced cycle of length 4 or more. A graph is a split
graph if its vertex set can be partitioned into a clique and an independent set. A universal vertex in a graph is a vertex that is adjacent to every other vertex in the graph. A clique in G is a
complete subgraph of G. The clique number of G, denoted ω(G), is the maximum cardinality of a clique in G.
Dominator and Total Dominator Colorings in Graphs
A set of vertices in a graph G is a packing if the vertices in S are pairwise at distance at least 3 apart, that is, if u and v are arbitrary distinct vertices in S, then d(u, v) ≥ 3. Equivalently, S
is a packing if the closed neighborhoods of vertices in S are pairwise disjoint. A subset S of vertices in a graph G is an open packing if the open neighborhoods of vertices in S are pairwise
disjoint. Further the set S is a perfect packing (respectively, a perfect open packing) if every vertex belongs to exactly one of the closed (respectively, open) neighborhoods of vertices in S. The
packing number ρ(G) (respectively, the open packing number ρ o (G)) is the maximum cardinality of a packing (respectively, open packing) in G. A vertex and an edge are said to cover each other in a
graph G if they are incident in G. A vertex cover in G is a set of vertices that covers all the edges of G. The vertex covering number τ (G) (also denoted by β(G) or vc(G) in the literature) is the
minimum cardinality of a vertex cover in G. The independence number α(G) of a graph G is the maximum cardinality of an independent set in G.
3 Dominator Colorings A vertex in a graph G dominates itself and all vertices adjacent to it. Further, a vertex is a dominator of a set S if it dominates every vertex in S. A dominator coloring of a
graph G is a proper coloring of G with the additional property that every vertex in V (G) dominates all vertices in at least one color class, that is, each vertex of the graph belongs to a singleton
color class or is adjacent to every vertex of some (other) color class. The dominator chromatic number χ d (G) of G is the minimum number of color classes in a dominator coloring of G. A χ d
-coloring of G is a dominator coloring of G that uses χ d (G) colors. The concept of a dominator coloring in a graph was birthed in the late 1970s when Cockayne, Hedetniemi, and Hedetniemi [9]
defined the domatic number of a graph involving partitions into dominating sets. In 2006, Hedetniemi, Hedetniemi, and McRae [14] further studied the concept of dominator colorings in graphs. (We
remark that these two papers are cited as [4] and [13], respectively, in the 2006 paper by Gera, Horton, and Rasmussen [13].) On March 15, 2004, Hedetniemi, Hedetniemi, Laskar, McRae, and Wallis [15]
submitted a paper on dominator partitions in graphs, but due to the backlog in the journal at the time, the paper only appeared 5 years later! In 2006, Gera et al. [13] published a paper on dominator
colorings in graphs, and in 2007, Gera [11, 12] continued the study of dominator colorings. Since every vertex is a dominator of itself, the coloring of G that assigns a unique color to each vertex
is a trivial dominator coloring of G. Thus, every graph G has a dominator coloring, and therefore the dominator chromatic number χ d (G) is well-defined. Since every dominator coloring of G is a
coloring of G, we have the following observation. Observation 1 For every graph G, we have χ (G) ≤ χ d (G).
M. A. Henning
Fig. 1 A χ d -coloring of a path P17
The simplest example to show that strict inequality may occur in Observation 1 is to take G to be a path P4 given by v1 v2 v3 v4 . We note that χ (G) = 2 and the unique 2-coloring of G has color
classes {v1 , v3 } and {v2 , v4 }. However, neither the vertex v1 nor v4 dominates any color class, implying that χ d (G) ≥ 3. However, the 3-coloring of G with color classes V1 = {v1 , v3 }, V2 =
{v2 }, and V3 = {v4 } is a dominator coloring of G, noting that the vertices v1 and v3 dominate the color class V2 , the vertex v2 dominates both color classes V1 and V2 , and the vertex v4 dominates
its own color class V3 . Thus, χ d (G) ≤ 3. Consequently, χ d (G) = 3. As shown by Theorems 2 and 3, the difference χ d (G) − χ (G) can be made arbitrarily large by taking, for example, G to be a
path Pn or cycle Cn of sufficiently large order n. We note that if G is a star K1,k where k ≥ 1, then a proper 2-coloring of G is also a dominator coloring of G, and so χ (G) = χ d (G) = 2. If G = Kn
, then χ (G) = χ d (G) = n. Hence, equality in Observation 1 is possible. Gera et al. [13] determined the dominator chromatic number of a path Pn on n vertices. We note that χ d (P2 ) = χ d (P3 ) =
2. As observed earlier, χ d (P4 ) = 3. It is a simple exercise to verify that χ d (P5 ) = 3. If G is the path Pn : v1 v2 . . . vn where n ≥ 6, let f : V (G) → {1, 2, . . . , 2 + n3 } be the dominator
coloring defined by ⎧ ⎪ ⎪ ⎨
1 when n (mod 6) ∈ {1, 3} 2 when n (mod 6) ∈ {0, 4} f (vi ) =
i ⎪ ⎪ ⎩ + 2 when n (mod 6) ∈ {2, 5}. 3 However if n ≡ 1 (mod 3), then we redefine f (vn ) to be the value n3 + 2. When n = 16, for example, the resulting dominator coloring is illustrated in Figure 1
where here f (v16 ) = 16 3 + 2 = 8 (and where color 1 is blue, color 2 is white, color 3 is green, etc.). Gera et al. [13] proved that the dominator coloring f defined above is a χ d coloring of the
path Pn . Theorem 2 ([13]) For n ≥ 2, we have χd (Pn ) =
1 + n3 if n ∈ {2, 3, 4, 5, 7} 2 + n3 otherwise.
Dominator and Total Dominator Colorings in Graphs
(a) χd (C5) = 3
(b) χd (P5) = 3
Fig. 2 χ d -coloring of C5 and P5
In 2007, Gera [12] determined the dominator chromatic number of a cycle Cn on n vertices. Theorem 3 ([12]) We have χ d (C3 ) = 3, χ d (C4 ) = 2, and χ d (C5 ) = 3, while for n ≥ 3 and n∈{4, 5}, we
have χd (Cn ) =
n 3
+ 2.
We note that if H is a spanning proper subgraph of G, then a χ d -coloring of G may not be a dominator coloring of H. As a simple example, the χ d -coloring of the cycle C5 shown in Figure 2(a) is
not a dominator coloring of P5 , even though χ d (C5 ) = χ d (P5 ) = 3. For disconnected graphs, we have the following upper and lower bounds on the dominator chromatic number. Theorem 4 ([13]) If G
is a disconnected graph with components G1 , G2 , . . . , Gk where k ≥ 2, then k − 1 + max {χd (Gi ) | i ∈ [k]} ≤ χd (G) ≤
χd (Gi ).
Proof Let Ci be a χ d -coloring of Gi for all i ∈ [k], where we can choose the colorings so that no two color classes uses the same color. Let C be the union of these k color classes, and so the
restriction of C to the component Gi yields the χ d coloring Ci for all i ∈ [k]. The coloring C is a chromatic dominator coloring of G, and so χd (G) ≤ |C| =
k i=1
|Ci | =
k i=1
χd (Gi ) =
χd (Gi ).
To prove the lower bound, consider a component of G with largest dominator chromatic number. Each of the remaining k − 1 components of G requires at least one additional color, since every vertex
must be a dominator of some color class. Hence, χd (G) ≥ k − 1 + max {χd (Gi ) | i ∈ [k]}. That the lower bound of Theorem 4 is tight may be seen by taking G to be the vertex-disjoint union of k ≥ 2
stars K1,n , for some n ≥ 2. Each component H of G has
M. A. Henning
χ d (H) = 2. Assigning to all leaves of G the same color and assigning to the central vertex of each of the k stars a unique color produce a dominator coloring of G using k + 1 = (k − 1) + 2 = k − 1
+ max {χd (H ) | H is a component of G} colors. That the upper bound of Theorem 4 is tight may be seen by taking G to be the vertex-disjoint union of k ≥ 2 copies of K2,n for some n ≥ 2. Let C be a
dominator coloring of G. Since every vertex must be a dominator of some color class, we note that each component of G has at least one color not used in any other color class. Suppose that some
component H of G uses exactly one color, say color 1, not used in any other color class. If two or more vertices in H are colored 1, then no vertex in the color class associated with the color 1 is a
dominator of any color class, a contradiction. Hence, exactly one vertex in H is colored 1. However, every vertex different from v and not adjacent to v in the component H is therefore not a
dominator of any color class, a contradiction. Hence, each component of Ghas at least two colors unique to that component, implying that χd (G) ≥ 2k = ki=1 χd (Gi ), where G1 , G2 , . . . , Gk denote
the components of G.
3.1 Bounds on the Dominator Chromatic Number By definition of a dominator coloring, we have the following observation. Observation 5 If v is an arbitrary vertex in a graph G, then in every dominator
coloring of G, the closed neighborhood N[v] of v contains a color class. Theorem 6 If G is a graph, then χ d (G) ≥ ρ(G), with strict inequality if there is no perfect packing in G. Proof If S is a
packing in G, then by Observation 5, the closed neighborhoods of vertices in S contain at least |S| color classes, and so χ d (G) ≥|S|. Choosing S to be a maximum packing, we have that χ d (G) ≥ ρ
(G). Further, if G does not have a perfect packing, then at least one additional color class is needed to contain the vertices that do not belong to the closed neighborhood of any vertex in S, and so
χ d (G) ≥ ρ(G) + 1. The dominator chromatic number of a graph is related to its independence number as follows, where the independence number α(G) of a graph G is the maximum cardinality of an
independent set in G. Theorem 7 ([13, 15]) If G is a connected graph of order n, then χ d (G) ≤ n + 1 − α(G). Proof If n = 1, then the result is trivial since in this case χ d (G) = n = α(G) = 1.
Hence, we may assume that n ≥ 2. Let I be a maximum independent set in G, and consider the coloring C that colors all vertices in I with the same color, and colors all remaining n − α(G) vertices
each with a different color. Each vertex in V (G) I dominates the color class that contains it, noting that it is the unique vertex in that color class. By the connectivity of G and by the
independence of the set I, every vertex in I has degree at least 1 and has all of its neighbor in V (G) I. Therefore,
Dominator and Total Dominator Colorings in Graphs
(a) χ(G) = 2
(a) χd (G) = 3
Fig. 3 A double star G = S(3, 3)
by our choice of the coloring C, every vertex in I dominates every color class that contains one of its neighbors. Hence, C is a dominator coloring of G, implying that χd (G) ≤ |C| = n + 1 − |I | = n
+ 1 − α(G). That the bound of Theorem 7 is sharp may be seen by taking, for example, a double star G = S(1 , 2 ). We note that the (unique) proper 2-coloring of the double star is a dominator
coloring of G, since no leaf dominates a color class. Hence, χ d (G) ≥ 3. However, coloring all the leaves with one color, and coloring the two central vertices (the non-leaf vertices) with distinct
colors, produces a proper 3coloring that is a dominator coloring. Hence, χ d (G) = 3. In this example, G has order n = 1 + 2 + 2 and α(G) = 1 + 2 = n − 2, and so χ d (G) = n + 1 − α(G). In the
special case when G = S(3, 3), we illustrate the χ -coloring and χ d -coloring of G in Figure 3(a) and 3(b), respectively. As observed earlier, the coloring of a graph G of order n that assigns a
unique color to each vertex is a trivial dominator coloring of G, and so χ d (G) ≤ n. By Observation 1, if G is a connected graph on at least two vertices, then χ d (G) ≥ χ (G) ≥ 2. We state these
observations formally as follows. Observation 8 If G is a connected graph of order n ≥ 2, then 2 ≤ χ d (G) ≤ n. A characterization of graphs achieving equality in the lower and upper bounds of
Observation 8 is given by the following result. Theorem 9 ([11, 15]) If G is a connected graph of order n ≥ 2, then the following holds. (a) χ d (G) = 2 if and only if G is a complete bipartite
graph. (b) χ d (G) = n if and only if G is a complete graph. Proof Suppose that χ d (G) = 2. By Observation 1, χ (G) = 2, implying that the 2coloring of G is a dominator coloring of G. Let V1 and V2
be the two color classes of G. If |Vi | = 1 for some i ∈ [2], then G = K1,n−1 , and the desired result follows. Hence, we may assume that |Vi |≥ 2 for i ∈ [2]. Thus, no vertex can be a dominator of
its own color class, implying that every vertex in Vi is a dominator of the color class V3−i for i ∈ [2], that is, G = Kn1 ,n2 where ni = |Vi |. Hence if χ d (G) = 2, then G is a complete bipartite
graph. The converse is immediate.
M. A. Henning
Suppose next that χ d (G) = n. By Theorem 7, χ d (G) ≤ n + 1 − α(G). If G is not a complete graph, then α(G) ≥ 2, implying that χ d (G) ≤ n − 1, a contradiction. Hence, G must be a complete graph.
The converse is immediate. The dominator chromatic number of a graph is related to its domination number. For a given graph G, let A(G) denote the set of all γ -sets in G. We next present an upper
bound on the dominator chromatic number of a graph. Theorem 10 If G is a connected graph, then χd (G) ≤ γ (G) + min {χ (G − S)}, S∈A(G)
and this bound is tight. Proof Let S be an arbitrary γ -set of G, and let C be a proper coloring of the graph G − S using χ (G − S) colors. We extend the coloring C to a coloring of the vertices of G
by assigning to each vertex in S a new and distinct color. Let C denote the resulting coloring of G, and note that C uses γ (G) + χ (G − S) colors. Since S is a dominating set of G, every vertex in V
(G) S is adjacent to at least one vertex of S. Since the color class of C containing a given vertex of S consists only of that vertex, each vertex in V (G) S is adjacent to every vertex of some color
class in the coloring C . Further, each vertex of S is a dominator of its own (singleton) color class. Hence, C is a dominator coloring of G using γ (G) + χ (G − S) colors. This is true for every γ
-set of G. The desired upper bound now follows by choosing S to be a γ -set of G that minimizes χ (G − S). The bound is achieved, for example, by taking G to be a complete graph. The proof of Theorem
10 yields the following more general result. Theorem 11 If G is a connected graph, and D(G) denotes the set of all dominating sets of G, then χd (G) ≤ min { |S| + χ (G − S) }. S∈D(G)
Gera [11, 12] established the following upper and lower bounds on the dominator chromatic number of an arbitrary graph in terms of its domination number and chromatic number. Theorem 12 ([11, 12])
Every graph G satisfies max{γ (G), χ (G)} ≤ χd (G) ≤ γ (G) + χ (G). Proof By Observation 1, recall that χ (G) ≤ χ d (G). To show that γ (G) ≤ χ d (G), consider a χ d -coloring of G with color classes
V1 , . . . , Vk , where k = χ d (G). Let vi be an arbitrary vertex in the color class Vi for i ∈ [k], and consider the set D = {v1 , . . . , vk }. Let v be an arbitrary vertex of G. By definition of
a dominator coloring, the vertex v is a dominator of the color class Vi for at least one i ∈ [k]. In particular,
Dominator and Total Dominator Colorings in Graphs
the vertex v = vk or the vertex v is adjacent to the vertex vk . This is true for every vertex v of G, implying that D is a dominating set of G. Hence, γ (G) ≤|D| = χ d (G). This establishes the
desired lower bound. The upper bound follows from Theorem 10, noting that χ (G − S) ≤ χ (G) for every proper subset S ⊂ V (G). Gera [12] established an intermediate value-type result for the
dominator chromatic number and showed that for every triple (a, b, c) of integers where 1 ≤ a ≤ c and 2 ≤ b ≤ c is a dominator realizable triple, there exists a connected graph G such that γ (G) = a,
χ (G) = b, and χ d (G) = c. That the lower bound of Theorem 12 is sharp may be seen by taking, for example, a complete bipartite graph G with both partite sets of cardinality at least 2. In this
case, γ (G) = χ (G) = χ d (G) = 2. To see that the upper bound is sharp, let G, for example, be a path Pn or a cycle Cn for some n ≥ 8 even. In this case, γ (G) = n3 and χ (G) = 2, and so by Theorems
2 and 3, we have χd (G) = 2 + n3 = χ (G) + γ (G).
3.2 Special Classes of Graphs In this section, we consider the dominator chromatic number of certain classes of graphs.
Bipartite Graphs
As a special case of Theorem 12 when G is a bipartite graph, we have the following result. Theorem 13 ([11, 12, 15]) If G is a bipartite graph, then γ (G) ≤ χ d (G) ≤ γ (G) + 2. In order to
characterize the graphs achieving equality in the lower bound of Theorem 13, we define a special subclass of bipartite graphs as follows. Definition 1 A bipartite graph G is a partially complete
bipartite graph if G can be obtained from the disjoint union of k ≥ 1 complete bipartite graphs Kxi ,yi with partite sets Xi and Yi where xi = |Xi |≥ 2 and yi = |Yi |≥ 2 for all i ∈ [k] by adding
edges between copies of these graphs so that the resulting graph is connected and the following conditions hold, where X = ∪ki=1 Xi and Y = ∪ki=1 Yi . (a) For each set Xi where i ∈ [k], there is no
set A ⊆ Y Yi such that |A ∩ Yj | = 1 for all j ∈ [k] {i} and the set A dominates the set Xi . (b) For each set Yi where i ∈ [k], there is no set A ⊆ X Xi such that |A ∩ Xj | = 1 for all j ∈ [k] {i}
and the set A dominates the set Yi . (c) For each set Xi where i ∈ [k], if A ⊆ Xi dominates of the partite sets in Y , then ≥|A|.
M. A. Henning
(d) For each set Yi where i ∈ [k], if A ⊆ Yi dominates of the partite sets in X, then ≥|A|. We note, for example, that every complete bipartite graph with both partite sets of cardinality at least 2
is a partially complete bipartite graph. In particular, K2,n is a partially complete bipartite graph for all n ≥ 2. Theorem 14 ([11]) If G is a connected bipartite graph of order at least 2, then γ
(G) = χ d (G) if and only if G is a partially complete bipartite graph. Let Qn be the n-dimensional hypercube, and so Qn can be represented as the nth power of K2 with respect to the Cartesian
product operation 2, that is, Q1 = K2 and Qn = Qn−1 2 K2 for n ≥ 2. Gera [11] established the following upper bound on the dominator chromatic number of an n-dimensional hypercube. The proof given in
[11] is algorithmic in nature. Theorem 15 ([11]) For n ≥ 2, χ d (Qn+1 ) ≤ χ d (Qn ) + γ (Qn ). The following result established an upper bound on the dominator chromatic number of a connected
bipartite graph in terms of its order. Theorem 16 ([11]) If G is a connected bipartite graph of order n ≥ 2, then χd (G) ≤ 12 (n + 2), and this bound is sharp. Proof Let X and Y be the partite sets
of G, where |X|≤|Y |. Coloring all vertices in Y with the same color and assigning a new color to each vertex of X produce a dominator coloring of G using |X| + 1 ≤ 12 n + 1 colors. This establishes
the desired upper bound. That this bound is sharp may be seen by taking G to be the corona of an arbitrary connected bipartite graph F, and so G = cor(F). The graph G has order n = 2|V (F)| and
satisfies γ (G) = |V (F)|. Coloring all added vertices of degree 1 with the same color and assigning a new color to every vertex of F produce a dominator coloring of T using |V (F)| + 1 colors. Thus,
χ d (G) ≤|V (F)| + 1. We note that each added vertex v of degree 1 either dominates its own class, in which case the vertex v is the only vertex of that color, or dominates the class of its unique
neighbor, in which case its neighbor in F is the only vertex of that color. This implies that at least |V (F)| vertices must receive a unique color. Since at least one additional color is needed for
the remaining vertices of G, every dominator coloring of G uses at least |V (F)| + 1 colors. Thus, χ d (G) ≥|V (F)| + 1. As observed earlier, χ d (G) ≤|V (F)| + 1. Consequently, χd (G) = |V (F )| + 1
= 12 n + 1. 3.2.2
Since no tree is a partially complete bipartite graph, we have the following consequence of Theorems 13 and 14. Theorem 17 ([11]) If T is a tree of order n ≥ 2, then χ d (T) = γ (T) + 1 or χ d (T) =
γ (T) + 2.
Dominator and Total Dominator Colorings in Graphs
By Theorem 2, if T is a path Pn where n ≥ 8, then χd (T ) = n3 + 2 = γ (T ) + 2. We note that if T is obtained from k ≥ 1 vertex-disjoint copies of K1,r where r ≥ 2 by adding a new vertex and joining
it to the central vertex of each star, then χ d (T) = k + 1 = γ (T) + 1. More generally, if a tree T contains a γ -set D such that V (T) D is an independent set, then χ d (T) = γ (T) + 1, noting that
we can color all vertices outside D with the same color and assign a new color to every vertex of D to produce a minimum dominator coloring of T using γ (T) + 1 colors. In particular, we note that
both values for the dominator chromatic number in Theorem 17 are achievable for infinitely many trees. We say that a tree belongs to dominator class i if χ d (T) = γ (T) + i for i ∈ [2]. It remains
an open problem to characterize the dominator class 1 and dominator class 2 trees. A sufficient condition for a tree to belong to dominator class 1 is the following. Proposition 18 ([5, 15]) If T is
a nontrivial tree such that γ (T) = τ (T), then T belongs to dominator class 1. Proof Let D be a minimum vertex cover in T, and so |D| = τ (T) = γ (T). Since D is a vertex cover, the set V D is an
independent set. Coloring all vertices in V D with the same color and assigning a new color to each vertex of D produce a dominator coloring of T using |D| + 1 = γ (T) + 1 colors. Thus, χ d (T) ≤ γ
(T) + 1. By Theorem 17, χ d (T) ≥ γ (T) + 1. Consequently, χ d (T) = γ (T) + 1. As observed in [5, 15], the converse of Proposition 18 is not true in general. For example, the tree T shown in Figure
4 belongs to dominator class 1, noting that χ d (T) = 5 = γ (T) + 1. However, γ (T) = 4 < 5 = τ (T). The 5-coloring shown in Figure 4 is a χ d -coloring of the tree T. In 2012, Boumediene Merouane
and Chellali [4] provide a characterization of trees that belongs to dominator class 1. Theorem 19 ([4]) If T is a nontrivial tree, then χ d (T) = γ (T) + 1 if and only if there exists a γ -set, D,
of T such that the set V (T) (D ∪ N(A)) is an independent set where A = {v ∈ D : pn(v, D) = {v}}, that is, A is the set of vertices in D, if any, that are isolated in T[D] and have no D-external
private neighbor. In practice, a tree may admit many minimum dominating sets, and it may not be easy to identify such a set satisfying the statement of Theorem 19. Therefore, in 2015, Boumediene
Merouane and Chellali [5] provide a different characterization, which is more pleasing in the sense that it resulted in a polynomial time algorithm Fig. 4 A χ d -coloring of a tree T
M. A. Henning
(a) The tree T
(b) The vertex cover X in the forest F = T − N[B]
Fig. 5 A tree T and its associated forest F
for computing the dominator chromatic number for every nontrivial tree. In order to state this result, we need some additional terminology. Recall that a leaf of a tree is a vertex of degree 1 and a
vertex with a leaf neighbor is a support vertex. Given a tree T, let L and S be the set of leaves and support vertices of T, respectively. Further, let A be the set of vertices of T that are neither
leaves nor support vertices, but have a support vertex as a neighbor, that is, if v ∈ A, then v∈L ∪ S, but the vertex v is adjacent to a vertex in S. Further, let B be the set of vertices that have
all their neighbors in A but do not belong to A, that is, B = {v ∈ V A : N(v) ⊆ A}. Let F be the forest obtained from T by deleting all vertices in N[B], that is, F = T − N[B], or, equivalently, F is
the subgraph of T induced by the set V (T) N[B]. Let X be a minimum vertex cover of F containing all support vertices of T (if X contains a leaf of T, we simply replace this leaf by its neighbor in
T). To illustrate these definitions, consider the tree T shown in Figure 5(a). The label of each vertex represents one of the sets, namely, L, S, A, or B, that it belongs to, as shown in Figure 5(a).
The associated forest F = T − N[B] is illustrated in Figure 5(b), where the vertices in the vertex cover X are given by the darkened vertices. We are now in a position to state the characterization
of trees that belong to dominator class 1 as given in [5]. Theorem 20 ([5]) If T is a nontrivial tree, then χ d (T) = γ (T) + 1 if and only if the following three conditions hold. (a) B is a packing.
(b) B ∪ X is a γ -set of T. (c) γ (F) = τ (F).
Dominator and Total Dominator Colorings in Graphs
Fig. 6 A χ d -coloring of the tree T
Suppose that T is a nontrivial tree satisfying the three conditions in the statement of Theorem 20. We color the vertices of T as follows. • For each vertex v in B, we give a unique color to all
vertices in N(v). • To each vertex in X, we give a unique (new) color. • To all remaining vertices (including all vertices in L ∪ B), we give the same, but new, color. By condition (b), the set X ∪ B
is a γ -set of T, and so γ (T) = |B| + |X|. Thus, the resulting coloring is a dominator coloring of T using |B| + |X| + 1 = γ (T) + 1 colors. Hence, γ (T) + 1 ≤ χ d (T) ≤|B| + |X| + 1 = γ (T) + 1.
Thus we must have equality throughout this inequality chain, implying that χ d (T) = γ (T) + 1. To illustrate this coloring, consider the tree T shown earlier in Figure 5(a), where a vertex is
labelled B or X if it belongs to the set B or X, respectively. To the one vertex in B, we color its two neighbors green, and to the other vertex in B, we color its two neighbors yellow. We color the
five vertices in X with five new colors, namely, red, white, black, orange, and pink. Thereafter, we color all remaining vertices of T with a new color, namely, blue. The resulting coloring,
illustrated in Figure 6, is a dominator coloring of T using |B| + |X| + 1 = γ (T) + 1 = 8 colors and is therefore a χ d -coloring of T. Based on Theorem 20, the authors in [5] give a quadratic time
algorithm computing the dominator chromatic number of any nontrivial tree.
Chordal Graphs and Split Graphs
By Theorem 12, every graph G satisfies χ d (G) ≥ γ (G). In 2012, Chellali and Maffray [6] improved this bound by imposing certain structural restrictions on the graph. Theorem 21 ([6]) If G is a
connected graph of order n ≥ 2 that is C4 -free or is claw-free and different from C4 , then χ d (G) ≥ γ (G) + 1. Since every chordal graph is C4 -free, as is every split graph, as an immediate
consequence of Theorem 21, we have the following result. Corollary 22 ([6]) If G is a connected graph of order n ≥ 2, then the following holds.
M. A. Henning
(a) If G is a chordal graph, then χ d (G) ≥ γ (G) + 1. (b) If G is a split graph, then χ d (G) ≥ γ (G) + 1. We also remark that Theorem 17 follows immediately from Theorem 21, noting that every tree
is, of course, C4 -free. Chellali and Maffray [6] characterized the split graphs that achieve equality in the bound of Corollary 22(b). Theorem 23 ([6]) If G is a connected split graph whose vertex
set is partitioned into a clique Q and an independent set I such that Q is minimal, then χ d (G) = γ (G) + 1 if and only if every vertex of Q is a support vertex.
Proper Interval Graphs and Block Graphs
In 2015, Panda and Pandey [32] study bounds on the dominator chromatic number for two important subclasses of chordal graphs, namely, proper interval graphs and block graphs. A graph G is an interval
graph if there exists a one-to-one correspondence between its vertex set and a family F of closed intervals in the real line, such that two vertices are adjacent if and only if their corresponding
intervals intersect. Further, if no interval in F contains another interval in F, then the graph G is called a proper interval graph. Panda and Pandey [32] establish the following lower and upper
bounds for the dominator chromatic number of a proper interval graph in terms of its domination number and chromatic number. We note that the upper bound is a restatement of the result in Theorem 12.
Theorem 24 ([32]) Every proper interval graph G satisfies χ (G) + γ (G) − 2 ≤ χd (G) ≤ γ (G) + χ (G). Moreover, all three values can be achieved by χ d (G). For a vertex v of G, the graph G − v is
the graph obtained from G by deleting v and deleting all edges of G incident with v. A vertex v is a cut-vertex of G if the number of components increases in G − v. A block of a graph G is a maximal
connected subgraph of G that has no cut-vertex of its own. Thus, a block is a maximal 2-connected subgraph of G. The number of vertices in a block is called the order of the block. Any two blocks of
a graph have at most one vertex in common, namely, a cut-vertex. If a connected graph contains a single block, we call the graph itself a block. A block graph is a connected graph in which every
block is a clique. A block containing exactly one cut-vertex is called an end block. A non-complete block graph has at least two end blocks. Panda and Pandey [32] generalized the result of Theorem 17
to the class of block graphs. (We note that every tree is a block graph, in which every block is a copy of K2 .) Theorem 25 ([32]) If G is a block graph of order at least 2 with k blocks where each
block has the same order, then
Dominator and Total Dominator Colorings in Graphs
χd (G) = γ (G) + χ (G) − 1
χd (G) = γ (G) + χ (G).
Further, both values can be achieved by χ d (G). To illustrate the tightness of the bounds, for k ≥ 1, let Gk,1 be a block graph with 2k blocks B1 , B2 , . . . , B2k , each having order s ≥ 3, and 2k
− 1 cut-vertices v1 , v2 , . . . , v2k−1 such that V (Bi ) ∩ V (Bi+1 ) = {vi } for all i ∈ [2k − 1]. The resulting graph G = Gk,1 satisfies γ (G) = k, χ (G) = s and χ d (G) = k + s − 1. When k = 3
and s = 3, the block graph Gk,1 is illustrated in Figure 7. We color each vertex of the γ -set, {v1 , v3 , v5 }, of G3,1 with a unique color (namely, green, yellow, and pink), and we 2-color the
remaining vertices with two new colors (namely, blue and red). The resulting 5-coloring is a χ d -coloring of G. For k ≥ 3, let G = Gk,2 be a block graph with 2k + 1 blocks, B1 , B2 , . . . , B2k+1 ,
each having order k, and 2k cut-vertices v1 , v2 , . . . , v2k . All vertices in block B2k+1 are cut-vertices, say vk+1 , vk+2 , . . . , v2k . For each i ∈ [k], Bi is an end block, having exactly one
cut-vertex vi . For each j where k + 1 ≤ j ≤ 2k, block Bj has exactly two cut-vertices vj and vj−k . The resulting graph G = Gk,2 satisfies γ (G) = k, χ (G) = k and χ d (G) = 2k. When k = 3, the
block graph Gk,2 is illustrated in Figure 8. We color each vertex of the γ -set, {v1 , v2 , v3 }, of G3,1 with a unique color (namely, green, yellow, and pink), and we 3-color the remaining vertices
with three new colors (namely, blue, red, and black). The resulting 6-coloring is a χ d -coloring of G. As a consequence of Theorem 25, we have the following result. We note that if G is a block
graph, then the clique number ω(G) of G is the maximum order among all blocks in G. Corollary 26 ([32]) If G is a non-complete block graph that contains an end block of order ω(G), then χ d (G) = γ
(G) + χ (G) − 1 or χ d (G) = γ (G) + χ (G). Panda and Pandey [32] characterize the block graphs G for which one of the end blocks is of maximum size (namely, ω(G)) and χ d (G) = γ (G) + χ (G) − 1.
P4 -Free Graphs
Chellali and Maffray [6] determined the dominator chromatic number of the class of graphs that are P4 -free by exploiting the structure of these graphs, namely, that if
B2 v1
Fig. 7 A block graph G3,1
B3 v2
B4 v3
B5 v4
B6 v5
M. A. Henning
B3 v3 B6 v6 B1
B4 v1
B7 v4
B5 v5
B2 v2
Fig. 8 A block graph G3,2
G is a P4 -free graph of order at least 2, then G or its complement G is disconnected (see Seinsche [34]). We mention that P4 -free graphs are also known as cographs. Theorem 27 ([6]) If G is a P4
-free graph, then the following holds. (a) If G is connected, then χ d (G) = χ (G). (b) If G is disconnected with k ≥ 2 components and h of these components have a universal vertex, then either G has
a component H with a universal vertex and satisfies χ (G) = χ (H), in which case χ d (G) = χ (G) + 2k − h − 1, or G has no such component, in which case χ d (G) = χ (G) + 2k − h − 2.
Other Classes
We mention that the dominator chromatic number of other classes of graphs has also been studied, including degree splitting graph of some graphs [22], dragon and lollipop graphs [30], wheel related
graphs [35], the generalized Petersen graph [31], and Mycielskian graphs [1]. However, we do not define these classes of graphs here.
3.3 Graph Products In this section, we present some results on the dominator chromatic number in Cartesian products of graphs. The Cartesian product G 2 H of graphs G and H is the graph with vertex
set V (G) × V (H) = {(g, h) : g ∈ V (G) and h ∈ V (H)}, where two vertices (g1 , h1 ) and (g2 , h2 ) in the Cartesian product G 2 H of graphs G and H are adjacent if either g1 = g2 and h1 h2 is an
edge in H or h1 = h2 and g1 g2 is an edge in G.
Dominator and Total Dominator Colorings in Graphs
Fig. 9 A χ d -coloring of P2 2 P4
Fig. 10 The circular ladder graph CL8
In 2017, Chen, Zhao, and Zhao [8] determined the dominator chromatic number of Cartesian products of certain paths and cycles. The Cartesian product Pm 2 Pn of paths Pm and Pn is known as a 2 × n
grid graph. Theorem 28 ([8]) χd (P2 2 P2 ) = 2, χd (P2 2 P3 ) = χd (P2 2 P3 ) = 4, and for all n ≥ 5,
n + 3. χd (P2 2 Pn ) = 2 A dominator coloring of the 2 × 4 grid graph, for example, using four colors is shown in Figure 9. The Cartesian product of a cycle Cn on n ≥ 3 vertices and a path P2 on two
vertices is called a circular ladder graph CLn of order 2n; that is, CLn = Cn 2 K2 (cf. [23]). A circular ladder graph is also called a cycle prism in the literature. We note that CLn is bipartite if
and only if n is even. The circular ladder graph CLn is also called the n-prism in the literature. For example, let G = C8 2 K2 be the circular ladder graph CL8 shown in Figure 10. We note that G is
a bipartite graph and γ (G) = 4, and so, by Theorem 13, χ d (G) ≤ 6. As shown in the proof of Theorem 12, we can find a dominator coloring of G using six colors as follows. We first 2-color the
vertices of G with the colors 1 and 2 (depicted as the colors blue and red in Figure 10), and thereafter we recolor the vertices of a γ -set of G with the colors 3, 4, 5, and 6 (depicted as the
colors green, yellow, white, and black, respectively, in Figure 10). The resulting 6-coloring is a dominator coloring of G. Thus, χ d (G) ≤ 6. Moreover, as shown in the proof of Theorem 29, χ d (G) ≥
γ (G) + 2 = 6. Consequently, χ d (CL8 ) = 6. In 2015, Manjula and Rajeswari [29] claimed to have proven that χ d (CLn ) = n + 1 for all n ≥ 9. This result is incorrect. The correct value for the
dominator chromatic
M. A. Henning
Fig. 11 A χ d -coloring of CL3 = P2 2 C3
number of a circular ladder graph is given in Theorem 29 by Chen, Zhao, and Zhao [8]. Theorem 29 The dominator chromatic number of the circular ladder graph CLn = Cn 2 K2 is given by χ d (CL3 ) = 3
and for all n ≥ 4 as follows. ⎧1 ⎨ 2 (n + 4) when n ≡ 0 (mod 4) χd (CLn ) = 12 (n + 5) when n (mod 4) ∈ {1, 3} ⎩1 2 (n + 6) when n ≡ 2 (mod 4). Proof Let G = Cn 2K2 be the circular ladder graph CLn
where n ≥ 3. A dominator coloring of CL3 using three colors is shown in Figure 11, showing that χ d (CL3 ) ≤ 3. By Observation 1, χ d (CL3 ) ≥ χ (CL3 ) = 3. Consequently, χ d (CL3 ) = 3. Hence in
what follows, we let n ≥ 4. Let x1 x2 . . . xn x1 and y1 x2 . . . yn y1 be the two disjoint copies of the cycle Cn used to construct CLn = Cn 2 K2 and where xi yi ∈ E(CLn ) for i ∈ [n]. We note that
γ (G) = n/2 + n/4−n/4, that is, γ (G) = n/2 if n ≡ 0 (mod 4), γ (G) = n/2 + 1 if n ≡ 2 (mod 4), and γ (G) = (n + 1)/2 if n (mod 4) ∈{1, 3}. We show firstly that χ d (G) ≤ γ (G) + 2. If n is even,
then we can apply Theorem 13 to yield χ d (G) ≤ γ (G) + 2. Suppose that n ≡ 3 (mod 4). Thus, n = 4k + 3 for some k ≥ 0. In this case, the set D=
{x4i+1 , y4i+3 }
is a γ -set of D, noting that D is a dominating set of G and |D| = 2(k + 1) = (n + 1)/2 = γ (G). We note that removing the set D from G produced a graph G − D∼ =P6k+4 . We now 2-color the vertices of
the path G − D, and thereafter we color each vertex of D with a unique color. The resulting coloring of G is a dominator coloring of G using γ (G) + 2 = 2k + 4 colors. Suppose next that n ≡ 1 (mod
4). Thus, n = 4k + 1 for some k ≥ 1. In this case, we consider the set S=
{x4i+2 , y4i+4 }.
We note that the set S is a packing in G. Further, the set S dominates all vertices of G, except for the two vertices y1 and xn (note that here xn = x4k+1 ). Further, we
Dominator and Total Dominator Colorings in Graphs
note that S ∪{x1 } is a γ -set of G, implying that |S| = γ (G) − 1. We note that with the set S as defined above, the graph G − S can be obtained from a path P6k on 6k vertices given by P : y1 y2 y3
x3 x4 x5 . . . y4(k−1)+1 y4(k−1)+2 y4(k−1)+3 x4(k−1)+3 x4(k−1)+4 x4(k−1)+5 that starts at the vertex y1 and ends at the vertex x4k+1 , by adding the two vertices x1 and y4k+1 and adding the four
edges x1 y1 , x1 x4k+1 , y1 y4k+1 , and x4k+1 y4k+1 . We now 2-color the vertices of the path P with the colors 1 and 2 and color x1 and y4k+1 with the same new color 3 to produce a 3-coloring of G −
S. Thereafter, we color the vertices of S with |S| = γ (G) − 1 new colors, one distinct color to each vertex. The resulting coloring is a dominator coloring of G using γ (G) + 2 colors. In
particular, we note that y1 and x4k+1 each are dominators of the color class {x1 , y4k+1 } (whose vertices are colored 3), while every other vertex in G − S is a dominator of the unique vertex in S
that it is adjacent to. Further, each vertex v of S is a dominator of the color class that contains the vertex v (and is a singleton set consisting only of the vertex v). To illustrate the above
coloring, consider the case, for example, when n = 9. In this case, the set S = {x2 , y4 , x6 , y8 } and is given by the set of darkened vertices in Figure 12(a). Further the 3-coloring of the graph
G − S is illustrated in Figure 12(b). We then extend this 3-coloring of G − S to a 7-coloring of G by adding four new colors, one distinct color to each vertex in the set S. The resulting 7-coloring
is a dominator coloring of G, implying that χ d (CL9 ) ≤ 7. In all the above cases, we have shown that χ d (G) ≤ γ (G) + 2. Further one can readily show that χ d (G) ≥ γ (G) + 2. We present a proof
of the simplest case when n ≡ 0 (mod 4) as an illustration. In this case, n = 4k for some k ≥ 1. Further, γ (G) = 2k, and every γ -set of G is a packing. Each vertex v ∈ D either dominates its own
class, in which case the vertex v is the only vertex of that color, or dominates a color class that is a subset of its neighborhood, N(v). Since the sets N[v] = N(v) ∪{v} are vertex-disjoint sets for
all v ∈ D, this implies that at least |D| = γ (G) vertices must receive a unique color. Since at least two additional colors are needed for the remaining vertices of G, every dominator coloring of G
uses at least γ (G) + 2 colors. Hence, χ d (G) ≥ γ (G) + 2 in this case when n ≡ 0 (mod 4). Analogous arguments show that χ d (G) ≥ γ (G) + 2 in the three other cases when n (mod 4) ∈{1, 2, 3}. We
x1 y1
(a) χd(G) ≤ 7
Fig. 12 A circular ladder graph G = CL9
x9 x1 y9
x5 y5
x7 y6
(b) The graph G−S
x9 y9
M. A. Henning
omit the details. Therefore, χ d (G) = γ (G) + 2. The desired result now follows from our earlier observation that γ (G) = n/2 + n/4−n/4. We remark that Chen [7] continued the study of dominator
chromatic number of Cartesian products of certain paths and cycles and considers the 3 × n grid, P3 2 Pn , the Cartesian product P3 2Cn . Two vertices (g1 , h1 ) and (g2 , h2 ) in the direct product
graph G × H of graphs G and H are adjacent if g1 g2 ∈ E(G) and h1 h2 ∈ E(H). They also consider the dominator chromatic number of the Cartesian product Km 2 Kn for m, n ≥ 2. Paulraja and Handrasekar
[33] determined the dominator chromatic number for some classes of graphs, such as the direct product Km × Kn for m, n ≥ 2, and the direct product (Km ◦ K1 ) × Kr for m, n ≥ 2. They also present
results on the Cartesian product Kn 2 Qr for r ≥ 3, where Qr is the r-dimensional hypercube. We omit these results here.
3.4 Dominator Partition Number We discuss briefly in this section the dominator partition number of a graph. In their introductory paper, Hedetniemi, Hedetniemi, Laskar, McRae, and Wallis [15] define
a dominator partition of a graph G as a coloring (not necessarily proper) with the property that every vertex in G is adjacent to all other vertices in some color class (including possibly its own).
The dominator partition number of G, which they denote by π d (G), is the minimum number of color classes in a dominator partition of G. We note that every dominator coloring is a dominator
partition, but not conversely. Thus, π d (G) ≤ χ d (G) for all graphs G. Hedetniemi et al. [15] provide the following lower and upper bounds on the dominator partition number of a graph in terms of
the minimum and maximum degrees. Theorem 30 ([15]) If G is a graph of order n, then n ≤ πd (G) ≤ n − δ(G). 1 + (G) Hedetniemi et al. [15] showed that the dominator partition number of a graph is
surprisingly one of two possible values. Theorem 31 ([15]) If G is a graph of order n, then π d (G) = γ (G) or π d (T) = γ (G) + 1. Proof The proof of the lower bound π d (G) ≥ γ (G) is identical to
the proof we presented earlier of Theorem 12. To prove the upper bound π d (G) ≤ γ (G) + 1, let D = {v1 , . . . , vk } be a γ -set of G. Since the partition π = {V1 , . . . , Vk+1 } of V , where Vi =
{vi } for i ∈ [k] and where Vk+1 = V D, is a dominator partition of G, we have that π d (T) ≤ γ (G) + 1.
Dominator and Total Dominator Colorings in Graphs
3.5 Algorithmic and Complexity Results In this section, we consider the algorithmic complexity of the problem of computing the dominator chromatic number of an arbitrary graph. Formally, we consider
the following decision problem:
GRAPH DOMINATOR k-COLORABILITY Input: A graph G, and an integer k ≥ 1. Question: Does G have a dominator k-coloring?
Hedetniemi et al. [17] showed that to determine if a graph G has a dominator 3-coloring can be computed in polynomial time. Theorem 32 ([17]) GRAPH DOMINATOR 3-COLORABILITY is solvable in polynomial
time. To show that the GRAPH DOMINATOR k-COLORABILITY is NP-complete for k ≥ 4, we give a transformation from GRAPH k-COLORABILITY:
GRAPH k-COLORABILITY Input: A graph G, and an integer k ≥ 4. Question: Does G have a k-coloring?
Theorem 33 ([13, 15]) Graph Dominator k-Colorability is NP-complete for general graphs, for k ≥ 4. Proof Let k be an integer greater than 3. GRAPH DOMINATOR k-COLORABILITY is clearly in the class NP
since we can efficiently verify that an assignment of colors to the vertices of G is both a proper coloring and that every vertex dominates some color class. Next we transform an instance of GRAPH (k
− 1)-COLORABILITY to an instance of GRAPH DOMINATOR k-COLORABILITY. Given an instance of GRAPH (k − 1)COLORABILITY, a graph G, and a k − 1 coloring of G, construct an instance of GRAPH DOMINATOR
k-COLORABILITY as follows. Let G be the graph obtained from G by adding a new vertex v to G and adding all edges joining v to every vertex of G. We now consider the instance given by the graph G and
a dominator k-coloring of G. Let C be a (k − 1)-coloring of G, and let C be the k-coloring of G obtained from the coloring C by assigning a new color to the vertex v. Thus, the color class containing
v consists only of the vertex v. Since {v} ⊆ NG [u] for every vertex in G ,
M. A. Henning
every vertex in G dominates some color class. Thus, C is a dominator k-coloring of G . Conversely, suppose that G has a dominator k-coloring C. Since v is adjacent to every other vertex in G , the
vertex v is the only vertex in its color class. The removal of v produces a (k − 1)-coloring of G. It follows that G is (k − 1)-colorable if and only if G is dominator k-colorable. By Theorem 33, it
is NP-complete to decide if a graph admits a dominator coloring with at most four colors. Chellali and Maffray [6] characterized the graphs G such that χ d (G) ≤ 3 and showed that their
characterization leads to a polynomial time recognition algorithm for such graphs. A rough estimate of the complexity of their algorithm is O(n8 ). We note that this result that the problem “χ d (G)
≤ 3” can be solved in polynomial time is in contrast the problem “χ (G) ≤ 3,” which in NP-complete. In 2009, Hedetniemi et al. [15] and in 2011 Arumugam, Raja Chandrasekar, Misra, Philip, and Saurabh
[2] studied algorithmic aspects of dominator colorings in graphs. They established the following complexity result. Theorem 34 ([2, 15]) For k ≥ 4 an integer, GRAPH DOMINATOR k-COLORABILITY, is
NP-complete for bipartite, chordal, planar, or split graphs. Arumugam et al. [2] complemented the above hardness results by showing that the GRAPH DOMINATOR COLORABILITY is fixed-parameter tractable
in certain classes. Informally, a parameterization of a problem assigns an integer k to each input instance, and a parameterized problem is fixed-parameter tractable, abbreviated FPT, if there is an
algorithm that solves the problem in time f (k) ·|I|O(1) , where |I| is the size of the input and f is an arbitrary computable function that depends only on the parameter k. (For a discussion on
parameterized complexity, we refer the reader to the 2013 book by Downey and Fellows [10].) A graph is an apex graph if there exists a vertex in G whose removal from G yields a planar graph. A family
F of graphs is apex minor-free if there is a specific apex graph H such that no graph in F has H as a minor. As an example, planar graphs are apex minor-free since no planar graph has K5 as a minor.
Apex graphs play an important role in aspects of graph minor theory and are closed under the operation of taking minors, that is, contracting an edge or removing an edge or vertex leads to another
apex graph. As remarked in [2], for k ≥ 4 an integer, GRAPH DOMINATOR k-COLORABILITY, is not fixed-parameter tractable in general graphs unless P= NP. However, the problem is fixed-parameter
tractable in apex minor-free graphs (which include planar graphs) and chordal graphs. Theorem 35 ([2]) For k ≥ 4 an integer, Graph Dominator k-Colorability, is fixedparameter tractable on apex
minor-free graphs and on chordal graphs.
Dominator and Total Dominator Colorings in Graphs
Arumugam et al. [2] show that for k ≥ 4 an integer, GRAPH DOMINATOR kCOLORABILITY, can be solved in “fast” fixed-parameter tractable time in split graphs. Theorem 36 ([2]) For k ≥ 4 an integer, Graph
Dominator k-Colorability, can be solved in O(2k · n2 ) time on a split graph on n vertices. Arumugam et al. [2] pose the problem of whether for k ≥ 4 an integer, GRAPH DOMINATOR k-COLORABILITY, can
be solved in polynomial time on interval graphs.
4 Total Dominator Colorings The total version of dominator coloring in a graph was studied by several authors. The concept of total dominator colorings in graphs was first defined in the manuscript
by Hedetniemi, Hedetniemi, McRae, Rall, and Hedetniemi [16] dated July 9, 2009. Subsequently, Hedetniemi, Hedetniemi, Hedetniemi, McRae, and Rall [17] continued the study of total dominator colorings
in graphs in their manuscript dated February 18, 2011. The first published papers on the topic appears to be the 2012 paper by Vijayalekshmi [36] and the 2015 paper by Kazemi [26]. Formally, a total
dominator coloring, abbreviated TD-coloring, of a graph G with no isolated vertex is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some other color class
(different from its own color class). The total dominator chromatic number of G which we denote by χ td (and denoted by χdt (G) in [18, 26]) is the minimum integer k for which G has a TDcoloring with
k colors. A χ td -coloring of G is a coloring of G that uses χ td (G) colors. Every total dominator coloring is a dominator coloring. Hence, we have the following observation. Observation 37 For
every graph G without isolated vertices, we have χ d (G) ≤ χ td (G). Consider an arbitrary χ td -coloring of G, and let S be a set consisting of one vertex from each of the resulting χ td (G) color
classes. Since every vertex in G is adjacent to every vertex of some color class (different from its own color class), the set S is a TD-set in G, implying that γ t (G) ≤|S| = χ td (G). Hence we have
the following result, first observed by Vijayalekshmi [36] and Kazemi [26]. Observation 38 ([26, 36]) For every graph G without isolated vertices, γ t (G) ≤ χ td (G). Analogous results to Observation
8 and Theorem 9 hold for the total dominator chromatic number. Theorem 39 ([26, 36]) If G is a connected graph of order n ≥ 2, then 2 ≤ χ td (G) ≤ n. Moreover, the following holds.
M. A. Henning
Fig. 13 A χ td -coloring of a path P14
(a) χ td (G) = 2 if and only if G is a complete bipartite graph. (b) χ td (G) = n if and only if G is a complete graph. For disconnected graphs, we have the following upper and lower bounds on the
total dominator chromatic number. Theorem 40 ([36]) If G is a disconnected graph with nontrivial components G1 , G2 , . . . , Gk where k ≥ 2, then 2k − 2 + max {χtd (Gi ) | i ∈ [k]} ≤ χtd (G) ≤
χtd (Gi ).
We remark that the total dominator chromatic number of a path and cycle is incorrectly determined in [26]. To state the total dominator chromatic number of a path Pn and a cycle Cn on n vertices, we
shall need the following well-known result (see [19]). Observation 41 For n ≥ 3, if G ∈{Pn , Cn }, then we have γt (G) = that is, γt (G) = n+1 2 for n odd.
n 2
n 2
n 4
if n ≡ 0 (mod 4), γt (G) =
n 2
n 4
+ 1 if n ≡ 2 (mod 4), and γt (G) =
Theorem 42 ([18]) For n ≥ 2, we have ⎧ for n ∈ {2, 3, 6} ⎨ γt (Pn ) χtd (Pn ) = γt (Pn ) + 1 for n ∈ {4, 5, 7, 9, 10, 11, 14} ⎩ γt (Pn ) + 2 otherwise. For example, a χ d -coloring of the path P14
(using γ t (P14 ) + 1 = 8 + 1 = 9 colors) is illustrated in Figure 13. Thus, by Observation 41 and Theorem 42, we have the following closed formula for the total dominator chromatic number of a path
of large order. Theorem 43 ([18]) For n ≥ 15, χtd (Pn ) = n2 + n4 − n4 + 2. For n ≥ 16, we define next a χ td (Pn )-coloring, Cn∗ , of a path Pn as follows. Let G be the path v1 v2 . . . vn , where n
≥ 16. For each vertex vi where i ≡ 2, 3 (mod 4), assign a unique color. For each vertex vi where i ≡ 1 (mod 4), assign a new additional color, say 1. For each vertex vi where i ≡ 0 (mod 4), assign a
further additional
Dominator and Total Dominator Colorings in Graphs
Fig. 14 A χ td -coloring of a paths P16 , P17 , P18 , and P19
color, say 2. Let Cn denote the resulting coloring. We now define a coloring Cn∗ as follows. If n ≡ 0, 3 (mod 4), let Cn∗ = Cn . If n ≡ 1 (mod 4), then recolor the vertex vn−1 (currently colored with
color 2) with a new distinct color, and let Cn∗ denote the resulting modified coloring. If n ≡ 2 (mod 4), then recolor the vertex vn−1 (currently colored with color 1) with a new distinct color, and
let Cn∗ denote the resulting modified coloring. The coloring Cn∗ when n ∈{16, 17, 18, 19}, for example, is illustrated in Figure 14. The darkened vertices in this coloring of Cn∗ in Figure 14 form a
γ t -set of the path. A new color is assigned to each darkened vertex in the path. Theorem 44 ([18]) χ td (C3 ) = 3, χ td (C4 ) = 2, and χ td (C11 ) = 8. For all other values of n ≥ 5, we have χ td
(Cn ) = χ td (Pn ).
4.1 Bounds on the Total Dominator Chromatic Number By definition of a total dominator coloring, we have the following observation. Observation 45 If v is an arbitrary vertex in a graph G without
isolated vertices, then in every dominator coloring of G, the open neighborhood N(v) of v contains a color class. Theorem 46 If G is a graph without isolated vertices, then χ td (G) ≥ ρ o (G), with
strict inequality if there is no perfect packing in G. Proof If S is an open packing in G, then by Observation 45, the open neighborhoods of vertices in S contain at least |S| color classes, and so χ
td (G) ≥|S|. Choosing S to be a maximum open packing, we have that χ td (G) ≥ ρ o (G). Further, if G does not have a perfect open packing, then at least one additional color class is needed to
contain the vertices that do not belong to the open neighborhood of any vertex in S, and so χ td (G) ≥ ρ o (G) + 1. If H is any connected graph of order k ≥ 1, then the 2-corona G = H ◦ P2 satisfies
ρ o (G) = 2k = χ td (G), illustrating the existence of graphs G that contain a perfect
M. A. Henning
Fig. 15 The graph C4 ◦ P2
open packing and satisfy ρ o (G) = χ td (G). The graph C4 ◦ P2 , for example, is shown in Figure 15 (here, H = C4 ). If a graph G contains a perfect open packing, then it is not necessarily true that
ρ o (G) = χ td (G). The simplest example illustrating this is a path G = P4 , with ρ o (G) = 2 and χ td (G) = 3. More generally, if G = Pn where n ≡ 0 (mod 4) and n ≥ 8, then G has a perfect open
packing and ρ o (G) = γ t (G). However in this case, by Theorem 42, we have χ td (G) = γ t (G) + 2 = ρ o (G) + 2. For a given graph G, let At (G) denote the set of all γ t -sets in G. We next present
an upper bound on the total dominator chromatic number of a graph. Theorem 47 ([18, 26]) If G is a connected graph without isolated vertices, then χtd (G) ≤ γt (G) + min {χ (G − S)}, S∈At (G)
and this bound is tight. Proof Let S be an arbitrary γ t -set of G, and let C be a proper coloring of the graph G − S using χ (G − S) colors. We now extend the coloring C to a coloring of the
vertices of G by assigning to each vertex in S a new and distinct color. Let C denote the resulting coloring of G, and note that C uses γ t (G) + χ (G − S) colors. Since S is a TD-set of G, every
vertex in G is adjacent to at least one vertex of S. Since the color class of C containing a given vertex of S consists only of that vertex, each vertex in G is adjacent to every vertex of some
(other) color class in the coloring C . Hence, C is a TD-coloring of G using γ t (G) + χ (G − S) colors. This is true for every γ t -set of G. The desired upper bound now follows by choosing S to be
a γ t -set of G that minimizes χ (G − S). The bound is achieved, for example, by taking G to be a complete graph. As shown in [18], the bound is also tight for infinitely many trees. The proof of
Theorem 47 yields the following more general result. Theorem 48 If G is a connected graph without isolated vertices, and TD(G) denotes the set of all total dominating sets of G, then χtd (G) ≤
min { |S| + χ (G − S) }.
We observe that χ (G − S) ≤ χ (G) for every proper subset S ⊂ V (G). This observation, together with the results of Observations 37 and 38, gives us the
Dominator and Total Dominator Colorings in Graphs
following analogous result to Theorem 12, thereby establishing upper and lower bounds on the total dominator chromatic number of an arbitrary graph in terms of its total domination number and
chromatic number. Theorem 49 ([26, 36]) Every graph G without isolated vertices satisfies max{γt (G), χ (G)} ≤ χtd (G) ≤ γt (G) + χ (G).
4.2 Special Classes of Graphs In this section, we consider the total dominator chromatic number of certain classes of graphs.
Bipartite Graphs
As a special case of Theorem 49 when G is a bipartite graph, we have the following result. Theorem 50 ([26, 36]) If G is a bipartite graph, then γ t (G) ≤ χ td (G) ≤ γ t (G) + 2. For each t ∈{0, 1,
2}, an infinite family Gt of bipartite graphs such that each graph G ∈ Gt satisfies χ td (G) = γ t (G) + t is constructed in [18] as follows. Let G0 be the family of graphs G without isolated
vertices that contain a TD-set S that is a perfect open packing in G and such that the neighborhood of each edge e in G[S] induces a complete bipartite graph in G, that is, if e = uv is an edge in G
[S], then the subgraph of G induced by the neighborhood, N[e], of e is a complete bipartite graph Kn1 ,n2 where d(u) = n1 and d(v) = n2 . Let G ∈ G0 . As an example, if H is an arbitrary graph, then
the graph G = H ◦ P2 belongs to the family G0 since the set S = V (G) V (H) is a TD-set that is a perfect open packing in G and the neighborhood of each edge e in G[S] induces a complete bipartite
graph K1,2 in G. Let G1 be the family of graphs that can be obtained from a graph H without isolated vertices by attaching any number of pendant edges, but at least one, to each vertex of H. For
example, if H is an arbitrary isolate-free graph, then the corona G = H ◦ P1 of H belongs to the family G1 . Let G2 be the family of all paths Pn and cycles Cn , where n ≡ 0 (mod 4) and n ≥ 8.
Theorem 51 ([18]) The following holds. (a) If G ∈ G0 , then χ td (G) = γ t (G). (b) If G ∈ G1 , then χ td (G) = γ t (G) + 1. (c) If G ∈ G2 , then χ td (G) = γ t (G) + 2.
M. A. Henning
Recall that the dominator chromatic number of a tree is one of two values (see Theorem 17). However, the total dominator chromatic number of a tree is one of three values. By Theorem 50, if T is a
tree, then γ t (T) ≤ χ td (T) ≤ γ t (T) + 2. Further, there are infinitely many trees T for which χ td (T) = γ t (T) + i for each i ∈ [2]0 = {0, 1, 2}. Theorem 52 ([26, 36]) If G is a tree, then γ t
(T) ≤ χ td (T) ≤ γ t (T) + 2. The following properties of χ td -colorings in a tree T are established in [18]. We say that a color class C in a given TD-coloring C of G is free if each vertex of G is
adjacent to every vertex of some color class different from C. Theorem 53 ([18]) If T is a nontrivial tree, then the following holds. (a) If γ t (T) = χ td (T), then no χ td (T)-coloring contains a
free color class. (b) If χ td (T) = γ t (T) + 1, then there exists a χ td (T)-coloring that contains a free color class. (c) If χ td (T) = γ t (T) + 2, then there exists a χ td (T)-coloring that
contains two free color classes. The trees T satisfying γ t (T) = χ td (T) are characterized in [18]. Let T be the family of trees constructed as follows. Let T consist of the tree P2 and all trees
that can be obtained from a disjoint union of k ≥ 1 stars each of order at least 3 by adding k − 1 edges joining leaf vertices in such a way that the resulting graph is connected and the center of
each of the original k stars remains a support vertex. Theorem 54 ([18]) If T is a nontrivial tree, then γ t (T) = χ td (T) if and only if T ∈ T. In [18] a tight upper bound on the total dominator
chromatic number of a tree in terms of its order is established, and the trees with maximum possible total dominator chromatic number are characterized. For this purpose, let F be the family of all
trees T that can be obtained from a tree H of order at least 2 by selecting an arbitrary edge e = uv in H and attaching a path of length 2 to each vertex of V (H) {u, v} so that the resulting paths
are vertex-disjoint. We call H the underlying tree of T. A tree in the family F with underlying tree H = P5 , for example, is illustrated in Figure 16 (here the vertices of H are depicted by the
darkened vertices). Theorem 55 ([18]) If T is a tree or order n ≥ 2, then χtd (T ) ≤ equality if and only if T ∈ F. Fig. 16 A tree in the family F
2 3 (n
+ 1), with
Dominator and Total Dominator Colorings in Graphs
Mycielskian of a Graph
Let G be a graph without isolated vertices and with V (G) = {v1 , v2 , . . . , vn }. The Mycielskian M(G) is the graph obtained from G by adding n new vertices u1 , u2 , . . . , un and an additional
vertex v and then adding the edges vui for all i ∈ [n]. Further, for each edge vi vj of G, we add the edges ui vj and vi uj to complete the construction of M(G). For example, if G = K2 , then M(G) =
C5 . If G = C5 , then M(G) is the Grötzsch graph. Kazemi [24] proved that the dominator chromatic number of the Mycielskian of a graph is one of two values. Theorem 56 ([24]) If G is a graph without
isolated vertices, then χtd (M(G)) = χtd (G) + 1
χtd (M(G)) = χtd (G) + 2.
Jalilolghadr, Kazemi, and Khodkar [20] studied total dominator colorings of circulant graphs Cn (a, b) with two “jump sequences.” For n ≥ 3, let 1 ≤ a1 < · · · ak ≤n/2, and let S = {a1 , . . . , ak
}. The graph G with vertex set V (G) = [n] and edge set E(G) = {{i, j } : |i − j | ≡ ai (mod n) for some i ∈ [k]} is called a circulant graph with jump sequence S and denoted Cn (S) or Cn (a1 , . . .
, ak ). We note that Cn (S) is a k-regular graph. Jalilolghadr et al. [20] prove the following result. Theorem 57 ([20]) If G is a circulant graph Cn (a, b) where n ≥ 6, gcd(a, n) = 1 and a−1 b ≡ 3
(mod n), then ⎧ n for n ∈ {8, 9, 10} ⎨ 2 8 χtd (G) = 2 n8 + 1 for n ≡ 1 (mod 8) or n = 11 ⎩ n 2 8 + 2 otherwise. 4.2.5
Central Graphs
Kazemi and Kazemnejad [28] studied the total dominator chromatic number of central graphs, where they define the central graph C(G) of a graph G as the graph obtained from G by subdividing every edge
of G exactly once and adding all edges joining two vertices that were not adjacent in G. Among other results, they proved the following. Theorem 58 ([28]) If G is a connected graph of order n ≥ 4,
then the following holds.
(a) (b) (c) (d)
M. A. Henning
χtd (C(G)) ≥ 23 n + 1. χtd (C(G)) ≤ n + k2 where k is the order of a longest path in G. χ td (C(G)) ≤ n + 1 if Δ(G) ≤ n − 2. χtd (C(G)) ≤ n + n2 , with equality if and only if G∼ =Kn .
4.3 Graph Products In this section, we present some results due to Kazemi [25] on the total dominator chromatic number in Cartesian products (2) and direct products (×) of two graphs. Theorem 59
([25]) If G and H are two graphs without isolated vertices, then χtd (G × H ) ≤ χtd (G) · χtd (H ). Theorem 60 ([25]) For q ≥ p ≥ 2, if G is a complete p-partite graph and H is a complete q-partite
graph, then χ td (G × H) = p + 2. In particular, χ td (Kp × Kq ) = p + 2. Theorem 61 ([25]) If G and H are two graphs without isolated vertices, then max{χtd (G), χtd (H )} ≤ χtd (G 2 H ) ≤ min{χtd
(G) · n(H ), χtd (H ) · n(G)}. Theorem 62 ([25]) If G is a graph without isolated vertices, then χtd (G) ≤ χtd (G2 K2 ) ≤ 2χtd (G).
4.4 Algorithmic and Complexity Results We consider in this section the problem of finding the total dominator coloring number of an arbitrary graph. Formally, we consider the following decision
GRAPH TOTAL DOMINATOR k-COLORABILITY Input: A graph G, and an integer k ≥ 4. Question: Does G have a total dominator k-coloring?
Dominator and Total Dominator Colorings in Graphs
An identical proof to that of Theorem 33 can be used to show that the GRAPH TOTAL k-DOMINATOR COLORABILITY is NP-complete for general graphs by transforming it from an instance of GRAPH DOMINATOR
k-COLORABILITY. Theorem 63 ([17, 26]) GRAPH TOTAL DOMINATOR k-COLORABILITY is NP-complete for general graphs, for k ≥ 4.
5 Concluding Comments In this chapter, we have surveyed selected results on the dominator chromatic number and total dominator chromatic number of a graph. Other results can be found, for example, in
[3, 21, 27]. We close with a small list of open problems. Problem 1 Find graphs, or classes of graphs, G satisfying the following. (a) (b) (c) (d) (e) (f)
χ d (G) = γ (G). χ d (G) = χ (G). χ d (G) = γ (G) + χ (G). χ td (G) = γ t (G). χ td (G) = χ (G). χ td (G) = γ t (G) + χ (G).
Problem 2 Characterize the nontrivial trees T satisfying the following. (a) γ t (T) = χ td (T) + 1. (b) γ t (T) = χ td (T) + 2. Problem 3 Characterize the graphs G satisfying χ d (G) = χ td (G).
Problem 4 Determined the dominator chromatic number and the total dominator chromatic number of the m × n grid graph, Pm 2 Pn , for all m, n ≥ 2. Problem 5 For any dominator (or total dominator)
coloring, one can construct a so-called dominator digraph (total dominator digraph, respectively) which is an orientation of some of the edges of G such that for every vertex u, you orient the edge
uv from u to v if u dominates the color class of vertex v. We note that for dominator colorings, this digraph will contain loops, if a vertex forms a singleton color class. However, the total
dominator digraph will have no loops. We also note that these digraphs will have some unoriented edges which can be deleted. Study the resulting dominator digraphs and total dominator digraphs.
References 1. A.M. Abid, T.R. Ramesh Rao, Dominator coloring of Mycielskian graphs. Australas. J. Combin. 73(2), 274–279 (2019) 2. S. Arumugam, K. Raja Chandrasekar, N. Misra, G. Philip, S. Saurabh,
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Hedetniemi, Dominating partitions of graphs. Tech. Report., 1979. Unpublished manuscript 10. R.G. Downey, M.R. Fellows, Fundamentals of Parameterized Complexity. Texts in Computer Science (Springer,
London, 2013), xxx+763 pp. ISBN: 978-1-4471-5558-4; 9781-4471-5559-1 11. R. Gera, On the dominator colorings in bipartite graphs. Inform. Technol. New Gen. ITNG’07, 947–952 (2007) 12. R. Gera, On
dominator colorings in graphs. Graph Theory Notes N. Y. LII, 25–30 (2007) 13. R. Gera, S. Horton, C. Rasmussen, Dominator colorings and safe clique partitions. Congress. Num. 181, 19–32 (2006) 14.
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Irredundance C. M. Mynhardt and A. Roux
The concept of irredundance in graphs was introduced in 1978 by Cockayne, Hedetniemi and Miller [42] because of its relevance to dominating sets. Informally, a set X of vertices in a graph G is
irredundant if each vertex in X dominates a vertex of G (perhaps itself) that is not dominated by any other vertex in X. More formally, in terms of private neighbours, X is irredundant if pn(x, X) =
N [x]−N [X−{x}] = ∅ for each x ∈ X, that is, if each x ∈ X has an X-private neighbour (which could be x itself). If a set X has a vertex x without private neighbours, that is, if N[x] ⊆ N[X −{x}], we
say that x is redundant in X (in which case X is not an irredundant set).An irredundant set is maximal irredundant if it has no irredundant proper superset. The lower and upper irredundant numbers ir
(G) and IR(G) are, respectively, the smallest and largest cardinalities of a maximal irredundant set of G. If X is a maximal irredundant set of cardinality ir(G), we call X an ir(G)-set or simply an
ir-set, depending on circumstances. An IR(G)-set or IR-set is defined similarly; the same holds for any other domination-type parameter. This chapter is organised as follows. To begin, we consider
the partition of V (G) associated with an irredundant set in Section 1. Here, we discuss the concepts of private neighbours, the private neighbour cube and generalised irredundance. The chain of
lower and upper domination, independence and irredundance numbers is
The author “C. M. Mynhardt” was supported by the Natural Sciences and Engineering Research Council of Canada. C. M. Mynhardt () Department of Mathematics and Statistics, University of Victoria,
Victoria, BC, Canada e-mail: [email protected] A. Roux Department of Mathematical Sciences, Stellenbosch University, Stellenbosch, South Africa e-mail: [email protected] © The Author(s), under
exclusive license to Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://doi.org/10.1007/978-3-030-58892-2_6
C. M. Mynhardt and A. Roux
presented in Section 2. We discuss equality of parameters in the domination chain in Section 3, where we cover lower and upper irredundance perfect graphs, as well as other cases of equality, such as
graphs with ir = IR, ir = γ , α = IR or = IR. Bounds involving other graph parameters, including Nordhaus-Gaddumand Gallai-type results, can be found in Section 4. Differences between and ratios of
parameters in the domination chain are covered in Section 5 , criticality and stability concepts in Section 6, irredundance on chessboards in Section 7 and irredundant Ramsey numbers in Section 8.
Finally, we discuss reconfiguration of irredundant sets in Section 9 and complexity in Section 10. We state open problems and conjectures throughout the text where appropriate.
1 Partition of V (G) Associated with an Irredundant Set Let us examine the properties of an irredundant set more closely. As noted by Cockayne, Grobler, Hedetniemi and McRae [41], we can associate a
weak partition1 of V (G) with each irredundant set X, namely, V (G) = X ∪ Y ∪ C ∪ R, where
Y consists of vertices in V (G) − X that belong to private neighbourhoods of vertices in X, C consists of vertices in V (G) − X that have at least two neighbours in X, and R is the set of vertices
not dominated by X. In Figure 1, the set X is indicated by coloured (red and yellow) discs, Y by blue squares, C by white discs and R by green triangles. The set X is further partitioned as X = Z ∪
I, where
I is the set of vertices that are isolated in G[X], indicated by yellow discs, and Z = X − I , indicated by red discs. The blue private neighbours in Y and the observation that 5 ∈ pn(5, X) confirm
that X is irredundant. Closer scrutiny however reveals that X is not maximal irredundant. For any y ∈ R, y is a private neighbour of itself in the set X ∪{y}, and for any z adjacent to y, y is a
private neighbour of z in X ∪{z}. Hence we must
1 In
a weak partition, some of the parts could be empty.
also examine the private neighbourhood of each x ∈ X in these supersets of X to determine whether or not X is maximal irredundant. Since pn(1, X) = {6, 7} ⊆ N (12), pn(1, X∪{12}) = ∅, which means
that X ∪{12} is not irredundant. However, pn(1, X ∪{6}) = {7} and pn(6, X∪{6}) = {12}, and since all other vertices in X also have X ∪{6}-private neighbours, X ∪{6} is irredundant. The set X ∪{14} is
likewise irredundant, as 4 ∈ pn(4, X ∪ {14}) and 14 ∈ pn(14, X ∪ {14}). The following result, which was used implicitly in earlier work and first formalised in [41], and which extends a result by
Bollobás and Cockayne [10], provides a certificate for an irredundant set to be maximal irredundant (or not). Theorem 1.1 ([41]) An irredundant set X of a graph G is maximal irredundant if and only
if, for each u ∈ R = V (G) − N[X] and each v ∈ N[u], there exists a vertex x ∈ X such that v dominates pn(x, X). Proof. Assume that X is a maximal irredundant set of G for which the conclusion in the
statement of the theorem does not hold. Then there exist vertices u ∈ R and v ∈ N[u] such that v does not dominate the private neighbourhood of any x ∈ X.
Consider the set X = X ∪{v}. The stated property of v implies that pn(x, X ) = ∅ for each x ∈ X. Moreover, since u is not dominated by any x ∈ X but u ∈ N[v], it
follows that u ∈ pn(v, X ). Therefore X is irredundant, contrary to the maximality of X. Conversely, assume that X is an irredundant set of G for which the conclusion of the statement holds. Consider
any v ∈ V (G) − X. If v is undominated by X or adjacent to a vertex u that is undominated by X, then, by assumption, there exists a vertex x ∈ X such that v dominates pn(x, X). This implies that x is
redundant in X ∪{v}. On the other hand, if v and all its neighbours are dominated by X, then v is redundant in X ∪{v}. In either case, it follows that X ∪{v} is not irredundant. Since v is arbitrary,
we conclude that X is maximal irredundant.
Z X
G Y
Fig. 1 The partition of the vertex set of a graph associated with an irredundant set X
C. M. Mynhardt and A. Roux
If v ∈ V (G) − X and pn(x, X) ⊆ N[v], we say that v annihilates x. Let G be the graph obtained from G in Figure 1 by adding the edges 6 7 and 9 14. Then 12, 6 and 7 annihilate 1; 8 annihilates 2; 9,
13 and 14 annihilate 3; and 10 and 11 annihilate 4.
Therefore X is maximal irredundant (but not dominating) in G .
1.1 Private Neighbours The elements of an irredundant set X could have one or both of two types of private neighbours: for x ∈ X, the vertex y is an (i) X-self-private neighbour (X-spn) of x if y = x
and x is isolated in G[X] (the vertices 4, 5 in Figure 1), (ii) X-external private neighbour (X-epn) of x if y ∈ V (G) − X and N(y) ∩ X = {x}. The set of X-external private neighbours of x ∈ X is
denoted by epn(x, X). In Figure 1, pn(4, X) = {4} ∪ epn(4, X) = {4, 10, 11}. Bollobás and Cockayne [10] proved the following fundamental result. Theorem 1.2 ([10]) Every graph G without isolated
vertices has a minimum dominating set X in which each vertex has an X-epn. Proof. Among all minimum dominating sets of G, let X be one that maximises the number of edges in G[X]. We show that X has
the desired property. Assume to the contrary that epn(x, X) = ∅ for some x ∈ X. By the minimality of X, pn(x, X) = ∅, and the only possibility is that x is isolated in G[X]. Since G is isolate-free,
x is adjacent to a vertex u ∈ V (G) − X. Since epn(x, X) = ∅, u is adjacent to a
vertex y ∈ X −{x}. Consider the set X = (X −{x}) ∪{u}. Since x is adjacent to u, and
each neighbour of x in V (G) − X is adjacent to a vertex in X −{x}, X dominates G.
However, u is adjacent to y ∈ X , whereas x is nonadjacent to all vertices in X. Hence
X , having the same cardinality as X, is a minimum dominating of G such that G[X ] contains more edges than G[X], contradicting the choice of X. A third type of private neighbour of x ∈ X is
considered in [76]: the vertex y is an (iii) X-internal private neighbour (X-ipn) of x if y ∈ X −{x} and N(y) ∩ X = {x}. In Figure 1, vertex 3 is an X-ipn of 2.
1.2 The Private Neighbour Cube and Generalised Irredundance Using all combinations of the three types of private neighbours, Fellows, Fricke, Hedetniemi and Jacobs [76] constructed the so-called
private neighbour cube (illus-
Irredundance Fig. 2 The private neighbour cube of Fellows, Fricke, Hedetniemi and Jacobs [76]
139 COIR 111
011 OOIR
OIR 001
trated in Figure 2) and obtained six additional types of irredundance. Combining the private neighbourhood types and their negations (e.g. each x ∈ X has an X-epn but neither an X-spn nor an X-ipn),
Cockayne [30] obtained further generalised irredundance concepts which were investigated in greater depth by Finbow [77]. Most of these generalisations, as well as others obtained by, for example,
imposing structural requirements on X, are beyond the scope of this chapter, and we mainly consider irredundance in undirected graphs as defined in [42]. Apart from brief definitions to explain the
private neighbour cube, we consider only two other concepts of irredundance, open or OC-irredundance, introduced by Farley and Shacham [62], and CO-irredundance, defined by Fellows et al. [76]. The
vertices of the private neighbour cube are, simultaneously, types of sets defined according to which types of private neighbours their vertices possess and the maximum cardinality of such a set X in
a graph G. The types of sets are represented by binary strings of length 3 as described below. Type 000: Each vertex in X has a private neighbour which is not an X-ipn, an Xepn or an X-spn. Since we
only consider these three types of private neighbours, the only set of this type is the empty set. Type 001: Each vertex in X has an X-ipn. This is only possible if the subgraph G[X] of G induced by
X consists of disjoint copies of K2 . Following [76], we call such a set X a strong matching set and denote the maximum cardinality of a strong matching set in G by α ∗ (G). Type 010: Each vertex in
X has an X-epn; that is, epn(x, X) = N(x) − N[X − {x}] = ∅ for each x ∈ X. Since this definition involves open and c losed neighbourhoods, we obtain the concept of OC-irredundance, usually called
open irredundance. The lower and upper open irredundant numbers oir(G) and OIR(G) of G are, respectively, the smallest and largest cardinalities of a maximal open irredundant set of G. Open
irredundant sets were studied by Farley and Shacham [62]. Type 011: Each vertex in X has an X-ipn or an X-epn; that is, N(x) − N(X − {x}) = ∅ for each x ∈ X. Since this definition involves two open
neighbourhoods, we obtain the concept of open-open and OO-irredundance. The lower and upper open irredundant numbers ooir(G) and OOIR(G) are defined in the obvious manner. OO-irredundant sets were
considered by Farley and Proskurowski [61] and Farley and Shacham [62].
C. M. Mynhardt and A. Roux
Fig. 3 A CO-irredundant set X
Type 100: Each vertex in X is an X-spn, that is, X is an independent set. The associated parameters, namely, the independence number α(G) and the independent domination number i(G), are well
established in graph theory. Type 101: Each vertex in X is an X-spn or has an X-ipn. For such a set X, known as a 1-dependent set, (G[X]) ≤ 1. The maximum cardinality of a 1-dependent set of G is the
1-dependence number α 1 (G). These sets were studied by Fink and Jacobson [80]. Type 110: Each vertex in X is an X-spn or has an X-epn, that is, X is an irredundant set. Since we require that pn(x,
X) = N[x] − N[X − {x}] = ∅ for each x ∈ X, the concept of closed neighbourhood occurs twice in this definition. Thus, irredundance can also be called CC-irredundance. Type 111: Each vertex in X is an
X-spn or has an X-epn or X-ipn, that is, N[x] − N (X − {x}) = ∅ for each x ∈ X. Since this definition involves closed and o pen neighbourhoods, we get the concept of CO-irredundance. That is, a set X
is COirredundant if each x ∈ X has a private neighbour of at least one of the three types mentioned in Section 1.1. For example, the set X in Figure 3 is CO-irredundant: 1 is an X-spn, epn(2, X) =
{6}, ipn(3, X) = {2, 4} and epn(4, X) = {8}. The lower and upper CO-irredundant numbers coir(G) and COIR(G) are defined in the obvious manner. The partial order on the set of parameters (indicated by
arrows, from large to small) in the private neighbourhood cube is defined by the lexicographic order of the binary strings that represent the respective sets.
2 The Domination Chain The relationships between dominating, independent and irredundant sets of vertices of a graph are well-known [92, Chapter 3], and, following [41], we summarise them below.
maximal independent
minimal dominating (I )
(I I )
⇒ ⇒ maximal irredundant independent and dominating and dominating irredundant
It also follows from the definitions that any irredundant set is CO-irredundant, and any open irredundant set is irredundant. Simmons [122] showed that a total dominating set is minimal total
dominating if and only if it is CO-irredundant. The implications (I) and (II) in (1) lead to the domination chain (2), first mentioned in [42]. For any graph G, ir(G) ≤ γ (G) ≤ i(G) ≤ α(G) ≤ (G) ≤ IR
For open and CO-irredundance the inequalities, oir(G) ≤ γ (G) ≤ OIR(G) ≤ IR(G) for any graph G without isolated vertices, and IR(G) ≤ COIR(G) for any graph G hold (see [92, pp. 91–92]). Farley and
Shacham [62] and Favaron [66] gave examples of graphs with OIR < i, < OIR, ir < oir and oir < ir. For the parameters in the domination chain (2), Allan and Laskar [1] and Bollobás and Cockayne [10]
further showed that γ (G) ≤ 2 ir(G) − 1 (see Theorem 2.1 below), but all the other ratios of parameters in (2) are unbounded for general graphs: for i/γ , consider Kn,n ; for α/i, consider K1,n ; for
/α, consider K2 Kn ; and for IR / , consider K2 Kn+1 and delete an edge between the two copies of Kn+1 . Theorem 2.1 ([1, 10]) For any graph G, γ (G) ≤ 2 ir(G) − 1. Proof. We use the notation defined
in Section 1 and illustrated in Figure 1. Let X be an ir-set of G. If X dominates G, we are done; hence assume R = V (G) − N [X] = ∅. Let u ∈ R. By Theorem 1.1, there exists a vertex x ∈ X such that
u annihilates x. Since X does not dominate u, necessarily ∅ = epn(x, X) ⊆ N(u). Define XR = {x ∈ X : epn(x, X) = ∅ and epn(x, X) ⊆ N(u) for some u ∈ R}. For
each x ∈ XR , choose a vertex x ∈ epn(x, X) and define YR = {x : x ∈ XR }. Note that YR ⊆ Y and |YR | = |XR |≤|X|. Moreover, X ∪ YR dominates G. Since X X ∪ YR , the maximality of X (as an
irredundant set) implies that X ∪ YR is not irredundant. As stated in (1), this implies that X ∪ YR is not a minimal dominating set. Therefore γ (G) ≤ |X ∪ YR | − 1 = |X| + |YR | − 1 ≤ 2 ir(G) − 1.
Subject to γ (G) ≤ 2 ir(G) − 1 and two other obvious restrictions, the differences between the parameters in (1) can be arbitrary in a connected graph, as proved in [47]. Theorem 2.2 ([47]) For any
positive integers k1 ≤· · · ≤ k6 such that (a) k1 = 1 ⇒ k2 = k3 = 1, (b) k4 = 1 ⇒ k1 = · · · = k6 = 1 and (c) k2 ≤ 2k1 − 1, there exists a connected graph G such that ir(G) = k1 , γ (G) = k2 , i(G) =
k3 , α(G) = k4 , (G) = k5 and IR(G) = k6 .
C. M. Mynhardt and A. Roux
Much work has been done to bound the ratios or prove equality of pairs of the parameters in the domination chain for special graph classes, or bound the parameters, their differences or their sums
using other graph parameters. Except for a few results on open and CO-irredundance, we only mention results involving ir and IR.
3 Equality of Parameters in the Domination Chain For graph parameters π , λ, we say that G is a (π , λ)-graph if π (G) = λ(G). Thus, the (i, α)-graphs are precisely the well-covered graphs, and the
(γ , )-graphs are the well-dominated graphs, terms which work well(!) for independence and domination, but less well for irredundance and not at all if π and λ refer to different concepts. Many
instances of (ir, γ )- and (α, IR)-graphs are mentioned in [77, pp. 23–24] and [92, pp. 77–84], and we do not repeat those results here. We write H G (H G) if H is an induced (proper) subgraph of G.
More recent research focuses instead on (π , λ)-perfect graphs: a graph G is (π , λ)-perfect if π (H) = λ(H) for every H G. For a trivial example, note that each component of a P3 -free graph is
complete; hence P3 -free graphs are (ir, IR)-perfect graphs. Graphs with maximum degree ≤ 2 are (ir, i)- and (α, IR)perfect graphs (but not (i, α)-graphs). Also, (χ , ω)-perfect graphs, where χ and ω
denote the chromatic and clique number, respectively, are the classical perfect graphs. We discuss (ir, γ )-perfect and (, IR)-perfect graphs in Sections 3.1 and 3.2, respectively. In Sections
3.3–3.5, we consider equality results that do not involve (π , λ)-perfect graphs.
3.1 Lower Irredundance Perfect Graphs A graph is (lower) irredundance perfect if it is (ir, γ )-perfect, and k-(ir, γ )-perfect if ir(H ) = γ (H ) for each induced subgraph H with ir(H ) ≤ k. A graph
G is minimally (ir, γ )-imperfect if G is not (ir, γ )-perfect and ir(H ) = γ (H ) for every H G. The (ir, i)-perfect graphs form a proper subset of the (ir, γ )-perfect graphs. If G does not contain
any of the graphs H1 , . . . , Hk as induced subgraphs, we say that G is (H1 , . . . , Hk )-free. Eight graphs T1 , G1 , . . . , G7 are depicted in Figure 4, and in the rest of this section, when we
refer to T1 or Gi , i = 1, . . . , 7, we mean a graph in this figure. A (generalised) spider Sp(1 , . . . , k ), i ≥ 1, k ≥ 2, is a tree obtained from the star K1,k with centre u by subdividing the
edge uvi i − 1 times, i = 1, . . . , k. An early result on (ir, γ )-perfect graphs by Faudree, Favaron and Li [63] concerns P4 -free graphs, also known as cographs, a subclass of chordal graphs.
Fig. 4 Graphs used in Section 3.1
Slater tree
We prove the first part of the following theorem as an illustration of how results of this nature are obtained. Theorem 3.1 ([63]) (i) Any P4 -free graph, that is, any cograph, is (ir, γ )-perfect.
(ii) Any (P4 , K3,3 )-free or (K1,3 , G1 )-free graph is (ir, i)-perfect. Proof of (i). Let G be a P4 -free graph and consider an ir-set X of G. We show that X is a dominating set of G. Suppose to
the contrary that R = V (G) − N[X] = ∅ and consider u ∈ R. As in the proof of Theorem 2.1, there exists a vertex x ∈ X such that u annihilates x; that is, x ∈ Z (see Figure 1) and ∅ = epn(x, X) ⊆ N
(u). Let z be a vertex in Z adjacent to x, and let w ∈ epn(x, X). Then G[{y, x, w, u}]∼ =P4 , contrary to our hypothesis. It follows that X dominates G; hence γ (G) = ir(G). Since each induced
subgraph of a P4 -free graph is P4 -free, γ (H ) = ir(H ) for each H G, that is, G is (ir, γ )-perfect. Favaron [64] showed that graphs that contain none of six forbidden induced subgraphs are (ir, γ
)-perfect and conjectured that (P6 , G2 , G3 )-free graphs are (ir, γ )-perfect. This conjecture was proved by Puech [115], who also proved a similar result which involves two forbidden subgraphs.
Theorem 3.2 ([115]) Every (P6 , G2 , G3 )-free and every (P6 , G4 )-free graph is (ir, γ )-perfect. Since both P6 and G4 contain P5 as an induced subgraph, the class of P5 -free graphs is included in
the class of (P6 , G4 )-free graphs. Therefore we obtain the following corollary, first conjectured by Faudree et al. [63]. Corollary 3.3 ([115]) Every P5 -free graph is (ir, γ )-perfect.
C. M. Mynhardt and A. Roux
Puech in turn conjectured that every (P6 , G5 , G6 )-free graph is (ir, γ )-perfect, a proof of which was given by Volkmann and Zverovich [127]. Note that Theorem 3.4 implies Theorem 3.2 and thus
also the truth of the conjectures in [63, 64, 115]. Theorem 3.4 ([127]) Every (P6 , G5 , G6 )-free graph is (ir, γ )-perfect. Puech [117] determined all pairs of connected graphs (X, Y ) such that
every sufficiently large graph containing neither X nor Y as induced subgraph is (ir, γ )perfect. Theorem 3.5 ([117]) Let (X, Y ) be a pair of connected graphs and let n0 be a given positive integer.
A graph G is (X, Y )-free implies that G is (ir, γ )-perfect for any connected graph of order at least n0 if and only if one of the following statements holds: • • • •
X X X X
P5 and Y is arbitrary P6 and Y G4 G1 and Y Sp(1, 1, 2) G7 and Y Sp(1, 1, 3).
Building on work by Bollobás and Cockayne [10], Favaron [64], and Laskar and Pfaff [106], Henning [96] stated the following necessary and sufficient condition for a chordal graph to be (ir, γ )
-perfect. Theorem 3.6 ([96, 106]) (i) A chordal graph is (ir, γ )-perfect if and only if it is (T1 , G1 )-free. (ii) A tree is (ir, γ )-perfect if and only if it is T1 -free. Henning [96]
characterised 2-(ir, γ )-perfect graphs in terms of 12 forbidden induced subgraphs and also conjectured that a graph G is (ir, γ )-perfect if and only if it is 4-(ir, γ )-perfect. However, Volkmann
and Zverovich [128], in a paper that contains an excellent summary of results on (ir, γ )-perfect graphs up to that point, constructed a minimal irredundance imperfect counterexample F∗ (shown in
Figure 5) to Henning’s conjecture; F∗ is 4-(ir, γ )-perfect but ir(F ∗ ) = 5 and γ (F∗ ) = 6. The analytical proof that F∗ is 4-(ir, γ )-perfect is quite long, but this fact can be verified by
computer. The set {2, 4, 7, 9, 13, 15} (red vertices in Figure 5) is a minimum dominating set of F∗ , while {3, 4, 8, 13, 14} (blue-circled vertices, with one private neighbour each indicated by
brown squares) is an ir-set. They, in turn, formulated the following conjectures, which remain unresolved to date. (For what it is worth, the authors of this survey surmise that both conjectures are
false.) Conjecture 3.7 Volkmann and Zverovich [128, 2002] (i) A graph is (ir, γ )-perfect if and only if it is 5-(ir, γ )-perfect. (ii) The number of minimally (ir, γ )-imperfect graphs is finite.
Fig. 5 A 4-(ir, γ )-perfect graph F∗ with ir-set {3, 4, 8, 13, 14} and γ -set {2, 4, 7, 9, 13, 15} [128]
Cographs are (χ , ω)-perfect (i.e. perfect in the original sense) and (ir, γ )-perfect. Not all perfect graphs are (ir, γ )-perfect, e.g. the Slater tree is perfect but not (ir, γ )perfect. On the
other hand, C5 is (ir, γ )-perfect but not perfect. Problem 3.8 Characterise the intersection of the two classes of (χ , ω)-perfect and (ir, γ )-perfect graphs.
3.2 Upper Irredundance Perfect Graphs Jacobson and Peters [102] defined a graph to be upper irredundance perfect if it is (α, IR)-perfect. On the other hand, Gutin and Zverovich [90] and Zverovich
and Zverovich [133] defined a graph to be upper domination perfect if it is (α, )perfect and upper irredundance perfect if it is (, IR)-perfect. To be consistent with the corresponding definition for
the lower parameters, we prefer the latter definition. Fortunately, the definition of (π , λ)-perfect graphs is unambiguous, and we use it here. Moreover, Gutin and Zverovich [90] showed that any (α,
)-perfect graph is also (, IR)-perfect, from which it follows that the classes of (α, )-perfect and (α, IR)-perfect graphs are identical. We present the short proof here. Theorem 3.9 ([90]) If G is a
(α, Γ )-perfect graph, then it is (, IR)-perfect. Proof. Assume G is (α, )-perfect and consider any H G. Then H is also (α, )-perfect. Let X be an IR(H )-set and consider the subgraph F of H induced
by N[X]. Then X is a dominating set of F. Since X is an IR-set of H, pnH (x, X) = ∅ for each x ∈ X. But pnH (x, X) ⊆ N[X] for each x ∈ X; hence pnH (x, X) = pnF (x, X) = ∅ for each x ∈ X. Therefore X
is irredundant in F, and (see (1)) it follows that X is a minimal dominating set of F. Hence (F ) ≥ |X| = IR(H ). Since H is (α, )-perfect, α(H) = (H) and α(F) = (F). Thus IR(H ) ≤ (F ) = α(F ) ≤ α(H
) = (H ) ≤ IR(H ), that is, (H ) = IR(H ). Hence G is (, IR)-perfect.
C. M. Mynhardt and A. Roux
To see that a (, IR)-perfect graph need not be (α, IR)-perfect, note that α(K2 K3 ) = 2 and (K2 K3 ) = IR(K2 K3 ) = 3. Therefore K2 K3 is not (α, IR)-perfect. For any vertex v, (K2 K3 − v) = IR(K2 K3
− v) = 2, and it now follows easily that for each H K2 K3 , either (H ) = IR(H ) = 2 or (H ) = IR(H ) = 1. Thus K2 K3 is (, IR)-perfect. Not much is known about (α, )-imperfect (, IR)-perfect graphs,
but Cockayne, Favaron, Goddard, Grobler and Mynhardt [36] showed that if IR(G) > α(G) = 2, then IR(G) = max{r : Kr K2 G}. Problem 3.10 Characterise (α, Γ )-imperfect (, IR)-perfect graphs. We next
define a few classes of graphs that occur in the discussion below. A Meyniel graph is a graph in which every odd cycle of length five or more has at least two chords. A parity graph is a graph in
which every two induced paths between the same two vertices have the same parity; these graphs include distance-hereditary graphs and bipartite graphs and can be shown to be Meyniel graphs. A vertex
v of a graph G is a simplicial vertex if N[v] induces a clique, and a clique is a simplex if it contains a simplicial vertex. A vertex is v peripheral if v is simplicial either in G or in G. A graph
G is called a peripheral graph if every induced subgraph of G has a peripheral vertex. This family contains all chordal graphs and co-chordal graphs (graphs whose complements are chordal). A graph is
perfectly orderable if its vertex set admits a linear order < such that no induced P4 : (a, b, c, d) has a < b and d < c. Let C be the collection of all maximal cliques of a graph G. A set S ⊆ V (G)
such that |S ∩ C| = 1 for each C ∈ C is called a stable transversal of G. (Note that a stable transversal is a maximal independent set.) If each induced subgraph of G (including G itself) has a
stable transversal, then G is strongly perfect. A graph G is absorbently perfect if every induced subgraph H of G contains a minimal dominating set that has a nonempty intersection with each maximal
clique of H. Since each maximal independent set is minimal dominating, each strongly perfect graph is absorbently perfect (but the converse is false), and each absorbently perfect graph is perfect
(i.e. (χ , ω)-perfect) [91]. Jacobson and Peters [102] characterised (α, IR)-perfect graphs, a result that encompasses those obtained in several other articles, e.g. [27, 38, 84, 101]. We provide a
proof of this result because it illustrates the use of the partition of the vertex set of a graph associated with an irredundant set, as given in Section 1. Two sets of vertices X = {x1 , . . . , xk
} and Y = {y1 , . . . , yk } of G are independently matched if the only edges between X and Y are xi yi , i = 1, 2, . . . , k. Property 1 A graph G has Property 1 if, for any pair of vertex subsets X
and Y that are independently matched, α(G[X ∪ Y ]) ≥|X|. A graph with Property 1 is called a Property 1 graph. Theorem 3.11 ([102]) A graph is (α, IR)-perfect if and only if it is a Property 1 graph.
Proof. Suppose G is a Property 1 graph and consider any H G. Let X be an IRset of H, I the set of vertices that are isolated in H[X], and Z = X − I. If Z = ∅, then
α(H ) ≥ |I | = IR(H ) and thus α(H ) = IR(H ). Assume Z = ∅. For each z ∈ Z,
let yz ∈ EPN(z, X) and Y = {yz : z ∈ Z}. By the private neighbour property, Z and Y are independently matched in H and thus also in G. Since G is a Property 1 graph,
G[Z ∪ Y ] has an independent set, say A, such that |A| = |Z|. Since each vertex in I
is nonadjacent to each vertex in Z (by the definition of I) and each vertex in Y (by the private neighbour property), A ∪ I is the desired independent set of cardinality |Z| + |I | = IR(H ). Thus α(H
) = IR(H ), and it follows that G is (α, IR)-perfect. Conversely, suppose that α(H ) = IR(H ) for each H G. Suppose that for some integer k ≥ 1, there exist disjoint sets X, Y ⊆ V (G) that are
independently matched. Let H = G[X ∪ Y ]. Then each vertex in X has an X-external private neighbour in Y ; hence IR(H ) ≥ k. By assumption α(H) ≥ k, hence G has Property 1. As shown in [102] and
[90], respectively, strongly perfect graphs and absorbently perfect graphs have Property 1 and therefore are (α, IR)-perfect. The following (not necessarily disjoint) classes of graphs have been
shown to be strongly perfect and thus (α, IR)-perfect: • • • • •
comparability graphs, chordal graphs and complements of chordal graphs [6], perfectly orderable graphs [28], peripheral graphs [107], Meyniel graphs (and thus parity graphs) [119], graphs such that
all odd cycles have a common vertex (Volkmann, as cited in [90]), • permutation graphs, cographs, bipartite graphs, grids (easy to verify). It is easy to see that not all perfect graphs are (, IR)
-perfect: let G be the graph obtained from K2 Kn , n ≥ 5, by deleting two nonadjacent vertices. Then G is perfect by the strong perfect graph theorem, but (G) = 2 and IR(G) = n − 2. Other classes of
(α, IR)-perfect graphs are P4 -free graphs and (P5 , K2 K3 )-free graphs [27, as cited in [102]]; the latter result is an improvement of one in [38]. In addition, circular arc graphs have Property 1
and are therefore (α, IR)-perfect, as was proved in [84]. Gutin and Zverovich [90] also proved the previously mentioned result on (P5 , K2 K3 )-free graphs and gave a forbidden subgraph
characterisation of (α, IR)-perfect graphs (in terms of infinitely many forbidden subgraphs). Dohmen, Rautenbach and Volkmann [56] generalised (α, )- and (, IR)-perfect graphs as follows: For k ≥ 0,
a graph G is k-(α, )-perfect (k-(, IR)-perfect, respectively) if (H) − α(H) ≤ k (IR(H ) − (H ) ≤ k, respectively) for each H G. They showed that if G is k-(α, )-perfect, it is also k-(, IR)-perfect;
hence we refer to these graphs as k-(α, IR)-perfect. They generalised Property 1 to Property A(k) and showed that G is k -(α, IR)-perfect if and only if it has Property A(k). Property A(k) A graph G
has Property A(k) if, for any pair of vertex subsets X and Y that are independently matched, α(G[X ∪ Y ]) ≥|X|− k.
C. M. Mynhardt and A. Roux
3.3 (ir, IR)-Graphs We begin with the characterisations of the classes of bipartite and chordal (ir, IR)graphs (a.k.a. well irredundant graphs) given by Topp and Vestergaard [125]. The corona of K3
with an end-vertex deleted is known as the bull, usually denoted B. Theorem 3.12 ([125]) Let G be a nontrivial connected graph. (i) If G is bipartite, then G is an (ir, IR)-graph if and only if G is
a (γ , Γ )-graph if and only if G = C4 or G = H ◦ K1 for some connected bipartite graph H. (ii) If G is chordal, then G is an (ir, IR)-graph if and only if every vertex of G belongs to exactly one
simplex, and if G has the bull B as induced subgraph, then the unique vertex of degree two in B is not a simplicial vertex of G. (iii) (Corollary to (ii)) If G is a block graph, then G is an (ir, IR)
-graph if and only if G is a generalised corona H ◦{Hv : v ∈ V (H)}, where H is a connected block graph and every graph of the family {Hv : v ∈ V (H)} is complete. Topp and Vestergaard [125] also
characterised the (ir, IR)-graphs belonging to a class of graphs containing 5-cycles, while Finbow and van Bommel [79] characterised the (ir, IR)-graphs belonging to a class of planar graphs
containing many copies of K4 ; these characterisations are quite lengthy, and we omit them here.
3.4 (ir, γ )-Graphs The inflated graph GI of a graph G is obtained by replacing every vertex xi of degree di by a clique Xi of order di , and each edge xi xj by an edge uv, where u ∈ V (Xi ), v ∈ V
(Xj ), and different edges of G are replaced by nonadjacent edges in GI . Dunbar and Haynes [58] conjectured that inflated graphs are (ir, γ )-graphs, but Favaron [68] (see Theorem 5.12) showed that
in general the difference between the parameters can be arbitrarily large. However, Puech [116] proved the conjecture for inflated trees. Theorem 3.13 ([116]) If T is a tree, then ir(TI ) = γ (TI ).
3.5 (α, IR)- and (Γ, IR)-Graphs Let (P, ≤) be a poset. The upper bound graph G of P has V (G) = P, and xy ∈ E(G) if and only if x = y, and there exists z ∈ P such that x, y ≤ z. Cheston, Hare,
Hedetniemi and Laskar [27, as cited in [102]] showed that if G is an upper bound graph, then α(G) = IR(G). However, G is not necessarily (α, IR)-perfect; an illustrating example is given in [102].
Fig. 6 A flower with large faces F1 and F2
Dunbar and Haynes [58] proved that inflated graphs of trees are (α, IR)-graphs, while Favaron [69] characterised 2-connected graphs whose inflated graphs are (α, IR)-graphs. A flower is a 2-connected
planar graph with a plane representation in which all edges lie on the boundary of one or both of two faces, called the large faces. The f − 2 other faces are called the petals. See Figure 6 for an
example of a flower. Theorem 3.14 Let G be a graph of order n ≥ 2. (i) [58] If T is a tree, then α(TI ) = IR(TI ) = n − 1. (ii) [69] If G is 2-connected, then α(GI ) = IR(GI ) if and only if G is a
subdivision of K4 in which the boundaries of all four faces have odd length or a flower in which the boundaries of all petals have odd length. (In either case α(GI ) = IR(GI ) = n.) Favaron [65] (see
Proposition 4.4) showed that IR(G) ≤ n − δ(G) for any n-vertex graph G. Cockayne and Mynhardt [51] characterised graphs for which equality holds, and it turns out that these graphs are (, IR)-graphs
and sometimes (α, IR)-graphs. Proposition 3.15 ([51]) Let G be a connected graph of order n and minimum degree δ. (i) If IR(G) = n − δ, then (G) = IR(G), and if, in addition, δ < α(G) = IR(G). (ii)
If IR(G) = n/2, then (G) = IR(G).
n 2,
4 Bounds Involving Other Graph Parameters As can be expected, there are many bounds for ir and IR in terms of other graph parameters. We mention bounds for ir and IR separately and distinguish
between bounds involving only one of these parameters among those in the domination chain, the sum of two parameters π (G) and λ(G) (called Gallai-type results after Gallai’s Theorem [23, Theorem
12.11]) and the sum π(G) + π(G) and product
C. M. Mynhardt and A. Roux
π(G) · π(G) of a parameter π of a graph and its complement (called NordhausGaddum-type results after the Nordhaus-Gaddum Theorem [23, Theorem 14.23]). We first consider bounds that hold for general
graphs (with perhaps some degree restrictions) and then bounds for specific classes of graphs.
4.1 General Graphs 4.1.1
Bounds for ir
The upper bounds ir(G) ≤ n/2 and ir(G) ≤ n − for any graph G of order n follow directly from the classical bounds γ (G) ≤ n/2 of Ore (see [92, Theorem 2.1]) and γ (G) ≤ n − of Berge (see [92, Theorem
2.11]). Domke, Dunbar and Markus [57] showed that it is possible to find graphs for which γ = n − and ir < n − . They construct an infinite class of such graphs, the smallest of which is the graph G
(depicted in Figure 7) obtained as follows: begin with P7 : (v1 , . . . , v7 ), join v3 and v5 , join a new vertex x to v2 , v3, v5 , v6 and another new vertex y to x. Then (G) =deg(x) = 5, γ (G) = 4
(many γ -sets) and ir(G) = 3 = |{y, v3 , v5 }|. Topp and Vestergaard [125] characterised graphs with ir = n/2, obtaining the same class of graphs as those for which γ = n/2, as determined in [113]
and later in [81], namely, graphs for which each component is either C4 or a corona H ◦ K1 , where H is an arbitrary connected graph. Favaron and Mynhardt [74] gave a fairly complicated
characterisation of graphs for which ir = n − .
y Fig. 7 A graph G satisfying γ (G) = n − = 4 (the black vertices form a γ -set) and ir(G) = 3 = |{y, v3 , v5 }|
Blidia, Chellali and Maffray [9] improved Berge’s bound for ir(G). For any vertex v of a graph G, let αv (G) denote the cardinality of a maximum matching
(G) = max{α (G) : v ∈ V (G) and deg(v) = }. of G − N[v] and α v
. Theorem 4.1 ([9]) For any graph G and μ ∈ {ir, γ , i}, μ(G) ≤ n − − α
Bollobás and Cockayne [11] obtained the lower bound ir(G) ≥ n/(2 − 1), which is attained by paths Pn , n ≡ 0 (mod 3). This bound was improved by Cockayne and Mynhardt [50]. Theorem 4.2 ([50]) If G
has order n and maximum degree Δ ≥ 2, then ir(G) ≥ The bound is sharp for each value of Δ ≥ 2.
2n 3 .
Burger, Henning and van Vuuren [17] found a lower bound for ir in terms of the lower packing number ρ L (G), which is the minimum cardinality of a maximal 2-packing of G. Theorem 4.3 ([17]) If G is a
connected graph such that ir(G) > 1, then ir(G) ≥ 2 3 (1 + ρL (G)). This bound is sharp (e.g. for the graph G1 in Figure 4). When is large relative to n, it can also be a better lower bound for ir
than the bound in Theorem 4.2. For example, the graph G1 can be generalised by replacing its triangle with F ∼ =Kr , where r ≥ 4, and appending a path of length 2 to each but one vertex of F. Denote
this graph by Hr . See Figure 8 for the case r = 6, where a ρ L -set is shown in green and an ir-set in blue. In general, ρL (Hr ) = ir(Hr ) = r − 1. Since Hr has order n = 3r − 2 and = 2, while
Theorem maximum degree = r, Theorem 4.2 gives ir(Hr ) ≥ 6r−4 3r 4.3 gives ir(Hr ) ≥
+ ρL (Hr )) = 2r3 . On the other hand, the former bound is 2n better for, e.g. paths Pn , since 3 = ir(Pn ) = n3 and ρL (Pn ) = n5 . 2 3 (1
Fig. 8 The graph H6 for which ρL (H6 ) = ir(H6 ) = 5
C. M. Mynhardt and A. Roux
Bounds for coir and oir
Finbow [78] proved lower bounds for the CO-irredundant number. For any graph G of order n and maximum degree ,
coir(G) ≥
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n 2
if = 2
4n 13
if = 3
2n if ≥ 4. 3 − 3
The bounds are best possible and the extremal graphs are characterised. Lower bounds for the open irredundance number were obtained in [35], and an extremal graph in each case was exhibited. For any
graph G of order n, maximum degree and without isolated vertices, if = 1, then oir(G) = n2 ; if = 2, then oir(G) ≥ n3 , otherwise
oir(G) ≥
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
2n 11
if = 3
n 8
if = 4
(3 − 1)n if ≥ 5. 23 − 52 + 8 − 1)
Bounds for IR
Since every vertex of a maximum independent set S of a graph G of order n has at least δ neighbours in V (G) − S, α(G) ≤ n − δ. Favaron [65] showed that the same bound holds for IR(G) (which can also
be deduced from results by [38] mentioned below), while Henning and Slater [100] and Cockayne and Mynhardt [51] bounded IR for regular graphs. Graphs for which equality holds in either case were
characterised in [51]. Proposition 4.4 (i) [65] For any graph G of order n, IR(G) ≤ n − δ. (ii) [51] IR(G) = n − δ if and only if G is one of the following graphs: G = Kn−δ ∨ H , where H is any graph
of order δ(G) and minimum degree at least 2δ(G) − n, or δ ≥ n/2 and G = (K2 Kn−δ ) ∨ F , where F is any graph of order 2δ(G) − n and minimum degree at least 3δ − 2n.
Proposition 4.5 ([51, 100]) For any r ≥ 1, if G is an r-regular graph of order n, then IR(G) ≤ n/2. Equality holds if and only if each component of G is either an r-regular (hence balanced) bipartite
graph or K2 H , where H is an r − 1-regular graph. Bacsó and Favaron [4] generalised the bound in Proposition 4.5 to non-regular graphs. The bound is better than the bound IR ≤ n − δ in Proposition
4.4 for small δ, namely, when δ + < n. n Proposition 4.6 ([4]) For any graph G, IR(G) ≤ 1+δ/ . Equality holds if and only if G is a bipartite graph such that all vertices in the same partite set have
the same degree, or G is a regular graph described in Proposition 4.5.
Aouchiche, Favaron and Hansen [3] obtained further upper bounds as well as a lower bound for IR in terms of order and maximum degree. Proposition 4.7 ([3])
√ (i) For any graph G of order n, IR(G) ≤ n − 2 n − 1 + , and there exists a graph that achieves equality in the bound. (ii) For any connected graph G of order n, IR(G) ≤ 12 · n2 , with equality if
and only if G is a path, an even cycle or a claw. (iii) Let√G be a connected graph of order n ∈{4, 6} or n ≥ 8. Then IR(G) ≥ 2 n − . The bound can be reached for any n ∈{4, 6} or n ≥ 8. Finally,
Hedetniemi, Jacobs and Laskar [94] showed that IR(G) ≤ r(N (G)) and OIR(G) ≤ r(G), where r(G) denotes the rank of the adjacency matrix A(G) of G and r(N(G)) the rank of the closed neighborhood matrix
N(G) = A(G) + I.
Nordhaus-Gaddum-Type Results
Cockayne and Mynhardt [45] showed that for every graph G of order n, IR(G) · IR(G) ≤
n(n + 2 . 4
The n graph G attains the bound if and only n if G or G consists of (i) a set X of independent vertices, (ii) a set Y of 2 2 vertices where G[Y ] is complete and X ∩ Y = {x} and (iii) an arbitrary
set E of edges that join vertices in X −{x} to vertices in Y −{x}. For CO-irredundance, it was shown in [44] that for any graph G of order n, COIR(G) + COIR(G) ≤ n + 2 and
COIR(G) · COIR(G) ≤ (n + 2)2 /4 ,
C. M. Mynhardt and A. Roux
and that the bound can be attained for all even values of n. For open irredundance, Cockayne [32] showed that for any graph G of order n ≥ 16, OIR(G) + OIR(G) ≤ 3n/4 and OIR(G) · OIR(G) < 9n2 /64.
The bound for OIR(G) + OIR(G) can be attained if n ≡ 0(mod 4) and extremal graphs were exhibited.
Gallai-Type Results for ir and IR
Cockayne et al. [38] proved that if G has no isolated vertices and X is an irredundant set, then V (G) − X is dominating. They deduced that γ (G) + IR(G) ≤ n and hence ir(G) + IR(G) ≤ n. If δ ≥ 1 and
γ (G) + IR(G) = n, then α(G) = (G) = IR(G) (the converse is false). If δ ≥ 2, then ir(G) + IR(G) √ ≤ γ (G) + IR(G) ≤ n − δ + 2. They conjectured that i(G)+IR(G) ≤ 2(n+δ − 2nδ). This conjecture was
proved by Wang [129] and by Favaron [65], who also presented graphs that attain equality. Proposition 4.8 For any graph G of order n, √ (i) [65, 129] i(G)√+ IR(G) ≤ 2(n + δ − 2nδ), (ii) [65] i(G) + 2
δ IR(G) ≤ n + δ. Chellali and Volkmann [24] used Brooks’s Theorem on the chromatic number χ and the result in Theorem 4.1 to bound χ (G) + μ(G) and μ(G) · χ (G), μ ∈ {ir, γ , i}. Recall that αv (G)
denotes the cardinality of a maximum matching of
(G) = max{α (G) : v ∈ V (G) and deg(v) = }. G − N[v] and α v Proposition 4.9 ([24]) For any graph G of order n and maximum degree Δ, and μ ∈ {ir, γ , i},
χ (G) + μ(G) ≤ n + 1 − α .
For μ ∈ {ir, γ }, equality holds if and only if G = H ∪ (t1 C4 ) ∪ (t2 K1 ) ∪
(Fi ◦ K1 ),
where H ∈{KΔ+1 , C5 , C7 } and Δ(H) = Δ, ti is a nonnegative integer for i = 1, 2, I is a (possibly empty) set of indices and Fi , i ∈ I, is a connected graph. Theorem 4.10 ([24]) Let G = C5 , C7 be
a connected graph of order n ≥ 4 and μ ∈ {ir, γ , i}. Then
# μ(G)χ (G) ≤
(G))2 (n − α 4
$ .
Equality holds if and only if • G ∈{K4 , C9 , C11 }, or
(G) with either n − α (G) − 2 = 0 when • χ (G) = Δ, μ(G) = n − − α
(G) is odd. n − α (G) is even, or n − α (G) − 2 = ±1 if n − α If G is bipartite, then equality holds if and only if G ∈{C4 , P4 , P5 , P7 }. Corollary 4.11 ([24]) If G is a connected graph of order n
≥ 4, then ir(G)χ (G) ≤ n2 /4.
4.2 Specific Graph Classes 4.2.1
As mentioned above, ir(G) ≤ n − for all graphs G. Domke et al. [57] showed that equality holds for a tree T if and only if T = K1 , K1,r , r ≥ 1, P4 or a spider obtained from K1,r by subdividing at
most r − 1 edges. 2n The lower bound ir(G) ≥ 3 given in Theorem 4.2 can be improved for trees, as shown by Cockayne [33] and Poschen and Volkmann [114]. Theorem 4.12 (i) [114] For a tree T with order
n and leaves, ir(T ) ≥ n+2− 3 . Equality holds if and only if the distance between each pair of distinct leaves in T is congruent to 2 (mod 3). (ii) [33] If T = K1,n−1 is a tree of order n and
maximum degree Δ ≥ 3, then ir(T ) ≥ 2(n+1) 2+3 . The bound in Theorem 4.12(i) is better if there are relatively few leaves, while the bound in Theorem 4.12(ii) is better if many non-leaf vertices
have degree . The trees for which equality holds in (ii) in the case where ir is even were also characterised in [33].
Claw-Free Graphs
Favaron [70] investigated upper bounds for IR in claw-free graphs, as well as graphs which attain or nearly attain equality in the bounds. Theorem 4.13 ([70]) Every connected claw-free graph G of
order n satisfies n+1 n IR(G) ≤ n+1 2 . If IR(G) = 2 , then α(G) = IR(G), and if IR(G) = 2 , then IR(G) = (G).
C. M. Mynhardt and A. Roux
Corollary 4.14 If G is a claw-free graph of order n and δ(G) ≥ 2, then IR(G) ≤ n 2 . Proof. As proved in [108], if G is a claw-free graph of order n and minimum degree δ, then α(G) ≤ 2n/(δ + 2).
Since δ ≥ 2, α(G) ≤ n/2. By Theorem 4.13, if IR(G) = n+1 n+1 2 , then α(G) = 2 , which is not the case. Favaron also described an infinite class F of claw-free graphs with IR(G) = n+1 2 and showed
that this class characterises claw-free graphs with IR(G) = n+1 . She 2 then investigated claw-free graphs G such that IR(G) = n2 . Theorem 4.15 ([70]) Let G be a connected claw-free graph of order n
≥ 7. (i) If n is even and IR(G) = n2 , then δ(G) = n2 or 1 ≤ δ(G) ≤ n4 . Moreover, for every integer r between 1 and n/4 or equal to n/2, there exists a connected claw-free graph G of order n such
that IR(G) = n/2 and δ(G) = r. n−1 n+3 (ii) If n is odd and IR(G) = n−1 δ(G) ≤ n+1 2 , then 2 ≤ 2 or 1 ≤ δ(G) ≤ 4 .
n+1 } ∪ { n−1 Moreover, for every integer r ∈ {1, . . . , n+3 4 2 , 2 }, there exists a connected claw-free graph G of order n such that IR(G) =
n−1 2
and δ(G) = r.
Other Graphs
For inflated graphs, Dunbar and Haynes [58] stated it as an open problem to bound IR(GI ). Favaron [68] proves that (i) IR(GI ) ≤ m(G) for every graph G of size m without isolated vertices, where
equality holds if G is bipartite, and (ii) IR(GI ) ≤ 2 n (G)/4 . Favaron and Puech [75] showed that the lower irredundance number ir of a plane, cylindrical or toroidal grid of order m × n (i.e. GH ,
where G ∈{Pm , Cm } and H ∈{Pn , Cn }) is at least mn/5 and is asymptotically equal to mn/5 when m and n tend to infinity.
5 Differences Between Parameters in the Domination Chain Many authors have obtained results on the differences between irredundance numbers (usually IR) and other parameters in the domination chain,
mostly in terms of order, or order and maximum degree, and occasionally also involving the chromatic number χ . Others have bounded the ratios of irredundance numbers to domination and independence
numbers, mostly for special graph classes.
5.1 Differences Between Lower Parameters Allan, Laskar and Hedetniemi [2] observed that the inequality 2 ir(G) − γ (G) ≥ 1 obtained in [1, 10] can be improved to 2 ir(G) − γ (G) ≥ (k + 1), where k is
the maximum number of isolated vertices in an ir-set. Zverovich [137] bounded γ − ir in terms of order, and i − ir in terms of maximum degree and order. Theorem 5.1 ([137])
. (i) For any graph G of order n ≥ 3, γ (G) − ir(G) ≤ n−3 4 (ii) For any graph G of order n with maximum degree Δ ≥ 3, i(G) − ir(G) ≤ min
% 2 − 3 −1 n , n− − 1. 2 − 1 2
5.2 Differences Between Upper Parameters Henning and Slater [100] conjectured that (G) = IR(G) if G is a cubic graph. The conjecture is false (see Theorem 5.14), but generated considerable interest
in bounding the differences IR −α and IR − for general graphs and the ratio IR / for special classes of graphs. Rautenbach [118] was the first to bound the abovementioned differences in terms of
order and to characterise the extremal graphs. Theorem 5.2 ([118]) (i) For any graph G of order n ≥ 4, IR(G) − α(G) ≤
n−4 2
. (ii) For any graph G of order n ≥ 6, IR(G) − (G) ≤ n−4 . 2
For even values of n ≥ 6, equality holds in (i) if and only if G = Kn/2 K2 . If n ≥ 7 and odd, equality holds in (i) if and only if G = K(n−1)/2 K2 together with a vertex u which is either an
isolated vertex, or u is adjacent to all vertices of one of the copies of K(n−1)/2 , or there is a pair of adjacent vertices x and y, one from each copy of K(n−1)/2 such that u is adjacent to all
vertices except x and y (and itself). For even values of n ≥ 8, equality holds in (ii) if and only if G is obtained from K(n+2)/2 K2 by deleting two nonadjacent vertices. If n ≥ 7 and odd, equality
holds in (ii) if and only if G is one of the following types of graphs: let H be the graph obtained from K(n+1)/2 K2 by deleting two nonadjacent vertices and denote the vertex sets of the two copies
of K(n−1)/2 in H by V1 and V2 . For i = 1, 2, let vi ∈ Vi be the vertex of degree n−3 2 , i.e. the vertex not adjacent to any vertex in Vj , j = i. Form the graph G by adding a new vertex w, where w
is either (a) isolated, or (b) adjacent precisely to all vertices in (say) V1 , or (c) adjacent precisely to all, except possibly one, vertices in (say) V1 −{v1 }, or (d) there is a pair of adjacent
vertices x ∈ V1 and y ∈ V2 such that w is adjacent to all vertices except x and y (and itself).
C. M. Mynhardt and A. Roux
Rautenbach also bounded IR(G) − α(G) (and hence IR(G) − (G) and (G) − α(G)) in terms of order and maximum degree and conjectured that if (G) ≤ 3, then IR(G) − α(G) ≤ n6 . Theorem
5.3 ([118]) For any graph G with maximum degree Δ, IR(G) − α(G) ≤ (−1)2 n . 22 Bacsó and Favaron [4] and Zverovich [137] obtained Rautenbach’s conjecture as a corollary to the following result.
Theorem 5.4 Let G be a connected graph of order n, chromatic number χ and maximum degree Δ ≥ 2. Then
(i) [4, 137] IR(G) − α(G) ≤ −2 2 n . (ii) [4] Equality holds if and only if (a) G is a path or a cycle or (b) Δ ≥ 3 and G = Kn/2 K2 , in which case IR = = n/2, α = n/ and χ = Δ. We see from Theorem
5.4(ii) that for fixed ≥ 3, there exist arbitrarily large connected -regular graphs that satisfy equality: for arbitrary r ≥ 1, join one copy of rKn/2 to another one by a perfect matching that
ensures the graph is connected. −2 Favaron also mentioned that the stronger result IR(G) − α(G) ≤ χ2χ n holds for χ ≥ 2. When χ < n/2, this bound is better than Rautenbach’s bound IR(G)−α(G) ≤ n−4 2
(Theorem 5.2(i)). The bound in Theorem 5.4(i), although asymptotically the same as Rautenbach’s bound, is better for specific values of and immediately proves Rautenbach’s conjecture. (The conjecture
was subsequently also proved in [109, 130].) Corollary 5.5 ([4, 137]) If Δ(G) = 3, then IR(G) − α(G) ≤ n6 . Zverovich also showed that this bound can be improved for triangle-free cubic graphs. A
generalised Petersen graph can be described as a cubic graph obtained by joining the vertices of a regular polygon to the corresponding vertices of a star polygon. Theorem 5.6 ([137]) (i) If G is a
triangle-free cubic graph of order n, then IR(G) − α(G) ≤
n 7
Equality is attained by any generalised Petersen graph of order 14. (ii) If G contains no Kq (q ≥ 3) and Δ ≥ 1, then IR(G) − α(G) ≤
+q −4 n . 2( + q)
Henning [97, p. 57] questioned whether there exists a cubic graph G such that ir(G) < γ (G) < i(G) < α(G) < (G) < IR(G). That this is indeed the case
was proved by Zverovich and Zverovich [135]. The graph in their construction has connectivity 2, and they asked whether there exists a 3-connected cubic graph with the same property. Theorem 5.7
([135]) For any nonnegative integers k1 , . . . , k5 , there exists a cubic graph G with connectivity 2 such that γ (G) − ir(G) ≥ k1 , i(G) − γ (G) ≥ k2 , α(G) − i(G) ≥ k3 , (G) − α(G) ≥ k4 and IR(G)
− (G) ≥ k5 .
5.3 Ratios of Lower Parameters As shown in [1, 10], γ (G)/ ir(G) < 2 for any graph G. To see that this ratio can be arbitrarily close to 2, construct the graph Gk , k ≥ 2, as follows. (See Figure 9
for k = 3.) Begin with the corona Kk ◦ K1 and subdivide each pendant edge. Let X = {x1 , . . . , xk } be the vertex set of Kk , Y = {y1 , . . . , yk } the vertices of degree 2 and R = {r1 , . . . ,
rk } the end-vertices; assume that each yi is adjacent to xi and ri . For each pair of distinct integers i, j ∈{1, . . . , k}, join a new vertex cij to xi and xj . This is the graph Gk . Now X is a
maximal irredundant set in which pn xi , X) = {yi } for each i, and it is not difficult to see that X is an ir-set and ir(Gk ) = k. Let D be a dominating set of Gk . To dominate the end-vertices, {yi
, ri } ∩ D = ∅ for each i; hence |D ∩ (Y ∪ R)|≥ k. To dominate cij , {xi , xj , cij } ∩ D = ∅. After some thought, this implies that |D ∩ (X ∪{cij : i, j ∈{1, . . . , k}})|≥ k − 1. Since X ∪ Y −{xk }
dominates Gk , γ (Gk ) = 2k − 1. Therefore limk→∞ γ (Gk )/ ir(Gk ) = 2. For many graph classes, though, the bound on the ratio can be improved. We summarise the results below. The cyclomatic number μ
(G) of G is given by μ(G) = |E(G)|−|V (G)| + k(G), where k(G) denotes the number of components of Fig. 9 The graph G3 with ir(G3 ) = 3 and γ (G3 ) = 5
C. M. Mynhardt and A. Roux
G. A claw-free block graph is a graph, all of whose blocks are claw-free. A blockcactus graph is a graph whose blocks are either complete or induced cycles. Theorem 5.8 (i) [55] For any tree T, γ (T
)/ ir(T ) < 32 . (ii) [126] For any block graph G and for any graph G with cyclomatic number μ(G) ≤ 2, γ (G)/ ir(G) ≤ 32 . The bound 32 is best possible for block graphs and does not hold if μ(G) ≥
3. (iii) [73] If G is a claw-free graph, an inflated graph or the line graph of a trianglefree or a bipartite graph, then γ (G)/ ir(G) ≤ 32 . (iv) [72] If G is a claw-free block graph, then γ (G)/ ir
(G) ≤ 74 . (v) [136] If G is a block-cactus graph having π (G) induced cycles of length 2 (mod 4), then γ (G)/ ir(G) ≤ (8π(G) + 6)/(5π(G) + 4). Volkmann [126] conjectured that γ (G)/ ir(G) < Theorem
5.8(v) implies this inequality.
for any cactus graph.
Corollary 5.9 ([136]) If G is a block-cactus graph, then γ (G)/ ir(G) < bound is asymptotically best possible.
8 5.
Henning and Slater [100] showed that the difference γ − ir can be arbitrary for cubic graphs. For the graphs they constructed, γ / ir ≥ 15 13 . This suggests the following problem. Denote the maximum
ratio of two parameters π and λ by & ' & ' max πλ . Hence, by Corollary 5.9, max γir → 85 for block-cactus graphs. & ' Problem 5.10 Determine or bound max γir for cubic graphs. By the abovemen& '
tioned construction, max γir ≥ 15 13 . Since a tree is (ir, γ )-perfect if and only if it is T1 -free, where T1 is the Slater tree in Figure 4, and γ (T1 )/ ir(T1 ) = 54 , we also have the following
problem. & ' & ' Problem 5.11 Determine max γir for trees. Is it true that max γir = 54 for trees? For inflated graphs, Dunbar and Haynes [58] conjectured that ir(GI ) = γ (GI ) for any graph G.
Puech proved this conjecture if G is a tree (see Theorem 3.13), but Favaron [68] gave a construction of 2-connected graphs G to show that the difference can be arbitrarily large. For the graphs Gk
constructed, the ratio 5 γ (GkI )/ ir(GkI ) ≥ limk→∞ 5k+2 4k+2 = 4 . Theorem 5.12 ([68]) For every positive integer k, there exists a 2-connected graph Gk such that γ (GkI ) − ir(GkI ) ≥ k. & '
Problem 5.13 Determine max γir for inflated graphs.
5.4 Ratios of Upper Parameters Henning and Slater [100] conjectured that (G) = IR(G) if G is cubic, but this was proved to be false in [51] and [118]. Theorem 5.14 (i) [51] For any positive integer
k, there exists a 2-connected cubic graph Hk such that IR(Hk ) − (Hk ) ≥ k. (ii) [118] For any positive integer k and any integer r ≥ 3, there exists a connected r-regular graph Hr,k such that IR
(Hr,k ) − (Hr,k ) ≥ k. The ratio IR / for the cubic graphs Hk in Theorem 5.14(i) is IR / = 76 , while the ratios for the graphs constructed by Rautenbach in Theorem 5.14(ii) are
IR / ≥
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
18 17 16 15 3r 2r+4 4r 3r+4
for cubic graphs, for 4-regular graphs for r-regular graphs where r ≥ 6 is even for r-regular graphs where r ≥ 5 is odd.
We know the ratio IR /α is unbounded for regular graphs (K2 KIR ), as is the ratio IR / for non-regular graphs (e.g. K2 KIR +1 − e, where e is an edge joining a vertex of one copy of KIR +1 to
another). Cockayne and Mynhardt [49] exhibited an infinite class of triangle-free graphs for which the difference between the upper irredundance and domination numbers is arbitrarily large, thus
answering a question of [63]. The ratio for the given graphs Gk is IR(Gk )/ (Gk ) = Problem 5.15 Determine max graphs.
4k . 3k + 2
for (i) r-regular graphs and (ii) triangle-free
6 Criticality and Stability We next consider how the upper and lower irredundance numbers change with the removal of a vertex or an edge or with the addition of an edge. For a graph parameter π , a
graph G is: • π -critical (π + -critical) if π (G − v) < π (G) (π (G − v) > π (G)) for every vertex v ∈ V (G),
C. M. Mynhardt and A. Roux
• π -edge-critical (π + -edge-critical) if π (G + e) < π (G) (π (G + e) > π (G)) for every edge e ∈ E(G), • π -edge-removal critical (π − -edge-removal critical), abbreviated to π -ERcritical (π −
-ER-critical), if π (G − e) > π (G) (π (G − e) < π (G)) for every edge e ∈ E(G).
6.1 Criticality All edgeless graphs with more than one vertex are π -critical and π -edge critical for π ∈ {ir, IR}. Furthermore, the complete graph Kn , n ≥ 2, is IR-ER-critical, while the star K1,n
, n ≥ 1, is ir-ER-critical. Topp [124] and Grobler [85] showed that there exist no π + -critical graphs for π ∈ {ir, IR}, no ir+ -edge-critical graphs and no IR− ER-critical graphs. Theorem 6.1 For
any nontrivial graph G, (i) (ii) (iii) (iv)
[124] IR(G − v) ≤ IR(G) for all v ∈ V (G); [85] ir(G − v) ≤ ir(G) for at least one v ∈ V (G); [85] if GKn , then ir(G + uv) ≤ ir(G) for at least one uv ∈ E(G); [85] if G ∼ Kn , then IR(G − uv) ≤ IR
(G) for at least one uv ∈ E(G). =
Proof. (i) An IR-set S of G − v is also irredundant in G and therefore IR(G) ≥ |S| ≥ IR(G − v). (ii) Let S be an ir-set of G. If S is also dominating, then for v ∈ V (G) − S, the set S is still
dominating in G − v. From the domination chain, it follows that ir(G − v) ≤ γ (G − v) ≤ |S| = ir(G). If S is not dominating, then S is an irredundant set of G − v for v ∈ R = V (G) − N[S]. If S is
not a maximal irredundant set of G − v, then there exists a vertex x ∈ V (G − v) − S such that S ∪{x} is an irredundant set of G − v. But then S ∪{x} is also an irredundant set of G, contradicting
the maximality of S in G. Hence S is a maximal irredundant set of G − v and ir(G − v) ≤ |S| = ir(G). (iii) If ir(G) = γ (G), then let S be a dominating ir-set of G. Since S is also a dominating set
of G + uv, it follows from the domination chain that ir(G + uv) ≤ γ (G + uv) ≤ |S| = ir(G). If ir < γ (G), let S be an ir-set of G. Then there exists an edge uv ∈ E(G) such that u, v ∈ R. If not, all
vertices in R are adjacent and S = S ∪{x}, with x ∈ R, is a dominating set of G. Since S is not irredundant, S is not a minimal dominating set and γ (G) < |S| + 1. That is, γ (G) ≤ |S| = ir(G), a
contradiction. From Theorem 1.1, it follows that S is a maximal irredundant set of G + uv and so ir(G + uv) ≤ |S| = ir(G).
(iv) Let S be an IR-set of G. If S is independent, then S is also an independent irredundant set of G − uv for every uv ∈ E(G). Hence IR(G) = |S| ≤ IR(G − v). If S is not independent, then there
exists an edge uv ∈ E(G) such that u, v ∈ S. Since S is an irredundant set of G − uv, IR(G) = |S| ≤ IR(G − uv).
Criticality of ir
First consider how the deletion of a vertex influences the lower irredundance number. While the deletion of a vertex can decrease the domination number by at most one, Favaron [67] showed that this
is not the case for the irredundance number of a graph. Theorem 6.2 ([67]) For every graph G and every vertex v of G such that ir(G−v) ≥ 2, ir(G − v) ≥ (ir(G) + 1)/2, and this bound is sharp. To
illustrate the sharpness of the bound, a graph G is constructed as follows.
Let A and A be two copies of K1,n , with V (A) = {x0 , . . . , xn } and V (A ) = {x0 , . . . , xn } where x0 and x0 are the centres. Add the edges xi xi for i = 0, . . . , n. Let r ≥ 2(n + 1). For
each each i = 0, . . . , n, add an independent set Yi = {yik : k = 1, . . . , r}, and join each vertex in Yi to xi . For each pair i, j with 0 ≤ i = j ≤ n, add an independent set Zij = {zijk : k = 1,
. . . , r}, and join each vertex in Zij to xi and xj . Finally, add a pendant vertex v to x0 . Then V (A) is an ir-set of G − v, so
ir(G − v) = n + 1, and the set (V (A) −{xn }) ∪ V (A ) is an ir-set of G (of cardinality 2n + 1). The case where n = 2 is illustrated in Figure 10. Since ir ≤ γ , it follows that if G is γ -critical
and ir = γ , then G is also ircritical. However, the classes of γ -critical and ir-critical graphs do not coincide. Grobler and Roux [89] constructed two classes of graphs that are γ -critical but
not ir-critical, while Roux [121] illustrated the existence of graphs that are ir-critical but not γ -critical.
A x1
A x1 Y1
Z01 v
Z12 Z02
x0 Y0
x2 Y2
Fig. 10 A graph G having ir(G)-set {x0 , x1 , x0 , x1 , x2 } and ir(G − v)-set {x0 , x1 , x2 }
C. M. Mynhardt and A. Roux
Similar to ir-critical graphs, if graph G is γ -edge-critical and γ = ir, then G is also ir-edge-critical. It is however still unknown whether there exist graphs which are γ -edge-critical but not
ir-edge-critical and vice versa. Problem 6.3 Investigate the intersection of the classes of γ -edge-critical and iredge-critical graphs. As for γ and i, the disjoint union of stars are also
ir-ER-critical. That these are not the only ir-critical graphs was shown in [87] and [37] where connected ir-ERcritical graphs for ir = 2 and ir = 3, respectively, were characterised. Whether ir−
-ER-critical graphs exist is still an open question. Problem 6.4 Does there exist a graph G for which ir(G − e) < ir(G) for all e ∈ E(G)?
Criticality of IR
Grobler and Mynhardt [86] showed that the classes of IR-critical and -critical graphs coincide and characterised these graphs as follows: Theorem 6.5 ([86]) If G is a connected graph with n vertices,
then the following statements are equivalent. (i) G is Γ -critical, (ii) n > 2, and for each Γ -set S and T = V (G) − S, S and T are independently matched, (iii) Γ (G) = n/2 and no Γ -set has
isolated vertices, (iv) G is IR-critical. IR-edge-critical graphs were independently characterised by Grobler and Mynhardt [88] and Dunbar, Monroe and Whitehead [59] as precisely the graphs Ka ∨ bK1
or Ka ∨ (bK1 ∪ (Kc K2 )) for a, b ≥ 0 and c ≥ 3, where V (K0 ) = ∅. Grobler and Mynhardt [88] also showed that the classes of IR-edge critical graphs and -edge-critical graphs coincide. Dunbar et al.
[59] conjectured that there do not exist any IR+ -edge-critical graphs. Cockayne, Favaron and Mynhardt [37] disproved this conjecture by exhibiting an infinite class of IR+ -edge-critical graphs for
which IR = 2. Theorem 6.6 ([37]) The graph Cm Cn is IR+ -edge-critical if and only if m, n∈{3, 4, 6}. Since Cm Cn is also + -edge-critical, we ask the following questions: Problem 6.7 (i) Do there
exist graphs that are IR+ -edge-critical but not Γ + -edge-critical? (ii) Find IR+ -edge-critical graphs with IR > 2.
Let S be an irredundant set of G and uv ∈ E(G). Then S is a uv-irredundant set if u ∈ S and v ∈ pn(u, S), and either u is an isolated vertex of S and v does not annihilate any vertex in S −{u}, or
there exists a vertex s ∈ N[v] − S that does not annihilate any vertex in S. Graphs which are IR-ER-critical were characterised in [87] as graphs for which there exists a uv -irredundant IR-set in G
for each edge uv ∈ E(G).
6.2 Stability When considering the effect on IR(G) of the addition of edges from G, a graph G is defined to be IR-insensitive if IR(G + e) = IR(G) for every e ∈ E(G). Dunbar et al. [59] characterised
IR-insensitive bipartite graphs without isolated vertices. Theorem 6.8 ([59]) A bipartite graph G containing no isolated vertex is IRinsensitive if and only if every independent set X ⊂ V (G)
satisfies the condition that |X|≤|N(X)|. Wang and Hua [131] introduced the stability number of a graph G, denoted by
SN(G), as the maximum cardinality among all sets of edges E ⊆ E(G) such that IR(G − E ) = IR(G). They showed that for any non-empty connected graph of order n ≥ 2, SN(G) ≤ n − 2, with equality when G
is a star. In a more general result, they showed that SN(G) ≤ (IR(G) − 1) − 1 for any non-empty connected graph G with IR(G) ≥ 2.
7 Chessboards Although we have not discussed the exact determination of ir and IR for specific classes of graphs (such as paths and cycles, for which this is easy), no survey on irredundance can be
complete without mentioning chessboard problems. Hedetniemi, Hedetniemi and Reynolds [93] gave a complete survey of results on domination, independence and irredundance on the different types of
chessboards up to 1998. Here we concentrate on the queens, kings and grid graphs, which, as far as we could ascertain, are the only chessboard graphs for which new irredundance results have appeared
since then. We summarise the known exact values in Table 1 and present bounds in Section 7.2.
7.1 Exact Values We only provide the provenance of results in Table 1 that do not appear in [93].
n Queens Qn Kings Kn Grids Gn
ir IR ir IR ir IR
4 2 4 3 4 4 [75] 8
6 3 [19] 7 4 9 18
5 3 [19] 5 4 9 13
Table 1 Known exact values for Qn , Kn and Gn
7 4 [52] 9 8 16 32
8 5 [103] 11 9 17 [103] 41
9 5 [103] 13 [103] 9 25 [103]
36 [103]
27 [103] 50
11 5 [103]
10 5 [103] 15 [103]
16 [71]
12 6 [103]
166 C. M. Mynhardt and A. Roux
For all values of n for which ir(Qn ) is known, namely, 1 ≤ n ≤ 13, ir(Qn ) = γ (Qn ). We do not surmise that equality holds for all n. The value IR(K8 ) = 17 is the only known case for which IR(Kn )
> α(Kn ), as α(K8 ) = 16 and α(Kn ) = IR(Kn ) for all n ≤ 7. Problem 7.1 (i) Find more values of ir(Qn ). What is the smallest n such that ir(Qn ) < γ (Qn )? (ii) Find more values of Γ (Qn ) and IR
(Qn ). What is the smallest n such that (Qn ) < IR(Qn )? (iii) What is the smallest n such that (Kn ) < IR(Kn )?
7.2 Bounds 7.2.1
Bounds for the Queens Graph
Apart from the exact values for ir(Qn ) in Table 1 and the value ir(Q13 ) = γ (Q13 ) = 7 [103], not much is known about irredundance numbers of queens graphs. The best bounds known are given by
n+1 4
≤ ir(Qn ) ≤ γ (Qn ) ≤
200 n + O(1), 393
where the lower bound follows from the bound ir(G ≥ (γ (G) + 1)/2 for all graphs and Spencer’s bound γ (Qn ) ≥ (n − 1)/2 as cited in [29] and the upper bound (for γ (Qn )) follows from a result in
[20] and a suitable dominating configuration for Q129 in [112]. For all known values of ir(Qn ), ir(Qn ) = γ (Qn ), so the lower bound above appears to be weak. Burger, Cockayne and Mynhardt [15]
showed that IR(Qn ) ≥ (Qn ) ≥ 2n − 5 for odd n ≥ 5, and IR(Qn ) ≥ (Qn ) ≥ 2n − 6 for even n ≥ 6. If n ≥ 18, then IR(Qn ) ≥ (Qn ) ≥ 5n 2 −O(1). They also obtained the upper bound IR(Qn ) ≤
( √ 6n + 6 − 8 n + n + 1 for n ≥ 9, improving the previous bound in [31]. Hedetniemi et al. [93, Open Problem 7] stated that it seems very likely that IR(Qn ) ≤ 5n or possibly even IR(Qn ) ≤ 4n. This
statement was disproved by Kearse and Gibbons [104], who obtained the bound IR(Qk 3 ) ≥ 6k 3 − 29k 2 − O(k) for every k ≥ 6, which implies that IR(Qn ) ≥ 6n − O(n2/3 ). They showed by computer that
for n = 17576 = 263 , IR(Qn ) > 5n, and concluded with the remark that it seems likely that 6n − O(n2/3 ) is also an upper bound for IR(Qn ). To summarise, the best known bounds for IR(Qn ), for n
large enough, are ) √ 6n − O(n2/3 ) ≤ IR(Qn ) ≤ 6n + 6 − 8 n + n + 1 .
C. M. Mynhardt and A. Roux
We end this section by mentioning that domination and irredundance numbers for queens on hexagonal boards were studied in [18].
Bounds for the Kings Graph
Favaron, Fricke, Pritikin and Puech [71] obtained the following lower and upper bounds for ir(Kn ) and IR(Kn ):
n2 9
≤ ir(Kn ) ≤
n+2 3
2 (3)
(n − 1)2 3
≤ IR(Kn ) ≤
n2 , 3
where upper bound in (4) holds for n ≥ 6 and all the other bounds hold for all n. For n ≡ 0 (mod 3), the bounds in (3) are equal; hence ir(Kn ) = n2 /9 in this case. The 2
− 1 if n ≡ 4 (mod 9). upper bound in (3) can be improved to ir(Kn ) ≤ n+2 3
Bounds for Grids
As mentioned in Section 4.2, Favaron and Puech [75] showed that ir(Gn ) ≥ n2 /5 and ir(Gn ) is asymptotically equal to n2 /5 when n tends to infinity. Since grids are (α, IR)-perfect, all their upper
parameters are equal (as given in [38]), namely,
α(Gn ) = (Gn ) = IR(Gn ) =
⎧ 2 n ⎪ ⎨ 2
if n is even
⎪ ⎩ n2 +1
if n is odd.
8 Irredundant Ramsey Numbers In order to solve a problem in formal logic, Frank Plumpton Ramsey (22 February 1903–19 January 1930) proved, in passing, his now-famous theorem as a “minor lemma”. This
result led to an area in extremal graph theory known as Ramsey Theory. A full version of the lemma, now known as Ramsey’s Theorem, can be found in [23, Theorem 20.1]; we state the well-known special
case for graphs
here. (In almost all cases, the edge colourings referred to below are not proper edge colourings.) Theorem 8.1 (Ramsey’s Theorem for Graphs) For any k ≥ 2 positive integers n1 , . . . , nk , there
exists a positive integer N such that any k-edge colouring of KN produces a monochromatic Kni for some i (1 ≤ i ≤ k). For fixed integers n1 , . . . , nk , the smallest integer N such that Ramsey’s
Theorem holds is called the Ramsey number r(n1 , . . . , nk ). Consider a k-edge colouring of KN in the colours 1, . . . , k, and let Hi be the spanning subgraph of KN whose edges are coloured i (1 ≤
i ≤ k). We restate Ramsey’s Theorem in terms of the clique numbers ω(Hi ). Ramsey’s Theorem in Terms of Clique Numbers For any k ≥ 2 positive integers n1 , . . . , nk , there exists a positive
integer N such that, for any edge-decomposition of KN into spanning subgraphs H1 , . . . , Hk , ω(Hi ) ≥ ni for some i (1 ≤ i ≤ k). Since the clique number of a graph equals the independence number
of its complement, we can rephrase Ramsey’s Theorem in terms of independence numbers. Ramsey’s Theorem in Terms of Independence Numbers For any k ≥ 2 positive integers n1 , . . . , nk , there exists
a positive integer N such that, for any edgedecomposition of KN into spanning subgraphs H1 , . . . , Hk , α(Hi ) ≥ ni for some i (1 ≤ i ≤ k). Since any independent set is irredundant, we obtain the
following result as a corollary to Ramsey’s Theorem. Corollary 8.2 (Ramsey’s Theorem for Irredundance) For any k ≥ 2 positive integers n1 , . . . , nk , there exists a positive integer N such that,
for any edgedecomposition of KN into spanning subgraphs H1 , . . . , Hk , IR(Hi ) ≥ ni for some i (1 ≤ i ≤ k). For fixed integers n1 , . . . , nk , the smallest integer N such that Corollary 8.2
holds is called the irredundant Ramsey number s(n1 , . . . , nk ). With the exception of s(3, 3, 3), irredundant Ramsey numbers have only been determined for cases where k = 2. Note that s(m, n) is
the smallest integer N such that any graph G of order N satisfies IR(G) ≥ m or IR(G) ≥ n. By “mixing” independence, upper domination and upper irredundance numbers in the definition of s(m, n), we
obtain four additional types of Ramsey numbers.
C. M. Mynhardt and A. Roux
⎧ ⎫ mixed ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ramsey numbert (m, n) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ upper domination ⎬ The Ramsey numberu(m, n) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ mixed domination ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ramsey numberv(m, n)irredundant −
domination ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ Ramsey numberw(m, n)
is the smallest integer N such that any graph G of order N satisfies
⎧ IR(G) ≥ m or ⎪ ⎪ ⎪ ⎪ ⎪ α(G) ≥ n ⎪ ⎪ ⎪ ⎪ (G) ≥ mor ⎪ ⎪ ⎪ ⎨ (G) ≥ n ⎪ (G) ≥ m or ⎪ ⎪ ⎪ ⎪ ⎪ α(G) ≥ n ⎪ ⎪ ⎪ ⎪ ⎪ IR(G) ≥ m or ⎪ ⎩ (G) ≥ n.
Since α(G) ≤ (G) ≤ IR(G), the following inequalities are immediate. For all positive integers m and n, s(m, n) ≤ w(m, n) ≤
t (m, n) u(m, n) ≤ v(m, n)
% ≤ r(m, n).
Hence all the usual upper bounds for r(m, n) (see, e.g. [23, Chapter 20]) and lower bounds for s(m, n) also hold for the other types of Ramsey numbers. The definitions imply that, like r(m, n), s(m,
n) = s(n, m) and u(m, n) = u(n, m) for all m and n, but this is not true for t, v and w. E. J. Cockayne conceived the concept of irredundant Ramsey numbers and supervised R. C. Brewster’s Masters
thesis [12] on the topic. Mixed Ramsey numbers were introduced by Cockayne, Hattingh, Kok and Mynhardt [39]; upper domination Ramsey numbers by Oellermann and Shreve, as cited in [98]; mixed
domination Ramsey numbers by Henning and Oellermann [99]; and irredundantdomination Ramsey numbers by Burger and van Vuuren [21].
8.1 Exact Values Exactly like the classical Ramsey numbers, it is easy to see that f (1, n) = f (n, 1) = 1 and f (2, n) = f (n, 2) = n for each n, where f ∈{r, s, t, u, v, w}, hence we are only
interested in f (m, n) where m, n ≥ 3. We list the known values of these Ramsey numbers in Table 2, where we only mention the provenance of the non-classical numbers, as those of r(m, n) are freely
available on the internet. There is only one known Ramsey number r(n1 , . . . , nk ), k ≥ 3, namely, r(3, 3, 3) = 17. There is similarly only one known irredundant Ramsey number s(n1 , . . . , nk ),
k ≥ 3, namely, s(3, 3, 3) = 13, determined by Cockayne and Mynhardt [46, 48]. The only other information we have on the numbers f (3, 3, 3) is the bounds 13 ≤ u(3, 3, 3) ≤ 14, as shown by Henning and
Oellermann [98].
Table 2 Known Ramsey numbers f (m, n), f ∈{r, s, t, u, v, w} n 3 4 5 6 s(3, n) 6 [13] 8 [13] 12 [13] 15 [14] w(3, n) 6 [21] 8 [21] 12 [21] 15 [21] w(n, 3) 8 [21] 12 [21] 15 [21] u(3, n) 6 [99] 8 [99]
12 [99] 15 [99] v(3, n) 6 [99] 9 [99] 12 [99] 15 [99] v(n, 3) 8 [21] 13 [21] 17 [21] t(3, n) 6 [39] 8 [39] 13 [39] 15 [99] t(n, 3) 9 [39] 12 [39] 17 [21] r(3, n) 6 9 14 18 s(4, n) 13 [34] w(4, n) 13
[21] u(4, n) 13 [21] v(4, n) 14 [21] t(4, n) 14 [21] r(4, n) 18 25 1 Also [26], without a computer search (which was used in [40]). 2 The Ramsey number r(3, 9) = 36 is also known.
7 18 [40]1 18 [16]
8 21 [22] 21 [22]
18 [21]
22 [21]
Table 2 reveals obvious gaps in the literature on irredundant Ramsey numbers. With increasing computing power, some missing results, stated below, should be within reach. Problem 8.3 (i) Determine w
(7, 3), u(3, 7), v(3, 7), v(7, 3) and t(7, 3). (ii) Determine or bound w(8, 3), u(3, 8), v(3, 8), v(8, 3) and t(8, 3). (iii) Burger and van Vuuren [22] showed that 24 ≤ s(3, 9) ≤ t(3, 9) ≤ 27.
Improve these bounds or determine s(3, 9) and t(3, 9). Determine or bound f (3, 9) and f (9, 3) for f ∈{t, u, v, w}. (iv) Determine f (3, 3, 3) for f ∈{t, u, v, w}. The CO-irredundant Ramsey number c
(m, n) is the smallest integer N such that every graph G of order N satisfies COIR(G) ≥ m or COIR(G) ≥ n. They were defined in [122] and also studied in [43, 53]. Since IR(G) ≤ COIR(G) for all
graphs, c(m, n) ≤ s(m, n) for all values of m and n. Hence c(1, n) = 1 for all n. Also, since COIR(K2 ) = COIR(K2 ) = 2, c(2, n) = 2 for all n. The following non-trivial numbers have been determined:
c(3, n) = n for all n ≥ 3, c(4, 4) = 6, c(4, 5) = 8, c(4, 6) = 11 and c(4, 7) = 14.
C. M. Mynhardt and A. Roux
As far as we know, there has been no hybridisation between CO-irredundant and the other types of Ramsey numbers mentioned above. Problem 8.4 Determine more CO-irredundant Ramsey numbers.
8.2 Bounds Analogies of bounds for r(m, n) hold for the other types of Ramsey numbers as well. For any f ∈{r, s, t, u, v, w} and integers m, n, the growth property f (m, n) ≤ min{f (m + 1, n), f (m,
n + 1)} holds, as does the following recursive upper bound. Proposition 8.5 For any f ∈{r, s, t, u, v, w} and integers m, n, f (m, n) ≤ f (m − 1, n) + f (m, n − 1); this inequality is strict if f (m
− 1, n) and f (m, n − 1) are both even. Chen, Hattingh and Rousseau [25] obtained an asymptotic lower bound for t(m, n) that was soon improved to a bound for s(m, n) by Erdös and Hattingh [60] and
Krivelevich [105]. Theorem 8.6 ([60, 105]) For every m ≥ 3, there exists a positive constant cm such that, for sufficiently large n, s(m, n) > cm
n log n
(m2 −m−1) 2(m−1)
Chen et al. [25] also obtained an upper bound for t(3, n), while Rousseau and Speed [120] bounded t(3, n), t(4, n) and t(m, 3); their bound for t(3, n) is an improvement of the bound in [25] for n
large enough. Theorem 8.7 For every positive integer n,
t (3, n) ≤
⎧⎪ 5 3 ⎪ ⎪ n 2 [25] ⎪ ⎪ ⎨ 2 ⎪ 3 ⎪ ⎪ 5n 2 ⎪ ⎪ [120] √ ⎩ log n
52 t (4, n) ≤ C √ [120]. log n Theorem 8.8 ([120]) For some positive constant c,
m c log m
2 < t (m, 3) < (1 + o(1))
m2 . log m
Since there exist graphs G such that α(G) = 2 and IR(G) = k for arbitrary k ≥ 2, the following interesting result is, perhaps, not entirely a surprise. Theorem 8.9 ([120]) limn→∞
t (3, n) = 0. r(3, n)
Problem 8.10 ([120]) Is it true that, for every m ≥ 3, limn→∞
t (m, n) = 0? r(m, n)
We can ask the same question for other types of Ramsey numbers. Let (f, g) ∈ {(c, s), (s, t), (s, w), (w, t), (w, u), (u, v), (w, r), (v, r)}. Problem 8.11 Is it true that, for every m ≥ 3, limn→∞
f (m, n) = 0? g(m, n)
9 Reconfiguration Reconfiguration problems are concerned with determining conditions under which a feasible solution to a given problem can be transformed into another such solution via a sequence of
feasible solutions in such a way that any two consecutive solutions are adjacent according to a specified adjacency relation. Reconfiguration problems model, for example, situations where we wish to
implement a sequence of predefined elementary changes in order to transform a given configuration to a more desirable one while the intermediate steps are also feasible. The reconfiguration of
dominating sets was introduced by Subramanian and Sridharan [123]. They defined the γ -graph γ · G of the graph G to be the graph
whose vertex set consists of all the γ (G)-sets, where two sets D and D are adjacent
if and only if |D ∩ D | = |D|− 1; that is, there exist vertices v ∈ D, v ∈ D such that
D = (D −{v}) ∪{v }. Here, v and v need not be adjacent in G. This version of the γ -graph is also known as the “single vertex replacement adjacency model” or the jump γ -graph. Fricke, Hedetniemi,
Hedetniemi and Hutson [82] studied the “slide adjacency model” or simply the slide γ -graph G(γ ), whose vertex set also consists
of all the γ (G)-sets, but two sets D and D are adjacent in G(γ ) if and only if there
exist adjacent vertices v ∈ D, v ∈ D such that D = (D −{v}) ∪{v }. An initial question of Fricke et al. was to determine exactly which graphs are γ -graphs; they showed that every tree is the slide γ
-graph of some graph and conjectured that every graph is such a graph. Connelly, Hedetniemi and Hutson [54] proved this conjecture. It is easy to see that if H is realisable as a γ -graph, then it is
the γ -graph of infinitely many graphs. Irredundance graphs were first considered by Mynhardt and Teshima [111]. For any parameter π ∈ {ir, i, α, , IR} (and some other domination parameters), they
defined the slide π -graph G(π ) of G similar to the slide γ -graph G(γ ) and showed that every graph is the (slide) π -graph of some graph, where π ∈ {ir, }, while not
C. M. Mynhardt and A. Roux
every graph is an i-graph. The graph GH constructed to show that a given graph H is the -graph of GH satisfies (GH ) = IR(GH ), but has more IR-sets than -sets; hence H is not an IR-graph of GH .
They left the problem of whether all graphs are IR-graphs open. However, Mynhardt and Roux [110] showed that, although all disconnected graphs can be realised as IR-graphs, this does not hold for
connected graphs. Theorem 9.1 ([110]) Every disconnected graph is the IR-graph of infinitely many graphs. To find connected graphs that are not IR-graphs, Mynhardt and Roux used the external private
neighbours in a given IR-set to find more IR-sets. The result is of interest in its own right, and we include the short proof. For an irredundant set X, we weakly partition X into subsets Z and I
(one of which may be empty), where each vertex in I is isolated in G[X] and each vertex in Z has at least one external private neighbour. (This partition is not necessarily unique. Isolated vertices
of G[X] with external private neighbours can be allocated arbitrarily to Z or I.) For each
z ∈ Z, let z ∈ epn(z, X) and define Z = {z : z ∈ Z}. Let X = (X − Z) ∪{Z }; note that
|X| = |X |. The set X is called a flip-set of X.
Proposition 9.2 If X is an IR-set of G, then so is any flip-set X of X.
Proof. Consider any x ∈ X . With notation as above, if x ∈ I = X − Z = X − Z , then
x is isolated in G[X]. Since each vertex in Z is an X-external private neighbour of
some z ∈ Z, no vertex in Z is adjacent to x. Therefore x is isolated in G[X ]. Hence
assume x ∈ Z . Then x = z for some z ∈ Z, so z is adjacent to z ∈ V (G) − X . Now z is non-adjacent to all vertices in I because the latter vertices are isolated in G[X],
and z is nonadjacent to all vertices in Z −{z}, because each v ∈ Z −{z } is an Xexternal private neighbour of some v ∈ Z −{z}. Therefore z ∈ epn(z , X ), that is,
z ∈ epn(x, X ). It follows that X is irredundant. Since |X | = |X|, X is an IR-set of G. Proposition 9.2 explains to some extent why a given connected graph H is not an
IR-graph: for a possible source graph G, an IR-set X and its flip-set X often belong to different components of G(IR) because G lacks the necessary IR-sets to form an
X − X path in G(IR). An IR-tree is a tree that is an IR-graph. All complete graphs are IR-graphs; hence K1 and K2 are IR-trees. To formulate results on IR-trees, we define some classes of trees. The
(generalised) spider Sp(1 , . . . , k ), i ≥ 1, k ≥ 2, is a tree obtained from the star K1,k with centre u by subdividing the edge uvi i − 1 times, i = 1, . . . , k. The double star S(k, n) is the
tree obtained by joining the centres of the stars K1,k and K1,n . The double spider Sp(1 , . . . , k ;m1 , . . . , mn ) is obtained from S(k, n) by subdividing the edges of the K1,k -subgraph i − 1
times, i = 1, . . . , k, and the edges of the K1,n -subgraph mi − 1 times, i = 1, . . . , n. Theorem 9.3 ([110]) (i) Stars K1,k , k ≥ 2 (trees of diameter 2), are not IR-trees.
(ii) The double star S(2, 2) is the unique smallest IR-tree with diameter three (and the unique smallest non-complete IR-tree). (iii) The double spider Sp(1, 1;1, 2) is the unique smallest IR-tree
with diameter four. (iv) The cycles C5 , C6 , C7 and the paths P3 , P4 , P5 are not IR-graphs. (v) The only connected IR-graphs of order four are K4 and C4 . As mentioned in [110], a direct proof
that P5 is not an IR-graph is somewhat simpler than the proof of Theorem 9.3(iii), but not simple enough to easily generalise to a proof that Pn or Cn , n ≥ 5, is not an IR-graph. The authors thus
stated the following conjecture and open problems. Conjecture 9.4 ([110]) Paths Pn , n ≥ 3, and cycles Cn , n ≥ 5, are not IR-graphs. Problem 9.5 ([110]) (i) Determine which spiders, double spiders
and double stars are IR-trees. (ii) Prove or disprove: Complete graphs and Km Kn , where m, n ≥ 2, are the only connected claw-free IR-graphs.
10 Complexity We conclude by briefly addressing complexity and algorithmic questions pertaining to irredundance. The decision problems that correspond to determining the lower and upper irredundance
numbers of general graphs are NP-complete [76, 95]. Computing ir(G) remains NP-complete on the classes of bipartite graphs [95], split (hence also chordal) graphs [106], partial k-trees [132], planar
cubic graphs [5] and irredundance perfect graphs [128]. It was however shown in [134] that graphs belonging to a subclass of irredundant perfect graphs, the class of locally well-dominated graphs,
are polynomial-time solvable. A linear-time algorithm also exists for determining the lower irredundance number of a tree [7]. The upper irredundance number coincides with the independence number for
bipartite and chordal graphs and can therefore be solved in polynomial time for these graph classes [38, 102]. For planar cubic graphs [5] and k-regular graphs with k ≥ 6 [83], the problem of
computing IR(G) remains NP-complete. In [8] Binkele et al. considered both exact and parameterised algorithms and showed that ir(G) can be computed in O∗ (1.99914n ) time and polynomial space.
Furthermore, an algorithm for determining IR(G) runs in O∗ (1.9369n ) time and polynomial space, where the running time can be improved to O∗ (1.8475n ) for exponential space.
C. M. Mynhardt and A. Roux
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The Private Neighbor Concept Stephen T. Hedetniemi, Alice A. McRae, and Raghuveer Mohan
1 Introduction Let G = (V, E) be a graph of order n = |V | and size m = |E|, and consider the family of all sets S ⊆ V of vertices having some desired property P. In graph theory, there are, of
course, many types of sets that are studied, for example, as a function of the types of subgraphs G[S] induced by S, the degrees of the vertices in S, and the relationships between the vertices in S
and the vertices in V \ S = S. But in particular, we would like to consider properties of sets which are called superhereditary. A property P is called superhereditary if whenever a set S has
property P, so does every superset of S. In a similar way, one defines a property P to be hereditary if whenever a set S has property P, so does every proper subset of S. It is easy to see, for
example, that the property of being an independent set is hereditary. Notice that if a set S ⊆ V has a superhereditary property P, then the entire vertex set V has property P. Thus, the largest
cardinality of a P-set is n = |V |. What is interesting, in this case, is (i) the minimum cardinality of a P-set, (ii) the maximum cardinality of a minimal P-set, and (iii) the nature of minimal
P-sets. A P-set S is called minimal if no proper subset S ⊂ S of S is a P-set. A P-set S is called 1minimal if for every vertex v ∈ S, the set S −{v} is not a P-set. It is important to note that for
superhereditary properties P, a set S is a minimal P-set if and only if S is a 1-minimal P-set.
S. T. Hedetniemi School of Computing, Clemson University, Clemson, SC, USA e-mail: [email protected] A. A. McRae () · R. Mohan Computer Science Department, Appalachian State University, Boone, NC
28608, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs,
Developments in Mathematics 66, https://doi.org/10.1007/978-3-030-58892-2_7
S. T. Hedetniemi et al.
In a 1-minimal P-set, every vertex contributes in some way to the set S having property P, and this contribution is absolutely necessary, else the set S −{v} does not have property P. It is the
nature of this private contribution that leads to the general concept of a private neighbor.
2 Private Neighbors To introduce this general concept, we present four examples of superhereditary properties P and for each, discuss the added property of being a minimal P-set. 1. A dominating set
in a graph G = (V, E) is a set S ⊆ V of vertices having the property that every vertex v ∈ S is adjacent to at least one vertex in S. This can also be stated as follows: dominating set : (∀v ∈ S)(∃u
∈ S)[u ∈ (N (v) ∩ S)]. In addition, a dominating set S is a minimal dominating set if every vertex v ∈ S dominates at least one vertex, either itself or a vertex in S, that no other vertex in S
dominates. This can also be stated as follows: minimal dominating set : (∀u ∈ S)(∃v ∈ V )[N [v] ∩ S = {u}]. The vertex v in the minimal dominating set definition above is called a private neighbor of
vertex u. The domination number γ (G) equals the minimum cardinality of a dominating set in G, while the upper domination number (G) equals the maximum cardinality of a minimal dominating set in G.
2. A total dominating set in a graph G = (V, E) is a set S ⊆ V of vertices having the property that every vertex v ∈ V is dominated by a vertex in S, other than itself. This can also be stated as
follows: total dominating set : (∀v ∈ V )(∃u ∈ S − {v})[u ∈ (N (v) ∩ S)]. In addition, a total dominating set S is a minimal total dominating set if every vertex u ∈ S dominates at least one vertex
in V , other than itself that no other vertex in S dominates and vertex u has at least one neighbor in S. This can also be stated as follows: minimal total dominating set : (∀u∈S)(∃v∈V )[N(u) ∩ S=∅ ∧
[N(v) ∩ S={u}]. The vertex v in the minimal total dominating set definition above is called a private neighbor of vertex u. 3. A vertex cover in a graph G = (V, E) is a set S ⊆ V of vertices having
the property that for every edge uv ∈ E, either u or v (or both u and v) is a vertex in S. In
The Private Neighbor Concept
this case, we say that both vertices u and v cover edge uv. This can also be stated as follows: vertex cover : (∀uv ∈ E)[{u, v} ∩ S = ∅]. In addition, a vertex cover S is a minimal vertex cover if
every vertex v ∈ S covers at least one edge that is not covered by any other vertex in S. This can also be stated as follows: minimal vertex cover : (∀u ∈ S)(∃uv ∈ E)[{u, v} ∩ S = {u}]. The edge uv v
in the minimal vertex cover definition above is called a private edge of vertex u. 4. A resolving set in a graph G = (V, E) is a set S ⊆ V of vertices having the property that for every pair of
vertices u, v ∈ S, there is a vertex w ∈ S such that d(u, w) = d(v, w). In this case, we say that vertex w resolves the vertex pair u and v and denote this as w u, v. This can also be stated as
follows: resolving set : (∀u, v ∈ S)(∃w ∈ S)[w " u, v]. In addition, a resolving set S is a minimal resolving set if every vertex u ∈ S resolves at least one vertex pair v, w ∈ S that is not resolved
by any other vertex in S. This can also be stated as follows: minimal resolving set : (∀w∈S)(∃u, v∈S)[w " u, v ∧ ( ∃x ∈ S)[x = w ∧ x " u, v]. The pair of vertices u and v in the minimal resolving set
definition above is called a private pair of vertex u.
3 Irredundant Sets In 1978, Cockayne, Hedetniemi, and Miller [15] introduced the concept of private neighbors, as defined above in the definition of a minimal dominating set, and made the important
distinction between a minimal dominating set and a set which isn’t necessarily dominating, but nevertheless satisfies the added condition of being minimal dominating set. They therefore introduced
the concept called irredundance in graphs as follows. A set S ⊂ V is called irredundant if (∀u ∈ S)(∃v ∈ V )[N[v] ∩ S = {u}]. Stated in words, a set S is called irredundant if and only if every
vertex u ∈ S has at least one private neighbor, either itself, if N[u] ∩ S = {u}, or a vertex v ∈ S such that N(v) ∩ S = {u}; such a vertex v ∈ S is called an external private neighbor of u. An
irredundant set S is called maximal if no proper superset of S is also an irredundant set.
S. T. Hedetniemi et al.
Fig. 1 The set {v2 , v3 } is irredundant, but not dominating
The minimum cardinality of a maximal irredundant set in a graph is called the lower irredundance number and denoted ir(G). The maximum cardinality of an irredundant set in a graph is called the upper
irredundance number and denoted IR(G). It is important to note that even though the concept of irredundant sets arises from the condition that defines a minimal dominating set, an irredundant set,
and indeed a maximal irredundant set, need not be a dominating set. Consider the example shown in Figure 1, the path P5 with vertices labeled in order v1 , v2 , v3 , v4 , v5 . The set S = {v2 , v3 }
is both an irredundant set and a maximal irredundant set, but it is not a dominating set. In this set, v2 is the only vertex in S that dominates v1 , and v3 is the only vertex in S that dominates v4
. Thus, since both vertices in S have an external private neighbor, S is an irredundant set, but S is not a dominating set, since no vertex in S dominates v5 . The idea of irredundance, and the
associated concept of private neighbors, applies to the added conditions which hold if a set S having some property P is, in addition, minimal with respect to property P. In general, irredundance
answers the question: why is a set minimal with respect to property P? Suppose, for example, that a set has property P2 if the subgraph G[S] induced by a set S has minimum degree δ(G[S]) ≥ 2. If S is
1-minimal with respect to this property, then any vertex v ∈ S, if removed, would result in a subgraph G[S −{v}] having a vertex of degree less than 2. And this means that every vertex in S must have
a neighbor of degree 2 in G[S]. We could say, therefore, that a set S is P2 irredundant if and only if every vertex v ∈ S has a neighbor in S of degree 2 in G[S]. Notice that a set S, such that G[S]
is a 2-regular graph (every vertex of which has degree 2), for example, a cycle Ck , is both 1-minimal and minimal with respect to property P2 . However, a set S, such that G[S] = K3 ∪ K3 , is a P2
-set that is 1minimal but is not minimal, since removing all three vertices in one K3 creates another, smaller P2 -set. Thus, we can observe that the reason an independent set is a maximal
independent set is because it is also a dominating set. In fact, a set is maximal independent if and only if it is both an independent and a minimal dominating set. And in the same way, the reason
that a dominating set is a minimal dominating set is because it is also an irredundant set. In fact, a set is a minimal dominating set if and only if it is a dominating set and an irredundant set.
Because of the generality of the notion of private neighbors, the corresponding notion of irredundance is very general. Although we will discuss a variety of different types of irredundance in this
chapter, a very comprehensive and in-depth study of the concept of irredundance in graphs can be found in the 2003 Ph.D. thesis
The Private Neighbor Concept
of Stephen Finbow [32]. In addition, see also the comprehensive chapter on the irredundance numbers ir(G) and IR(G) by Mynhardt and Roux [49] in this volume.
4 The Basic Private Neighbors and Corresponding Irredundance Numbers Given a vertex set S ⊆ V in a graph G = (V, E), we can define three kinds of private neighbors of a vertex v ∈ S. If vertex v is
not adjacent to any vertex in S, which is equivalent to saying that v is an isolated vertex in the subgraph G[S] induced by S, or that N(v) ∩ S = ∅, then we say that v is its own private neighbor
with respect to the set S, or that v is a self-private neighbor or an spn. If vertex v is adjacent to a vertex w ∈ V − S and w is not adjacent to any other vertex in S, then we say that w is an
external private neighbor, or an epn, of v. This is equivalent to saying that N(w) ∩ S = {v}. The set of private neighbors of a vertex v ∈ S is the set pn[v] = N[v] − N[S −{v}]. If vertex v is
adjacent to a vertex w ∈ S and w is not adjacent to any other vertex in S, then we say that w is an internal private neighbor, or an ipn, of v. This is equivalent to saying that for some vertex w ∈
S, N(w) ∩ S = {v}. 1. independence numbers i(G) and α(G), the minimum and maximum cardinalities of a maximal independent set. Equivalently, the minimum and maximum cardinalities of maximal sets S
such that every vertex v ∈ S is its own private neighbor. 2. irredundance numbers ir(G) and IR(G), the minimum and maximum cardinalities of a maximal irredundant set. A set S is an irredundant set if
for every vertex v ∈ S, pn[v, S] = N[v] − N[S −{v}] = ∅, that is, every vertex v ∈ S either is an spn or has an epn with respect to the set S. Irredundant sets were first defined and studied by
Cockayne, Hedetniemi, and Miller in 1978 [15]. See also an early survey paper by Hedetniemi, Laskar, and Pfaff [44] and a comprehensive paper showing the full generality of irredundance in graphs by
Cockayne and Finbow [12]. 3. open irredundance numbers oir(G) and OIR(G), the minimum and maximum cardinalities of a maximal open irredundant set in G. A set S is open irredundant if every vertex u ∈
S has an external private neighbor. This is equivalent to saying that for every vertex v ∈ S, N(v) − N[S −{v}] = ∅. Open irredundance was first studied by Farley and Schacham [28] in 1983; see also
Farley and Proskurowski [27] in 1984, Cockayne et al. [11] in 2003, and Cockayne et al. [22] in 2008. 4. open-open irredundance numbers ooir(G) and OOIR(G), the minimum and maximum cardinalities of a
maximal open-open irredundant set in G. A set S is open-open irredundant if every vertex u ∈ S has either an external or an internal private neighbor. This is equivalent to saying that for every
vertex v ∈ S, N(v) − N(S −{v}) = ∅. Open-open irredundance was introduced by Cockayne, Finbow, and Swarts [23] in 2010.
S. T. Hedetniemi et al.
5. closed-open irredundance numbers coir(G) and COIR(G), the minimum and maximum cardinalities of a maximal closed-open irredundant set in G. A set S is closed-open irredundant if every vertex u ∈ S
has either itself as a private neighbor, an external private neighbor, or an internal private neighbor. This is equivalent to saying that for every v ∈ S, N[v] − N(S −{v}) = ∅. 6. strong matching
number α ∗ (G), the maximum cardinality of a set S ⊆ V such that every vertex in S has an internal private neighbor. For such a set S, the induced subgraph G[S] consists of a disjoint union of
complete graphs K2 ; the set of edges in the subgraph induced by such sets are called strong or induced matchings. These sets were introduced independently by Cameron [6] in 1989 and later by
Golumbic and Laskar [41] in 1993. 7. 1-dependence number α 1 (G), the maximum cardinality of a set S ⊆ V such that every vertex has either itself as a private neighbor or has an internal private
neighbor. For such a set S, the induced subgraph G[S] consists of a disjoint collection of K1 ’s or K2 ’s, or equivalently, the subgraph G[S] has maximum degree = 1. These sets were introduced by
Fink and Jacobson [33] in 1985, who called these 1-dependent sets. In general, a k-dependent set is a set S such that (G[S]) ≤ k. These seven types of irredundance numbers all fit into a natural cube
of inequalities, as was shown by Fellows, Fricke, Hedetniemi, and Jacobs [31] in 1994, cf. Figure 2, where an arrow u → v indicates that the parameter associated with vertex u is greater than or
equal to the parameter associated with vertex v. As the inequalities in Figure 2 also show, we have the following sequence, which is called the Domination Chain, which was first observed by Cockayne,
Hedetniemi, and Miller in 1978 [15]. ir ≤ γ ≤ i ≤ α ≤ ≤ IR
Fig. 2 The private neighbor cube
The Private Neighbor Concept
5 Generalized Irredundance by Cockayne In 1999, Cockayne [10] considerably generalized the private neighbor cube of Fellows et al. [31]. He considered the 32 classes of sets S, which can be defined
in terms of the types of private neighbors that vertices in a set S may or may not have. Let p mean that a vertex v is its own private neighbor (an spn), meaning that v is not adjacent to any vertex
in S. Let q mean that v has a private neighbor inside S (an ipn), meaning that v is adjacent to a vertex u in S and no other vertex in S is adjacent to u. Let r mean that v has an external private
neighbor (an epn), meaning that it has a neighbor u ∈ V − S and no other vertex in S is adjacent to u or, equivalently, N(u) ∩ S = {v}. Therefore, assuming that all vertices must have at least one
type of private neighbor, there are five types of vertices, as follows. Note that it is not possible for a vertex to have both a self-private neighbor and an internal private neighbor; thus, there
are no vertices of types 1,1,0 or 1,1,1.
Type 1: Type 2: Type 3: Type 4: Type 5:
p 0 0 0 1 1
q 0 1 1 0 0
r 1 0 1 0 1
vertex v must have an epn, but has no spn or ipn. vertex v must have an ipn but has no spn or epn. vertex v must have both an ipn and an epn, but has no spn. vertex v must have an spn but has no ipn
or epn. vertex v must have an spn and an epn, but has no ipn.
There are, therefore, 25 = 32 types of sets of vertices S (all possible subsets of these five types), with prescribed private neighbor requirements for the vertices in S, as given in the table below.
1. 2. 3. 4. 5.
1. 2. 3. 4. 5.
p 0 0 0 1 1
p 0 0 0 1 1
q 0 1 1 0 0
q 0 1 1 0 0
r 1 0 1 0 1
r 1 0 1 0 1
S. T. Hedetniemi et al.
Consider the types of sets that are defined by each column. We will mention just a few. Column 0 defines a set in which no vertex can have a private neighbor of any kind. So, for example, the set of
all vertices in a cycle Cn constitutes such a set, since no vertex in a cycle has a private neighbor of any kind. Column 2 defines a set in which every vertex must be an spn but can obviously have no
ipn and can have no epn either. We can call this a {4}-set, since all vertices are of type 4. This is an independent set in which no vertex has an external private neighbor. Column 12 is a {2,3}-set,
which defines an induced matching in which every vertex must have an ipn and may or may not also have an epn. Most of these sets have not been studied, but several are of particular interest.
Cockayne [13] shows that of these 32 types of sets, only 12 define sets that are hereditary and therefore might warrant further study. They are the following: Column 1 defines a set in which every
vertex has both a self-private neighbor and an external private neighbor. This is called a {5}-set. This is an independent set in which every vertex has an external private neighbor. Column 3 is a
{4,5}-set, in which every vertex has an spn and may or may not have an epn. This is a standard independent set. Column 5 is a {3,5}-set, in which every vertex has an epn and either an spn or an ipn.
Column 7 is a {3,4,5}-set, in which every vertex either has an spn, with or without an epn, or does not have an spn, but has both an ipn and an epn. Column 9 is a {2,5}-set, in which every vertex
either has an ipn, but no spn or epn, or has no ipn, but has both an spn and an epn. Column 11 is a {2,4,5}-set, in which every vertex either has an spn, with or without an epn, or has an ipn with no
spn or epn. Column 13 is a {2,3,5}-set, in which every vertex either has an ipn, and may or may not have an epn, or has both an spn and an epn. Column 15 is a {2,3,4,5}-set, which is a 1-dependent
set, that is, a set in which every vertex has either an spn or an ipn. Column 21 is a {1,3,5}-set, which is an open irredundant set, in which every vertex has an epn. Column 23 is a {1,3,4,5}-set,
which is an irredundant set, in which every vertex has either an spn or an epn. Column 29 is a {1,2,3,5}-set, which is an open-open irredundant set, in which every vertex has either an ipn or an epn.
Column 31 is a {1,2,3,4,5}-set, which is a closed-open irredundant set, in which every vertex has at least one type of private neighbor.
The Private Neighbor Concept
6 Total Irredundance Numbers The Domination Chain, which follows from the definitions of independent sets, dominating sets, and irredundant sets, raises the question of whether there is a type of
irredundance related to total dominating sets. Total irredundance was introduced in 2002 by Favaron, Haynes, Hedetniemi, Henning, and Knisley [30]. See also Hedetniemi, Hedetniemi, and Jacobs [45] in
1993. A set S is total irredundant if and only if for every vertex v ∈ V , N[v] − N[S −{v}] = ∅. The total irredundance numbers, irt (G) and IRt (G), equal the minimum and maximum cardinalities of a
maximal total irredundant set. Notice that the irredundance numbers ir(G) and IR(G) are defined in terms of two conditions, at least one of which must hold for every vertex v ∈ S. By contrast, the
total irredundance numbers irt (G) and IRt (G) are defined in terms of two conditions, at least one of which must hold for every vertex v ∈ V . We note in passing that other types of total
irredundance can be defined in terms of the conditions that must hold for every vertex v ∈ V instead of every vertex v ∈ S. Thus, for example, one can define: total open irredundance: for every
vertex v ∈ V , N(v) − N[S −{v}] = ∅. This means that every vertex v, either in S or in V − S, is adjacent to some vertex in V − S that no other vertex in S is adjacent to. To the best of our
knowledge, none of these types of total irredundance have been defined and studied. Not only this, but one can define other types of irredundance in terms of private neighbor conditions that hold
only for vertices in V − S. For example, one could define: irredundance: for every v in S, N[v] − N[S −{v}] = ∅. total irredundance: for every v in V, N[v] − N[S −{v}] = ∅. external irredundance: for
every v in V − S, N[v] − N[S −{v}] = ∅. This means that for an external irredundant set S, every vertex in V − S either is not adjacent to any vertex in S or is adjacent to some vertex in V − S that
no vertex in S is adjacent to. To the best of our knowledge, external irredundance has not been studied.
7 The Covering Chain, a Dual of the Domination Chain In 2015, Arumugam, Hedetniemi, Hedetniemi, Sathikala, and Sudha [3] showed that there is an inequality chain that is complementary to the
well-known Domination Chain, ir ≤ γ ≤ i ≤ α ≤ ≤ IR
S. T. Hedetniemi et al.
This idea starts with the following well-known Theorem of Gallai [38], in which α(G) denotes the maximum cardinality of an independent set of vertices and β(G) denotes the minimum cardinality of a
vertex cover. Theorem 1 (Gallai) For any graph G of order n, α(G) + β(G) = n. The idea introduced by Arumugam et al. is that if the Domination Chain begins with the concept of independence and α(G),
then there might be an inequality chain that begins with the concept of a vertex cover and β(G). Notice that independence is a hereditary property (every subset of an independent set is also an
independent set), while the property of being a vertex cover is superhereditary (every superset of a vertex cover is also a vertex cover). In order to develop this idea, we will need to quickly
review a few definitions. 1. β(G), the vertex covering number, equals the minimum number of vertices in a vertex cover, that is, a set S ⊆ V having the property that for every edge uv ∈ E, either u ∈
S or v ∈ S, or both. 2. β + (G), the upper vertex covering number, equals the maximum number of vertices in a minimal vertex cover of G. 3. β (G), the edge covering number, equals the minimum number
of edges in an edge cover, that is, a set F ⊆ E having the property that every vertex v ∈ V is incident to at least one edge in F.
4. β + (G), the upper edge covering number, equals the maximum number of edges in a minimal edge cover of G. 5. α (G), the matching number, equals the maximum number of edges in a matching, that is,
a set F ⊂ E, no two edges in F have a vertex in common. 6. (G), the upper enclaveless number, equals the maximum number of vertices in a set S such that S has no enclave, that is, a vertex v ∈ S such
that N[v] ⊆ S. 7. ψ(G), the lower enclaveless number, equals the minimum number of vertices in a maximal enclaveless set S. It can be observed that since the property of being a vertex cover is
superhereditary, it follows that a vertex cover S is minimal if and only if S is 1-minimal. Before presenting the next several results, it is worthwhile pointing out that if a set S is a (minimal)
vertex cover, then the complement V − S must be a (maximal) independent set. Conversely, if S is a (maximal) independent set, then the complement V − S must be a (minimal) vertex cover. Proposition 1
(Arumugan et al.) A vertex cover S of a graph G is a minimal vertex cover if and only if S is a vertex cover and is enclaveless. Proof. Let S be a minimal vertex cover of G. Then for every vertex v ∈
S, S −{v} is not a vertex cover of G, and this must mean that v has at least one neighbor, say w ∈ V − S, so that the edge vw is not covered by S −{v}. Hence, N[v]S. Thus, S is an enclaveless set.
Conversely, let S be an enclaveless vertex cover. If S is not a minimal vertex cover, then there exists a vertex v ∈ S such that S −{v} is a vertex cover. Hence, N[v] ⊆ S, so that v is an enclave in
S, which is a contradiction. Thus, S is a minimal vertex cover.
The Private Neighbor Concept
Proposition 2 (Arumugam et al.) Every minimal vertex cover S in a graph G is a maximal enclaveless set of G. Proof Let S be a minimal vertex cover in G. It follows from Proposition 1 that S is
enclaveless. If S is not a maximal enclaveless set, then there exists a vertex u ∈ V − S such that S ∪{u} is enclaveless. Hence, there exists a vertex w ∈ N(u) ∩ (V − S), and the edge uw has both its
ends in V − S, which is a contradiction. Thus, S is a maximal enclaveless set. Corollary 1 For any graph G, ψ ≤ β ≤ β + ≤ .
Since the property of being an enclaveless set is hereditary, an enclaveless set S is maximal if and only if S is a 1-maximal enclaveless set. Proposition 3 (Arumugam et al.) An enclaveless set S is
maximal enclaveless if and only if S is enclaveless and V − S is irredundant. Proof Let S be a maximal enclaveless set. Then for any u ∈ V − S, S ∪{u} contains an enclave, say v. Hence, N[v] ⊆ S ∪
{u}, but N[v]S. If v = u, then N[v] ∩ (V − S) = {u}, and thus, v is an external private neighbor of u with respect to the set V − S. However, if v = u, then N(u) ⊆ S, and therefore u is not adjacent
to any vertex of V − S. Thus, in either case, u has a private neighbor, either external or itself, with respect to the set V − S, and therefore, V − S is an irredundant set. Conversely, let S be
enclaveless and V − S be an irredundant set. Then for any u ∈ V − S, there exists v ∈ N[u] such that N[v] ∩ (V − S) = {u}. If u = v, then u is an enclave of S ∪{u}. If v = u, then v ∈ S and v is an
enclave of S ∪{u}. Therefore, S is a maximal enclaveless set. This gives rise to a new definition. Definition 1 A subset S of V is called a co-irredundant set if V − S is an irredundant set of G. The
co-irredundance number cir(G) equals the minimum cardinality of a co-irredundant set in G. The upper co-irredundance number CIR(G) equals the maximum cardinality of a minimal co-irredundant set in G.
Note that the property of being a co-irredundant set is superhereditary, and thus a set S is a minimal co-irredundant set if and only if it is a 1-minimal co-irredundant set. Proposition 4 (Arumugam
et al.) Every maximal enclaveless set S in a graph G is a minimal co-irredundant set. Proof If S is a maximal enclaveness set, then it follows from Proposition 3 that V − S is irredundant and hence,
S is co-irredundant. If S is not a minimal coirredundant set, then there exists u ∈ S such that S −{u} is co-irredundant. Hence,
S. T. Hedetniemi et al.
(V − S) ∪{u} is an irredundant set. Let v be a private neighbor of u with respect to (V − S) ∪{u}. Clearly, v = u, since then N[u] ⊆ S is an enclave in S, a contradiction. However, if v ∈ S and N[v]
⊆ S. Thus, v is an enclave in S, which is a contradiction. Thus, S is a minimal co-irredundant set. Corollary 2
For any graph G, cir ≤ ψ ≤ β ≤ β + ≤ ≤ CI R.
This inequality chain above is called the Covering Chain of a graph G. But much more can be said about this, since it is closely related to the Domination Chain, as follows. Theorem 2 (Arumugam et
al.) For any graph G, (i) ir(G) + CIR(G) = n, (ii) IR(G) + cir(G) = n. Thus, the Covering Chain of a graph G can also be written as follows. n − IR ≤ n − ≤ n − α ≤ n − i ≤ n − γ ≤ n − ir Hence, the
Covering Chain is the dual of the Domination Chain. Proposition 5 (Arumugam et al.) For any graph G without isolated vertices, γ (G) ≤ cir(G). Proposition 6 (Arumugam et al.) For any graph G without
isolated vertices, IR(G) ≤ Ψ (G). Proof We know that (i) cir(G) + IR(G) = n, (ii) γ (G) ≤ cir(G), (iii) γ (G) + (G) = n. From this it follows that IR(G) ≤ (G).
We can combine the inequalities given in Propositions 5 and 6 into the Domination Chain and Covering Chain to get two larger chains, which we can call the Extended Domination Chain and the Extended
Covering Chain, respectively. ir + CIR = n
≤ ≥
γ + = n
≤ ≥
i + β+ = n
≤ ≥
α + β = n
≤ ≥
+ ψ = n
≤ ≥
IR + cir = n
≤ ≥
+ γ = n
≤ ≥
CIR + ir = n
The Private Neighbor Concept
Arumugam et al. also add the following interesting comparison of two inequality chains:
ir CIR
≤ ≥
γ γ
≤ ≤
i cir
≤ ≤
α ψ
≤ ≤
≤ ≤
IR β+
≤ ≤
≤ ≥
cir IR
≤ ≥
≤ ≥
β α
≤ ≥
β+ i
≤ ≥
≤ ≥
CIR ir
8 Domination in Terms of Perfection in Graphs To the definitions given above, of independent sets, dominating sets, and irredundant sets, we now introduce several concepts and parameters having to do
with what is called perfection in graphs. These concepts were first introduced in 1999 by Fricke, Haynes, Hedetniemi, Hedetniemi, and Henning [35] and later developed in detail by J. T. Hedetniemi,
S. M. Hedetniemi, and S. T. Hedetniemi in 2013 [47]. In the remainder of this section, we review the results given in this 2013 paper. In the notation that follows, a subscript (α S ) refers to a
parameter α having some condition on the vertices in S, for example, the vertex independence number is defined in terms of a condition on the vertices in S, that no two of them are adjacent.
Similarly, a superscript (α V −S ) refers to a parameter having some condition on the vertices in V − S, for example, the domination number is defined in terms of a condition on the vertices in V −
S, that every vertex in V − S has at least one neighbor in S. If a parameter α requires some condition on all vertices in V , no subscript or superscript appears. Definition 2 Given a set S ⊆ V in a
graph G = (V, E), a vertex v ∈ V is said to be S-perfect if |N[v] ∩ S| = 1, that is, the closed neighborhood N[v] contains exactly one vertex in S. Notice that if a vertex v ∈ S is S-perfect, then it
has no neighbors in S, and if a vertex v ∈ V − S is S-perfect, then it has exactly one neighbor in S. Definition 3 Given a set S ⊆ V in a graph G, a vertex v is almost S-perfect if it is either
S-perfect or is adjacent to an S-perfect vertex.
S. T. Hedetniemi et al.
Fig. 3 A perfect neighborhood set
When a set S has been given and is assumed, we simply say that a vertex is perfect or almost perfect, without referring to the set S. Definition 4 A set S ⊂ V is perfect if every vertex v ∈ S is
S-perfect and is almost perfect if every vertex v ∈ S is almost S-perfect; for brevity we say that an almost perfect set is an ap-set. Let θ ap (G) and Θ ap (G) equal the minimum and maximum
cardinalities of a maximal ap-set in G. Definition 5 A set S is externally perfect if every vertex in V − S is S-perfect and is externally almost perfect if every vertex in V − S is either S-perfect
or is adjacent to an S-perfect vertex; for brevity we say that an externally almost perfect set is an eap-set. Let θ ap (G) and Θ ap (G) equal the minimum and maximum cardinality of a minimal eap-set
in G. In the graph in Figure 3, given in the paper by Hedetniemi et al., a vertex labeled “p” is perfect, while a vertex labeled “ap” is almost perfect. The three shaded vertices form a set S that is
almost perfect (two vertices in S are almost perfect and the third is perfect) and is externally almost perfect (every vertex in V − S is either perfect or is adjacent to a perfect vertex).
Definition 6 A set S is a perfect neighborhood set if every vertex v ∈ V is either perfect or is adjacent to a perfect vertex. Let θ (G) and Θ(G) equal the minimum ap and maximum cardinalities of a
perfect neighborhood set in G, and let θp (G) and ap p (G) equal the minimum and maximum cardinalities of an independent perfect neighborhood set in G. Notice that the three shaded vertices in Figure
2 form a perfect neighborhood set. Definition 7 A set S is an eap irredundant, eap dominating, or eap independent set if it is a maximal irredundant, minimal dominating, or maximal independent set
that is also eap. Thus, every vertex v ∈ V − S is either perfect or is adjacent to a perfect vertex. Let irap (G), γ ap (G), iap (G), α ap (G), Γ ap (G), and IRap (G) denote the minimum and maximum
cardinalities of such sets.
The Private Neighbor Concept
Given these definitions, we can relate them to independent, dominating, and irredundant sets, for example, the concept of a set S being perfect is equivalent to the concept of a set being
independent. Proposition 7 A set S is perfect if and only if it is independent. ap
Corollary 3 For any graph G, θp (G) ≤ i(G) = i ap (G). ap
Corollary 4 For any graph G, α(G) = α ap (G) = p (G). Recall that a dominating set S is called perfect if every vertex v ∈ V − S has exactly one neighbor in S and is called an efficient dominating
set if S is independent, dominating, and perfect. Proposition 8 A set S is externally perfect if and only if S is a perfect dominating set. Proposition 9 A set S is completely perfect if and only if
S is an efficient dominating set (or a perfect code). The concept of being almost perfect (ap) is equivalent to the concept of being irredundant. Proposition 10 A set S is almost perfect if and only
if S is irredundant. Proof If a set S is almost perfect, then every vertex u ∈ S is either perfect or adjacent to a perfect vertex. Either u is an isolated vertex in G[S], in which case it is perfect
and is its own private neighbor, or u is adjacent to a perfect vertex, say w. But w cannot be in S. Thus, w ∈ V − S and w is perfect because |N[w] ∩ S| = |{u}| = 1. This means that w is an external
private neighbor of u. Thus, every vertex u ∈ S has a private neighbor, and hence S is irredundant. Conversely, if S is irredundant, then every vertex u ∈ S either is its own private neighbor, in
which case it is perfect, or has an external private neighbor, say w. But in this case w is perfect and therefore u is adjacent to a perfect vertex. Therefore, S is almost perfect. Corollary 5 For
any graph G, ir = θap ≤ ap = I R(G) Distance-2 dominating sets are closely related to externally almost perfect sets. Proposition 11 If a set S is eap, then it is a distance-2 dominating set. The
Domination Chain can now be considerably expanded in terms of the concept of perfection. Theorem 3 For any graph G, the following system of inequalities holds.
S. T. Hedetniemi et al.
IR ap
i =
γ ap
Θap p
≥ γ≤2
A similar pair of inequality chains holds when independent sets are considered. Proposition 12 For any graph G, the following inequalities hold. ap
γ≤2 ≤ γ≤2 ≤ i≤2 ≤ i ap = i.
γ≤2 ≤ i≤2 ≤ i≤2 ≤ i ap = i.
If we define γd (G) to equal the minimum cardinality of a dominating set that is externally almost perfect, as distinct from γ ap (G), which equals the minimum cardinality of a minimal dominating set
that is externally almost perfect, then we get the following refinement. Proposition 13 For any graph G, ap
γ ≤ γd
≤ γ ap ≤ i.
As given in Hedetniemi et al., the fact that each of these inequalities can be strict is illustrated by the unicyclic graph G in Figure 4. For this graph, the set S1 = {1, 3, ap 4, 6} shows that γ
(G) = 4; for γd (G) = 6, let S2 = {1, 2, 3, 4, 5, 6}; for γ ap (G) = 8, let S3 = {1, 6, 7, 8, 9, 10, 11, 12}; and for i(G) = 9, let S4 = {1, 7, 8, 9, 4, 13, 14, 15, 16}. In summary, the concept of
perfection in graphs provides a framework for unifying the concepts of independent sets, dominating sets, irredundant sets, perfect and efficient dominating sets, and perfect neighborhood sets. For
example: (i) A set is independent if and only if it is a perfect set. (ii) The independence parameters i(G) and α(G) can be expressed as maximal independent sets whose complements are almost perfect,
that is, i(G) = iap (G) and α(G) = α ap (G). (iii) A set is an irredundant set if and only if it is an almost perfect set. (iv) The parameters ir(G) and IR(G) can be expressed in terms of almost
perfect sets, namely, ir(G) = θ ap (G) and IR(G) = ap (G).
The Private Neighbor Concept
Fig. 4 γ < γd < γ ap < i
(v) The theorem in Fricke et al. [35], that (G) = (G), established an equality between two seemingly unrelated parameters. This result is now clearer. In particular, the inequality chain, α(G) ≤ (G)
≤ IR(G), can now be stated ap ap equivalently as p (G) ≤ (G) ≤ ap (G), since p (G) = α(G), (G) = (G), and ap (G) = IR(G). (vi) An expanded inequality chain exists between the domination and
independence parameters: ap
γ ≤ γd
≤ γ ap ≤ i ≤ α ≤ ap ≤ .
Further papers on perfect neighborhood sets in graphs can be found by Cockayne, Hedetniemi, Hedetniemi, and Mynhardt [20], by Favaron and Puech [29], and by Hedetniemi, Hedetniemi, and Henning [46].
9 Partitions Involving Irredundant Sets A (proper) coloring of a graph G is a vertex partition V = {V1 , V2 , . . . , Vk } such that for every 1 ≤ i ≤ k, Vi is an independent set. Since the property
of being independent is a hereditary property, one seeks the minimum order of a partition into independent sets. For example, the chromatic number χ (G) equals the minimum order of a partition of V
into independent sets. Continuing in the same manner, the property of being an irredundant set is hereditary. Therefore, it would be natural to consider the minimum order of a partition of V into
irredundant sets. As introduced by Haynes, Hedetniemi, Hedetniemi, McRae, and Slater in 2008 [43], the irratic number χ ir (G) equals the minimum order of a partition of V into irredundant sets.
Clearly, since every independent set is irredundant, for any graph G, χ ir (G) ≤ χ (G).
S. T. Hedetniemi et al.
This inequality immediately raises the question: can you prove that for any planar graph G, χ ir (G) ≤ 4, without appealing to the Four Color Theorem? Haynes et al. show the following. 1. 2. 3. 4. 5.
χ ir (G) = 1 if and only if G = Kn . χ ir (G) = n if and only if G = Kn . If χ (G) = 2, then χ ir (G) = 2. χ ir (G ◦ K1 ) = 2. χir (G2K2 ) = 2.
The authors provide bounds for χ ir (G) in terms of other well-known graphical parameters and take a closer look at graphs with χ ir (G) = 2, called bi-irratic graphs. Although every nonempty
bipartite graph is bi-irratic, the problem of characterizing the class of bi-irratic graphs remains open. The authors also study complexity questions and establish the NP-completeness of the problem
of determining if a given graph is bi-irratic. In 2012, Arumugam and Chandrasekar [2] prove that the problem of deciding if the vertices of a graph can be partitioned into two open irredundant sets,
that is, whether χ oir (G) = 2, is NP-complete. A complete coloring of a graph G is a proper vertex coloring of G having the property that for every two distinct colors i and j, there exist adjacent
vertices colored i and j. The maximum positive integer k for which G has a complete kcoloring is called the achromatic number (G) of G. A Grundy coloring of a graph G is a proper vertex coloring of G
having the property that for every two colors (positive integers) i and j with i < j, every vertex colored j has a neighbor colored i. The maximum positive integer k for which a graph G has a Grundy
k-coloring is the Grundy number r(G) of G. For every graph G, these four coloring parameters satisfy the inequalities: χ ir
For these four coloring numbers, Chartrand, Hedetniemi, Okamoto, and Zhang [7] showed in 2011 that if a, b, c, and d are integers with 2 ≤ a ≤ b ≤ c ≤ d, then there exists a nontrivial connected
graph G with χ ir (G) = a, χ (G) = b, r(G) = c, and (G) = d if and only if d = 2 or c = 2.
10 The Mystery of the Domination Chain: ?? ≤ ir(G) ≤ γ (G) ≤ i(G) ≤ α(G) ≤ (G) ≤ IR(G) ≤?? Notice that the parameters i(G) and α(G) are the min and max parameters associated with the hereditary
property, say P1 , of being an independent set.
The Private Neighbor Concept
Notice next that the parameters γ (G) and (G) are the min and max parameters associated with the superhereditary property, P2 , of being a dominating set. The third pair of parameters ir(G) and IR(G)
are the min and max parameters associated with the hereditary property P3 of being a maximal irredundant set. This would lead one to think that there should exist a pair of parameters, call them ψ(G)
and (G), that are associated with some superhereditary property, P4 , and we would have the following inequality chain. ψ? ≤ ir ≤ γ ≤ i ≤ α ≤ ≤ I R ≤ ? . . . and maybe still another similar pair of
parameters, call them φ and , that are associated with some hereditary property, P5 , and we would have the following inequality chain. φ? ≤ ψ? ≤ ir ≤ γ ≤ i ≤ α ≤ ≤ I R ≤ ? ≤ ? Does this chain of
inequalities continue indefinitely, alternating between hereditary and superhereditary properties, or does it terminate? Much of the discussion in this section, but not all, can be found in the 1997
paper by Cockayne, Hattingh, Hedetniemi, Hedetniemi, and McRae [17]. As discussed in this paper, the domination chain starts with the concept of an independent set, which is a hereditary property.
Because of this, one can say that an independent set S is maximal if and only if it is 1-maximal, which means that for every vertex v ∈ V − S, the set S ∪{v} is not an independent set. This, in turn,
is equivalent to saying that for every vertex v ∈ V − S, there exists at least one vertex u ∈ S such that v is adjacent to u. And this is the definition of a dominating set. So we can say that the
maximality condition for an independent set is the definition of a dominating set. In much the same way, the concept of a dominating set is a superhereditary property. Because of this, we can say
that a dominating set S is minimal if and only if it is 1-minimal, which means that for every vertex v ∈ S, the set S −{v} is not a dominating set. This, in turn, is equivalent to saying that either
vertex v has no neighbor in S, and therefore is not dominated by any vertex in S −{v}, in which case we say that v is its own self-private neighbor, or spn, or v is the only vertex that dominates
some vertex w ∈ V − S, in which case we say that w is an external private neighbor or epn of v. This condition is the definition of an irredundant set. Because of these three definitions, the
Domination Chain emerges. But now we come to the property of being an irredundant set. It is easy to see that this, like the property of being an independent set, is a hereditary property, since
every subset of an irredundant set is also an irredundant set. Thus, one can say that a set S is a maximal irredundant if and only if it is a 1-maximal irredundant set. And this means that a set S is
a maximal irredundant set if and only if for every vertex v ∈ V − S, S ∪{v} is not an irredundant set. This means that when you add v to the set S, some vertex in S ∪{v} does not have a private
S. T. Hedetniemi et al.
This is equivalent to saying that a maximal irredundant set S is a set that is irredundant and has the added property that for every vertex v ∈ V − S, either Condition (i): v does not have a private
neighbor in the set S ∪{v} or Condition (ii): v has a private neighbor in S ∪{v}, but there exists some vertex u ∈ S that has a private neighbor with respect to S but does not have a private neighbor
with respect to S ∪{v}. Conditions (i) and (ii) give rise to the following type of sets. Definition 8 A set S is called external redundant if for every vertex v ∈ V − S, either (i) v does not have a
private neighbor in the set S ∪{v} or (ii) v has a private neighbor in S ∪{v}, but there exists some vertex u ∈ S that has a private neighbor in S but does not have a private neighbor in S ∪{v}. Let
er(G) and ER(G) equal the minimum and maximum cardinality of a minimal external redundant set in G. Notice that an external redundant set need not be irredundant, but, by definition, every maximal
irredundant set is external redundant. Another way of thinking about the maximality condition of an irredundant set is the following. For any set S ⊆ V , let pnc(S) = |{v ∈ S : N[v] − N[S −{v}] = ∅}
|; this is called the private neighbor count of set S, which equals the number of vertices in S having either an spn or an epn. Thus, a set S is irredundant if and only if pnc(S) = |S|. We say that a
set S is pnc-maximal if for every vertex v ∈ V − S, pnc(S ∪{v}) ≤ pnc(S). Thus, an irredundant set is a maximal irredundant set if and only if pnc(S) = |S| and S is pnc-maximal or adding a vertex in
V − S to S cannot increase the private neighbor count. A concept closely related to external redundance was introduced in 1997 by Cockayne, Grobler, Hedetniemi, and McRae [16] and later studied by
Cockayne, Favaron, Puech, and Mynhardt in 1998 [18] and [19], and by Puech in 2000 [50]. Given a set S ⊂ V , let R = V − N[S] be the set of vertices not dominated by any vertex in S. Let u ∈ S be a
vertex for which pn[u, S] = N[u] − N[S −{u}] = ∅, that is, vertex u has at least one private neighbor (spn or epn) with respect to S. A vertex v ∈ V − S is said to annihilate u if v is adjacent to
every vertex in pn[u, S]. This means that u has a private neighbor with respect to the set S, but does not have a private neighbor with respect to the set S ∪{v}. A set S is said to be R-annihilated,
or an Ra-set, if every vertex in R annihilates some vertex in S. This is a property satisfied by every maximal irredundant set. The R-annihilated number ra(G) equals the minimum cardinality of an
Ra-set in G. Let A = {v ∈ V − S|v annihilates some u ∈ S}. For some subset U ⊂ V − S, we say that S is U-annihilated if U ⊆ A. Theorem 4 (Cockayne, Grobler et al.) A set S ⊂ V in a graph G = (V, E)
is maximal irredundant if and only if S is irredundant and N[R]-annihilated. Notice that N[R]-annihilated sets are equivalent to eternal redundant sets. Cockayne, Grobler et al. then define the
following parameters: γ ≤2 (G), minimum cardinality of a distance-2 dominating set;
The Private Neighbor Concept
ra(G), the minimum cardinality of an R-annihilated set; rai(G), the minimum cardinality of an irredundant R-annihilated set. θ (G), the (lower) perfect neighborhood number. θ i (G), the (lower)
independent perfect neighborhood number. ρ L (G), the lower 2-packing number. ρ(G), the upper 2-packing number. Theorem 5 (Cockayne, Grobler et al.) For any connected graph G,
% ⎧ ⎫ ⎨ ra ≤ rai ≤ ir ⎬ ≤ ≤ γ ≤ i. er ⎩ ⎭ θ ≤ θi ≤ ρL ≤ ρ
It so happens that the property of being external redundant, pnc-maximal, or N[R]-annihilated is neither superhereditary nor hereditary. Recall that the maximality condition of the hereditary
property of being independent is the definition of the superhereditary property of being a dominating set. And the minimality condition of the superhereditary property of being a dominating set is
the definition of the hereditary property of being an irredundant set. However, the maximality condition of the hereditary property of being an irredundant set seems to be the property of being an
external redundant set. Yet, somewhat surprisingly, the property of being an external redundant set is not superhereditary. In Figure 5 below, the set S = {1, 3, 4, 5, 8, 11} is a 1-minimal external
redundant set, since removing any one of these vertices results in a set that is no longer external redundant. However, you can remove both vertices 1 and 3 and still have an external redundant set.
This is not expected. And we are puzzled. Why isn’t the maximality condition of an irredundant set superhereditary? We don’t know. Fig. 5 A 1-minimal external redundant set
S. T. Hedetniemi et al.
Nevertheless, this has given rise to the definitions of er(G) and ER(G), and with these two definitions, we get the following extended inequality chain: er ≤ ir ≤ γ ≤ i ≤ α ≤ ≤ I R ≤ ER. Once again,
we are led to ask: is there still another pair of parameters which extend this inequality chain? As of now, none are known. Here is still another thought about the sequence: independent set,
dominating set, irredundant set, and external redundant set. The property of being an independent set is what could be called 1-local. In order to decide if a set S is independent, all you have to do
is to look at the neighbors N(v) of every vertex v ∈ S and see if another member of S appears. In order to decide if a set S is a dominating set, all you have to do is to look at every vertex w ∈ V −
S and make sure that N(w) ∩ S = ∅. This is also a 1-local check. But now, in order to decide if a set S is an irredundant set, you have to work a bit harder. Once again, you have to look at every
vertex in v ∈ S and see if N(v) ∩ S = ∅. If this is true, then vertex v is its own private neighbor. But if v is not its own private neighbor, then you must see if v has at least one neighbor, say y
∈ N(v) ∩ (V − S), for which N(y) ∩ S = {v}. Such a vertex y is then an external private neighbor of v. This then becomes a 2-local, or distance-2, check, in that you have to look the neighbors of
your neighbors. But now, consider the problem of deciding if an irredundant set is a maximal irredundant set, or an external redundant set, you must look at every vertex in V − S. Here, it is
possible that there could be a vertex z ∈ V − S, such that d(z, v) ≥ 3 for all vertices v ∈ S, in which case the set S ∪{z} would be an irredundant set. Thus, you could say that the property of being
a maximal irredundant set, or, equivalently, an external redundant set, is 3-local. And consider this. There is a very simple, greedy, linear algorithm for computing the vertex independence number α
(T) for any tree T (cf. Mitchell, Hedetniemi, and Goodman in 1975 [48] and a more general algorithm for chordal graphs in 1972 by Gavril [39]). A second, slightly more complex, but still linear,
algorithm exists for computing the independent domination number i(T), for any tree T (cf. Beyer, Proskurowski, Hedetniemi, and Mitchell in 1977 [5]). A third, more complex, but still linear
algorithm exists for computing the domination number γ (T), for any tree T (cf. Cockayne, Goodman, and Hedetniemi in 1975 [14]). And finally, there exists a much more complex, but still linear,
algorithm for computing the lower irredundance number ir(T), for any tree T (cf. Bern, Lawler, and Wong in 1985 [4]). But this algorithm consists of a 20-by-20 table of some 400 possible combinations
of the irredundance states of a vertex v and its parent in a rooted tree T. In [40], Goddard and Hedetniemi propose, without proof, an algorithm for computing er(T), for any tree T, consisting of a
23-by-23 table of combinations of external redundant states of a vertex v and its parent in a rooted tree T. The authors state, “We believe that the table for external redundance is correct. For a
proof of this, it would be
The Private Neighbor Concept
sufficient to prove that none of the [23] classes needs to be divided. This is a lengthy and tedious argument and is omitted.”
11 Broadcast Irredundance in Graphs In 2015, Ahmadi, Fricke, Schroeder, Hedetniemi, and Laskar [1] introduced the concept of broadcast irredundance in graphs. In this section, we review the basic
definitions and results of this model of irredundance in graphs. The following concepts and definitions of broadcasts in graphs were introduced by Erwin in 2004 [26] and developed further by Dunbar,
Erwin, Haynes, Hedetniemi, and Hedetniemi in 2006 [25]. A function f : V →{0, 1, 2, . . . } defined on the vertex set V of a graph G = (V, E) is called a broadcast if for every vertex v ∈ V , f (v) ≤
ecc(v). Intuitively this means, for example, that if a vertex v is assigned a broadcast power of 4, f (v) = 4, then all vertices within distance 4 or less of v can hear a broadcast from vertex v. The
cost f (V ) of a broadcast f is defined as f (V ) = v ∈ V f (v). Given a broadcast function f, let Vf0 = {v | f (v) = 0} and Vf+ = V − Vf0 = {u | f (u) > 0}. The vertices in Vf+ are called broadcast
vertices. Given a broadcast f and a broadcast vertex v, the broadcast neighborhood of v is the set Nf [v] = {u | d(u, v) ≤ f (v)}. We say that every vertex in the broadcast neighborhood Nf [v] can
hear a broadcast from v or is broadcast dominated by v. Thus, a vertex u with f (u) = 0 hears a broadcast if there exists a vertex v for which d(u, v) ≤ f (v). The set of vertices that a vertex u
hears is the set H (u) = {v ∈ Vf+ | d(u, v) ≤ f (v)}. Define H(f ) ⊆ V to equal the set of vertices that hear a broadcast defined by f. Finally, we say that a broadcast g satisfies g ≤ f, if for
every vertex v ∈ V , g(v) ≤ f (v). A broadcast f is called a dominating broadcast if for every vertex u ∈ V with f (u) = 0, H(u) = ∅, or equivalently if H(f ) = V . The broadcast domination number γ
b (G) of a graph G equals the minimum weight f (V ) of a dominating broadcast f in G. We say that a dominating broadcast f is minimal if there does not exist a dominating broadcast g for which g ≤ f.
The following characterization of minimal dominating broadcasts is due to Erwin [26]. Theorem 6 (Erwin [26]) A dominating broadcast f on a graph G is minimal if and only if the following two
conditions are satisfied: 1. for every broadcast vertex v with f (v) ≥ 2, there exists a vertex u ∈ Vf0 such that H(u) = {v} and d(u, v) = f (v), 2. for every broadcast vertex v with f (v) = 1, there
exists a vertex u ∈ N[v] such that H(u) = {v}.
S. T. Hedetniemi et al.
Let S ⊆ V be a minimal dominating set in a graph G. The characteristic function fS of S is the broadcast function defined as follows: fS (v) = 0 if v∈S, and fS (v) = 1 if v ∈ S. Proposition 14 (Erwin
[26]) If S ⊆ V is a minimal dominating set in a graph G, then the characteristic function fS is a minimal dominating broadcast. Corollary 6 For any graph G, γ b (G) ≤ γ (G). It is worth noting that
the broadcast domination number of a graph can be considerably smaller than its domination number. For example, let S(K1,n ) denote a subdivided star, that is, a graph having one central vertex of
degree n, to which are attached n paths of length 2. It is easy to see that γ b (S(K1,n )) = 2 < γ (S(K1,n )) = n. One can also speak of independent broadcasts. A broadcast f is called independent if
for every broadcast vertex v ∈ Vf+ , |H(v)| = 1, that is, H(v) = {v}. In other words, no broadcast vertex can hear a broadcast from any other vertex. The broadcast independence number α b (G) of a
graph G equals the maximum cost f (V ) of an independent broadcast in G, while the lower broadcast independence number, ib (G), equals the minimum cost of a maximal independent broadcast in G.
Proposition 15 The characteristic function fS of every maximal independent set S is (i) an independent broadcast, but not necessarily a maximal independent broadcast, and (ii) a minimal dominating
broadcast. Note that for a path P4 with vertices labeled in order v1 , v2 , v3 , v4 , the independent broadcast function f defined by f (v1 ) = f (v4 ) = 1 and f (v2 ) = f (v3 ) = 0 is the
characteristic function of the maximum independent set S = {v1 , v4 }. But this is not a maximal independent broadcast, since the independent broadcast function g(v1 ) = g(v4 ) = 2 and g(v2 ) = g(v3
) = 0 satisfies g = f and g ≥ f. In fact, α b (P4 ) = 4. Corollary 7 For any graph G, α(G) ≤ α b (G). Corollary 8 For any graph G, γb ≤ ib ≤ α ≤ αb ≤ b . It is interesting to note that the two
parameters i(G) and ib (G) are not comparable. Erwin’s characterization of minimal dominating broadcasts in effect defines irredundant broadcasts, which were introduced in 2015 by Ahmadi, Fricke,
Schroeder, Hedetniemi, and Laskar [1] as follows. Definition 9 A broadcast function f : V →{0, 1, . . . . . . } is called irredundant if it satisfies the following two conditions: (i) for every
broadcast vertex v with f (v) ≥ 2, there exists a vertex u ∈ Vf0 such that H(u) = {v} and d(u, v) = f (v), (ii) for every broadcast vertex v with f (v) = 1, there exists a vertex u ∈ N[v] such that H
(u) = {v}.
The Private Neighbor Concept
Stated equivalently, a broadcast function f is irredundant if reducing the broadcast value assigned to any broadcast vertex strictly decreases the number of vertices that hear a broadcast, that is,
for any broadcast g ≤ f, |H(g)| < |H(f )|. This is the analog of saying that every vertex in an irredundant set S has a private neighbor or that the number of vertices dominated by an irredundant set
S is strictly greater than the number of vertices dominated by any proper subset S ⊂ S. Given an irredundant broadcast f, every vertex w ∈ Nf [v] for which H(w) = {v} is called a private broadcast
neighbor of v or a private f-neighbor of v. Note that a broadcast vertex v can be its own private f -neighbor, while any other private f neighbor of v must be a vertex w ∈ Vf0 . An irredundant
broadcast f is maximal if there does not exist an irredundant broadcast g such that (i) g = f and (ii) for every vertex v ∈ V , g(v) ≥ f (v). Definition 10 The broadcast irredundance number irb (G)
equals the minimum cost of a maximal irredundant broadcast in G. Similarly, the upper broadcast irredundance number IRb (G) equals the maximum cost of an irredundant broadcast. To this definition we
can add the following. An irredundant broadcast f is called 1-maximal if increasing the value of f (v) of any one vertex v ∈ V creates a function f that is no longer an irredundant broadcast, either
because it is no longer a broadcast function or because some broadcast vertex no longer meets Condition (i) or Condition (ii) of the definition of an irredundant broadcast. To illustrate this
definition, Ahmadi et al. consider the path P8 , with vertices labeled in order v1 , . . . , v8 . Let f be the irredundant broadcast function defined by f (v4 ) = f (v5 ) = 2 and f (v1 ) = f (v2 ) =
f (v3 ) = f (v6 ) = f (v7 ) = f (v8 ) = 0. This broadcast function f is irredundant because broadcast vertex v4 has vertex v2 as a private broadcast neighbor, while broadcast vertex v5 has vertex v7
as a private broadcast neighbor. It can be seen that if the value of any of vertex other than v4 and v5 is increased, then either vertex v4 or v5 will no longer have a private broadcast neighbor.
Similarly, if the value of either v4 or v5 is increased, then vertex v5 or v4 will no longer have a private broadcast neighbor. Thus, f is a 1-maximal irredundant broadcast function. However, f is
not a maximal irredundant broadcast, since the function g defined by g(v4 ) = g(v5 ) = 3 and g(v1 ) = g(v2 ) = g(v3 ) = g(v6 ) = g(v7 ) = g(v8 ) = 0 is an irredundant broadcast for which g ≥ f.
Proposition 16 An irredundant broadcast function can be 1-maximal but not maximal, but every maximal irredundant broadcast is 1-maximal. Consider the path P6 , with vertices labeled v1 , . . . , v6 ,
the maximal irredundant set S = {v3 , v4 }, and the broadcast function fS defined by fS (v3 ) = fS (v4 ) = 1 and fS (v1 ) = f (v2 ) = f (v5 ) = fS (v6 ) = 0. It is easy to see that fS is an
irredundant broadcast. However, it is not a maximal irredundant broadcast. The function g defined by
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fS (v3 ) = fS (v4 ) = 2 and fS (v1 ) = f (v2 ) = f (v5 ) = fS (v6 ) = 0 is an irredundant broadcast function for which g ≥ fS . Proposition 17 The characteristic function of every maximal irredundant
set is an irredundant broadcast, but is not necessarily a maximal irredundant broadcast function. Proposition 18 (Ahmadi et al.) Every γ b -broadcast function is a maximal irredundant broadcast.
Proof Let f be a γ b -broadcast for an arbitrary graph G of order n. It follows that any broadcast g = f with g ≤ f cannot be a dominating broadcast, since f is a minimal dominating broadcast. This
means that f is an irredundant broadcast since for any such a broadcast g, with g ≤ f, |H(g)| < |H(f )| = n. It follows that f is a maximal irredundant broadcast since any broadcast h with h ≥ f must
have |H(h)| = n, and therefore since f ≤ h, and |H(f )| = n = |H(h)|, h cannot be an irredundant broadcast. Corollary 9 For any graph G, irb ≤ γb ≤ b ≤ I Rb . Finally, Ahmadi et al. establish the
following Broadcast Domination Chain: irb ≤ γb ≤ ib ≤ αb ≤ b ≤ I Rb . In addition they show the following. Proposition 19 (Ahmadi et al.) For any graph G, IR(G) ≤ IRb (G). At the end of their paper,
Ahmadi et al. ask the following questions: 1. What is the relationship between ir(G) and irb (G)? 2. Can irb (T) and IRb (T) be computed in polynomial time for a tree T? 3. What is the complexity of
the following decision problem? MAXIMAL IRREDUNDANT BROADCAST INSTANCE: Graph G = (V, E), function f : V →{0, 1, 2, . . . . . . } QUESTION: Is f a maximal irredundant broadcast function?
12 Roman Irredundance in Graphs In 2016 Chellali, Haynes, S.M. Hedetniemi, S. T. Hedetniemi, and McRae [8] introduced a Roman Domination Chain, comparable in many respects to the Domination Chain and
the Broadcast Domination Chain. In this section, we present the necessary definitions for this chain of inequalities.
The Private Neighbor Concept
A function f : V →{0, 1, 2} is a Roman dominating function on a graph G = (V, E) if for every vertex v ∈ V with f (v) = 0, there exists a neighbor u ∈ N(v) with f (u) = 2. The weight of a Roman
dominating function is f (V ) = v ∈ V f (v). The Roman domination number γ R (G) equals the minimum weight of a Roman dominating function on G, and the upper Roman domination number R (G) equals the
maximum weight of a minimal Roman dominating function on G. The graph theoretical introduction of Roman domination was introduced in 2004 by Cockayne, Dreyer, S. M. Hedetniemi, and S. T. Hedetniemi
[21]. Given any function of the form f : V →{0, 1, 2}, it is convenient to define the following three sets, for i ∈{0, 1, 2}, Vi = {v ∈ V : f (v) = i}. Thus, we can denote such a function f by f =
{V0 , V1 , V2 }. And the weight of such a function is f (V ) = |V1 | + 2|V2 |. A function f = {V0 , V1 , V2 } is called irredundant if 1. V1 is an independent set, 2. no vertex in V1 is adjacent to a
vertex in V2 , 3. every vertex v ∈ V2 has a private neighbor in V0 with respect to the set V2 , that is, there exists a vertex w ∈ V0 such that N(w) ∩ V2 = {v}. A Roman irredundant function is
maximal if increasing the value assigned to any vertex results in a function that is no longer irredundant. The (lower) Roman irredundance number irR (G) equals the minimum weight of a maximal Roman
irredundant function on G. The upper Roman irredundance number IRR (G) equals the maximum weight of an irredundant function on G. This leaves us to define the Roman independence numbers. A Roman
dominating function is called independent if V1 ∪ V2 is an independent set. The independent Roman domination number iR (G) equals the minimum weight of an independent Roman dominating function on G.
The Roman independence number α R (G) equals the maximum weight of an irredundant, independent Roman dominating function on G. With these definitions, Chellali et al. were able to prove the following
theorem. Theorem 7 (Chellali et al.) For any graph G, irR ≤ γR ≤ iR ≤ αR ≤ R ≤ I RR . Proof Sketch. 1. iR (G) ≤ α R (G) follows from the definitions of iR (G) and α R (G) and the fact that every iR
-function on a graph G is a Roman irredundant function. 2. γ R (G) ≤ iR (G) and α R (G) ≤ R (G) follow the fact that iR (G) and α R (G) are both realized by Roman independent, irredundant dominating
functions, while γ R (G) and R (G) are realized by Roman irredundant dominating functions. 3. R (G) ≤ IRR (G) follows from the definition of R (G) that every R -function is an irredundant function.
4. irR (G) ≤ γ R (G) follows, first of all, from the observation that every graph G has a γ R -function that is irredundant. It only remains to show that every irredundant
S. T. Hedetniemi et al.
γ R -function is maximal irredundant. This follows from the observation that if any vertex is assigned a larger value, then the resulting function will no longer be an irredundant function: a. by
definition no vertex in V2 can have its value increased or it won’t be a Roman dominating function, b. no vertex in V0 can be increased to 1 since it would be adjacent to a vertex in V2 and no longer
be an irredundant function, and c. no vertex in V0 or V1 can be increased to 2 since it could not have a private neighbor in V0 . Chellali et al. were also able to establish the following
inequalities: 1. 2. 3. 4.
ir(G) ≤ irR (G). α(G) ≤ α R (G) ≤ 2α(G). (G) ≤ R (G). IR(G) ≤ IRR (G).
13 Fractional Irredundance Many discrete, or integer-valued, graph theory concepts have fractional counterparts, and irredundance is no exception. This can be explained as follows. Let G = (V, E) be
a graph and let Y be an arbitrary set of real numbers, finite or infinite, positive or negative. A function f : V → Y is called a Y -dominating function if for every v ∈ V , f (N[v]) = u ∈ N[v] f (u)
≥ 1. In other words, the closed neighborhood sum f (N[v]) of every vertex v ∈ V is at least one. The weight of a Y -dominating function f is w(f ) = f (V ) = u ∈ V f (u). The Y domination number γ Y
(G) equals the minimum weight of a Y -dominating function f on G. A Y -dominating function f is called minimal if there does not exist another Y -dominating function g, f = g, with g(v) ≤ f (v) for
every v ∈ V . The upper Y -domination number Y (G) equals the maximum weight w(f ) of a minimal Y dominating function f on G. When Y = {0, 1}, γ {0,1} (G) = γ (G), the standard domination number of a
graph G, and {0,1} (G) = (G), the upper domination number of G. When Y = [0, 1] is the closed unit interval, γ [0,1] (G) = γ f (G), the fractional domination number of a graph G, and [0,1] (G) = f
(G), the upper fractional domination number of G. In 2006 Fricke, Hedetniemi and Jacobs [36] introduced the following. A function g : V → [0, 1] is called an irredundant function if for every vertex
v ∈ V with g(v) > 0, there exists a vertex w ∈ N[v] such that g(N[w]) = 1. An irredundant function f is called maximal if there does not exist an irredundant function h, h = g, with h(v) ≥ g(v) for
every v ∈ V .
The Private Neighbor Concept
It is easy to see that the characteristic function χ S of a (maximal) irredundant set S is a (maximal) irredundant function. Definition 11 The fractional irredundance number irf (G) is the infimum,
irf (G) = inf {g(V ) : g is a maximal irredundant function on G}, and the fractional upper irredundance number IRf (G) is the supremum, IRf (G) = sup{g(V ) : g is a maximal irredundant function on
G}. In 1988, Domke, Hedetniemi, and Laskar [24] point out that for the Hajós graph G, γ f (G) < γ (G) (cf. Figure 6), where γ f (G) = 3/2 < γ (G) = 2. In 1996, Fricke, Hedetniemi, and Jacobs [34]
point out that for the path P7 , irf (P7 ) = 2 < ir(P7 ) = 3. They also note that while the infimum of g(P7 ) = 2 over all maximal irredundant functions, no maximal irredundant function has this
value (cf. Figure 7). In 1990, Cheston, Fricke, Hedetniemi, and Jacobs [9] present a graph G with (G) < f (G) (cf. Figure 8). Given that these strict inequalities can hold for some graphs G, γ f (G)
< γ (G), (G) < f (G), and irf (G) < ir(G), the following result due to Fricke, Hedetniemi, and Jacobs [36] is somewhat surprising. Theorem 8 For any graph G, IR(G) = IRf (G). Fig. 6 γf =
0, there exists a vertex w ∈ N[v] such that g(N[w]) ≤ 1. Note that all irredundant functions are irreducible. Lemma 5 Every maximal irreducible function is maximal irredundant. Lemma 6 For any
irreducible function g, there exists a maximal irreducible function g for which g ≤ g . Lemma 7 If g is irreducible and g(V ) = IRf (G), then g is irredundant. Lemma 8 The set M = {f ∈ I : f (V ) = I
Rf (G)} is closed. For a function f on V , let z(f ) = {v ∈ V : f (v) = 0}, and let Z be the functions f ∈ M which maximize z(f ). The objective is to show that the functions in Z are 0–1 functions.
Lemma 9 There exists a function g ∈ Z having the smallest positive value b over all functions in Z. Let g and b be as in the previous lemma, and let vb be a vertex for which g(vb ) = b. Lemma 10 For
any vertex w = vb with g(w) = 0, there exists a vertex x ∈ N[w] − N[vb ] with g(N[x]) = 1. The proof of Theorem 8 follows by showing that a maximum irredundant function g having smallest value g(vb )
= b > 0 among the functions in Z is a 0–1 function. A similar theorem was proved in 2016 by Fricke, O’Brien, Schroeder, and Hedetniemi [37]. A real-valued function g : V → [0, 1] is called open
irredundant if for every vertex v ∈ V with g(v) > 0, there exists a vertex w adjacent to v such that g(N[w]) = 1.
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An open irredundant function g is maximal if there does not exist an open irredundant function h such that g = h and g(v) ≤ h(v), for every v ∈ V . Definition 12 The fractional open irredundance
number oirf (G) is the infimum, oirf (G) = inf {g(V ) : g is a maximal open irredundant function on G}, and the fractional upper open irredundance number OIRf (G) is the supremum, OIRf (G) = sup{g(V
) : g is a maximal open irredundant function on G}. Notice the slight distinction between open irredundant functions and irredundant functions; for irredundant functions, there must be a vertex w ∈ N
[v] (closed neighborhood) with g(N[w]) = 1, while for open irredundant functions, there must be a such a vertex w ∈ N(v) (open neighborhood) with g(N[w]) = 1, that is, w = v. Theorem 9 (Fricke et
al.) For any graph G, OIR(G) = OIRf (G). This result is proved roughly as follows. A function g : V → [0, 1] is open irreducible or oiru if for every vertex v ∈ V with g(v) > 0, there exists a vertex
w ∈ N(v) (open neighborhood) such that g(N[w]) ≤ 1. In the special case that for every v ∈ V with g(v) > 0, there exists a vertex w ∈ N(v) such that g(N[w]) = 1, we say that g is fractional open
irredundant. Furthermore, if g is a fractional open irredundant function such that g : V →{0, 1}, then g is open irredundant. Examples of each type of function are shown by Fricke et al. [37] in
Figure 9. Thus, OI RUf (G) = sup{g(V )|g is a fractional oiru function}, OI Rf (G) = sup{g(V )|g is a fractional open irredundant function} OI R(G) = sup{g(V )|g is an open irredundant function}.
Note that since all open irredundant functions are fractional open irredundant, and all fractional open irredundant functions are oiru, we immediately have that OI R(G) ≤ OI Rf (G) ≤ OI RUf (G).
Fig. 9 (i) An oiru function. (ii) A fractional open irredundant function. (iii) An open irredundant function
The Private Neighbor Concept
Fricke et al. show that, in fact, equality holds, a key step of which is showing that the supremum OIRUf (G) is a maximum, that is, there exists an oiru function g such that g(V ) = OIRUf (G).
14 Open Problems Involving Irredundance The inequalities in the Domination Chain and Covering Chain suggest that these parameters might have relationships to others that have been studied in the
literature. We mention just a few of these possibilities. 1. It has been shown that IR(G) ≤ ER(G), where ER(G) equals the maximum cardinality of an external redundant set (cf. p. 97 of [42]). How
does ER(G) compare with (G) and CIR(G)? 2. What can you say about the parameter cir(G)? We have observed that γ (G) ≤ cir(G) ≤ ψ(G). 3. It has been observed that (i) γ (G) ≤ α 1 (G); (ii) γ (G) ≤ β 1
(G); (iii) γ (G) ≤ β 2 (G), where β 2 (G) is the 2-maximal matching number (cf. p. 59 of [42]); and (iv) γ (G) ≤ 2ir(G) − 1. How do these bounds compare with either i(G) or cir(G)? 4. It has been
observed that for many classes of graphs, including bipartite, chordal, circular arc, cographs, and permutation graphs, just to name a few, the upper three parameters are all equal, that is, α(G) =
(G) = IR(G) (cf. p. 81 of [42]). Can these equalities be extended for these classes of graphs to: α(G) = (G) = IR(G) = (G) = CIR(G)? 5. The concept of irredundance is inherently about the concepts of
private neighbors. In searching for the next concept after independence, domination, and irredundance, several authors proposed the study of external redundance, Rannihilated sets, private neighbor
counts, and pnc-maximal sets. With this in mind, one can study the maximum number of private neighbors of a given type, or of given types, in sets S, not the number of vertices in S which have at
least one private neighbor, but the total number of vertices in V which are a private neighbor of some vertex in S. One can make the following definitions: IR∗ (S), the number of vertices in V that
are a private neighbor (self or external) of a vertex in S; IR(G) ≤ IR∗ (G) = max{IR∗ (S) : S ⊆ V }. OIR∗ (S), the number of vertices in V that are an external private neighbor of a vertex in S; OIR
(G) ≤ OIR∗ (G) = max{OIR∗ (S) : S ⊆ V }. OOIR∗ (S), the number of vertices in V that are either an external or an internal private neighbor of a vertex in S; OOIR(G) ≤ OOIR∗ (G) = max{OOIR∗ (S) : S ⊆
V }. COIR∗ (S), the number of vertices in V that are a private neighbor (self, external, or internal) of a vertex in S; COIR(G) ≤ COIR∗ (G) = max{COIR∗ (S) : S ⊆ V }. One can extend this to other
parameters, such as the following:
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I Rα∗ (S), the number of vertices in V that are a private neighbor (self or external) of a vertex in an independent set S; I Rα∗ (G) = max{I Rα∗ (S) : S an independent set in G}≥ α(G). I Rγ∗ (S), the
number of vertices in V that are a private neighbor (self or external) of a vertex in a dominating set S; I Rγ∗ (G) = max{I Rγ∗ (S) : S a dominating set in G}≥ (G). ∗ (S), the number of vertices in V
that are a private neighbor (self or I Rir ∗ (G) = max{I R ∗ (S) : S external) of a vertex in an irredundant set S; I Rir ir an irredundant set in G}≥ IR(G).
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An Introduction to Game Domination in Graphs Michael A. Henning
AMS Subject Classification: 05C65, 05C69
1 Introduction Although competitive optimization graph games are well studied in the literature, the domination game which we discuss in this chapter is relatively new and was only formally birthed
in 2010 by Brešar, Klavžar, and Rall [4]. We remark that this domination game introduced in [4], which we formally define in Section 2.1, is very different from the competition-enclaveless game
introduced in 2001 by Philips and Slater [46, 47] which is played by two players who take turns in constructing a maximal enclaveless set in a graph. (Some of the significant differences between
these two games are explained in [24, Chapter 5].) We also remark that the domination game introduced in [4] is very different from the domination game introduced in 2002 by Alon, Balogh, Bollobas,
and Szabo [1]. In [1], Alon et al. define the oriented game domination number of a graph G for which two players alternately orient an edge of G until all of the edges are oriented, their goals being
to minimize and maximize the domination number of the resulting oriented graph. Since Brešar et al. [4] first studied the concept of the domination game in graphs, it has subsequently attracted
considerable interest. Our aim in this chapter is to introduce and familiarize the reader with three domination-type games and to present selected results on these games with the hope to encourage
and stimulate continued research of the topic.
M. A. Henning () Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: [email protected] © The Author(s), under exclusive license to
Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://doi.org/10.1007/978-3-030-58892-2_8
M. A. Henning
For a more comprehensive and thorough treatment of the domination game, we refer the reader to the forthcoming book entitled “Domination Games Played in Graphs” by Brešar, Henning, Klavžar, and Rall
2 Domination-Type Games In this section, we define three domination-type games, namely the domination game, the total domination game, and the independent domination game. We remark that many other
domination-type games are studied in the literature, including the connected domination game, the paired-domination game, the competitionenclaveless game, the oriented domination game, the
maker-breaker domination game, the disjoint domination game, the fractional domination game, the Grundy domination game, the Grundy total domination game, the Z-Grundy domination game, the L-Grundy
domination game, to name a few. A survey of these dominationtype games in graphs can be found in [11]. In this chapter we model the “Domination in Graphs: Core Concepts” book by Haynes, Hedetniemi,
and Henning [23] in the sense that we focus exclusively on the three main domination parameters, namely the domination number, the total domination number, and the independent domination number. That
is, we restrict our attention to the domination game, the total domination game, and the independent domination game. We begin with a formal definition of the domination game played in graphs.
2.1 The Domination Game We recall that a vertex dominates itself and its neighbors. A dominating set of a graph G is a set S of vertices of G such that every vertex in G is dominated by a vertex in
S. The domination number of G, denoted γ (G), is the minimum cardinality of a dominating set in G. The domination game on a graph G consists of two players, Dominator and Staller, who take turns
choosing a vertex from G. Each vertex chosen must dominate at least one vertex not dominated by the vertices previously chosen. We call such a vertex a playable vertex. A move in the game, sometimes
referred to in the literature as a legal move for emphasis, is a vertex chosen by a player. The game ends when there are no more moves available. Upon completion of the game, the set of chosen
(played) vertices is a dominating set in G, but is not necessarily a minimal dominating set. The goal of Dominator is to end the game with a minimum number of vertices chosen, while Staller has the
opposite goal and wishes to end the game with as many vertices chosen as possible. The Dominator-start domination game and the Staller-start domination game are the domination game when Dominator and
Staller, respectively, choose the first
An Introduction to Game Domination in Graphs
221 s2
(a) γg (G) = 3
(b) γg (G) = 4
Fig. 1 A graph G with γ g (G) = 3 and γg (G) = 4
vertex. These games are called the D-game and S-game, respectively. The D-game domination number, γ g (G), of G is the minimum possible number of moves in a Dgame when both players play according to
the rules, while the S-game domination number, γg (G), of G is defined analogously for the S-game. A sequence of moves by Dominator that achieves the minimum possible number of moves is called an
optimal sequence for Dominator, while a sequence of moves by Staller that achieves the maximum possible number of moves is called an optimal sequence for Staller. We denote the sequence of moves
played in the D-game by d1 , s1 , d2 , s2 , . . . , where di is the vertex chosen on Dominator’s ith move, and si is the vertex chosen on Staller’s ith move in response to Dominator’s ith move. The
sequence of moves played in the S-game is denoted by s1 , d1 , s2 , d2 , . . ., where si is the vertex played by Staller on her ith move and di is the vertex played by Dominator on his ith move in
response to Staller’s ith move. As an illustration, if G is the graph shown in Figure 1, then γ g (G) = 3 and one optimal sequence in the D-game is given by d1 , s1 , d2 as shown in Figure 1a.
Moreover, γg (G) = 4 and one optimal sequence in the S-game is given by s1 , d1 , s2 , d2 as shown in Figure 1b. As remarked earlier, the domination game in graphs which we discuss in the chapter was
formally birthed in 2010 by Brešar, Klavžar, and Rall [4]. The domination game has subsequently been extensively studied in [5–10, 12, 13, 15, 19, 20, 24, 26, 28, 35, 36, 38–45, 49, 50, 53–56] and
other papers.
2.2 The Total Domination Game A vertex totally dominates another vertex if they are neighbors. A total dominating set of a graph G is a set S of vertices such that every vertex of G is totally
dominated by a vertex in S. The total domination number of G, denoted γ t (G), is the minimum cardinality of a total dominating set in G. The total domination game is defined analogously as the
domination game, except that in the total version each vertex chosen must totally dominate at least one vertex not totally dominated by the set of vertices previously chosen. Such a
M. A. Henning
(a) γtg (G) = 4
(b) γtg (G) = 4
Fig. 2 A graph G with γ tg (G) = 4 and γtg (G) = 4
chosen vertex is called a move (sometimes referred to as a legal move for emphasis) in the total domination game. The game ends when there is no legal move available. In this case, the set of
vertices chosen is a total dominating set in G. Dominator’s objective is to minimize the number of vertices chosen, while the goal of Staller is just the opposite, namely to end the game with as many
vertices chosen as possible. The Dominator-start (resp., Staller-start) total domination game is the total domination game when Dominator (resp., Staller) has the first move. As with the domination
game, we refer to these simply as the D-gama and S-game, respectively. The D-game total domination number, γ tg (G), of G is the minimum number of moves in the D-game when both players follow a
strategy to achieve their goals, while the S-game total domination number, γtg (G), is the maximum number of moves in a S-game when both players play optimally. We adopt the same notation as in the
domination game, and denote the sequence of moves played in the total version of the D-game by d1 , s1 , d2 , s2 , . . . , and the sequence of moves played in the total version of the S-game by s1 ,
d1 , s2 , d2 , . . .. As an illustration, if G is the graph shown in Figure 2, then γ tg (G) = 4 and one optimal sequence in the D-game is given by d1 , s1 , d2 , s2 as shown in Figure 2a. Moreover,
γtg (G) = 4 and one optimal sequence in the S-game is given by s1 , d1 , s2 , d2 as shown in Figure 2b. The total version of the domination game was first investigated in 2015 by Henning, Klavžar,
and Rall in [31], where it was demonstrated that these two versions differ significantly. The total domination game has subsequently been studied in [3, 14, 16–18, 27, 29–34, 37] and other papers.
2.3 The Independent Domination Game An independent dominating set in G is a dominating set of G that is independent. The independent domination number, denoted i(G), of G is the minimum cardinality
of an independent dominating set in G. An independent set of vertices in G is a dominating set of G if and only if it is a maximal independent set. Thus, i(G) is equivalently the minimum cardinality
of a maximal independent set of vertices in G.
An Introduction to Game Domination in Graphs
The independent domination game, called the competition-independence game by Philips and Slater [46, 47], is defined analogously as the domination game, except that in the independent version each
vertex chosen must not be adjacent to any vertex previously chosen. More formally, adopting the notation coined by Goddard and Henning [22], the game is played by two players, Diminisher and Sweller,
on some graph G. They take turns in constructing a maximal independent set I of G. That is, each turn a player chooses a vertex that is not adjacent to any of the vertices already chosen until there
is no such vertex. Such a chosen vertex is called a move (or legal move, for emphasis) in the independent domination game. The game ends when there is no legal move available. In this case, the set I
of vertices chosen is an independent dominating set in G. The goal of Diminisher is to make the final set I as small as possible and for Sweller to make the final set I as large as possible. The
Diminisher-start independent domination game and the Sweller-start independent domination game are the independent domination game when Diminisher and Sweller, respectively, choose the first vertex.
As with the domination and total domination game, these games are called the D-game and S-game, respectively. The D-game independent domination number, Id (G), of G is the minimum possible number of
moves in a D-game when both players follow a strategy to achieve their goals, while the S-game independent domination number, Is (G), is the number of moves in a S-game when both players play
optimally. As before, we denote the sequence of moves played in the independent domination version of the D-game by d1 , s1 , d2 , s2 , . . . , and the sequence of moves played in the independent
domination version of the S-game by s1 , d1 , s2 , d2 , . . .. As an illustration, if G is the graph shown in Figure 3, then Id (G) = 5 and one optimal sequence in the D-game is given by d1 , s1 , d2
, s2 , d3 as shown in Figure 3a. Moreover, Is (G) = 5 and one optimal sequence in the S-game is given by s1 , d1 , s2 , d2 , s3 as shown in Figure 3b. In 2001 Philips and Slater [46, 47] introduced
the independent domination game, which they called the competition-independence game. The independent domination game has not attracted the same amount of interest as the domination and total
domination games, and has been studied in [22, 52] and other papers. The most significant difference between the domination and total domination game compared with the independent domination game
(and the competition-enclaveless game defined in [46, 47]) is that the so-called Continuation Principle, which we present in Section 5, holds for both the domination game and total domination game,
but does not hold for the independent domination game. This makes it very difficult to obtain general results on the independent domination game (and the competitionenclaveless game).
M. A. Henning d2
s2 d1
d3 (a) Id(G) = 5
s3 (b) Is(G) = 5
Fig. 3 A graph G with Id (G) = 5 and Is (G) = 5
3 Basic Properties In their introductory paper on the domination game, Brešar, Klavžar, and Rall [4] established the following relationship between the domination number and the game domination
number. Theorem 1 ([4]) For every graph G, we have γ (G) ≤ γ g (G) ≤ 2γ (G) − 1. An analogous relationship between the total domination number and the game total domination number was established in
the introductory paper by Henning, Klavžar, and Rall in [31] on the total domination game. The proof of Theorems 1 and 2 are along similar lines. We therefore present here only the proof of Theorem
2. Theorem 2 ([31]) If G is a graph with no isolated vertex, then γ t (G) ≤ γ tg (G) ≤ 2γ t (G) − 1. Proof. Upon completion of the Dominator-start total domination game played on G, the vertices
played by Dominator and Staller together form a total dominating set of G, implying that γ t (G) ≤ γ tg (G). To prove that γ tg (G) ≤ 2γ t (G) − 1, Dominator adopts the following strategy. He selects
an arbitrary minimum total dominating set D of G and orders the vertices of D. On each of his moves he plays a vertex from the set D sequentially according to this ordering that has not yet been
played and is a legal move (and therefore totally dominates at least one vertex not totally by the set of vertices previously played by the two players). We note that when Dominator considers a
vertex v in the ordered set D, either the vertex v is a legal move, in which case he plays the vertex v, or the vertex v is not a legal move, in which case he considers the next vertex in the
ordering, if such a vertex exists. In both cases, after Dominator has played his move, the vertex v can never be a legal move in the remaining part of the game and Dominator therefore never considers
the vertex v again. Once Dominator has considered all vertices according to his ordering of the set D, every vertex is totally dominated by the set of vertices previously played by Dominator and
Staller, and hence no more moves are legal. Thus, Dominator plays at most |D| moves and Staller at most |D|− 1 moves. In this way, Dominator can
An Introduction to Game Domination in Graphs
guarantee that the game finishes in at most 2|D|− 1 = 2γ t (G) − 1 moves, implying that γ tg (G) ≤ 2γ t (G) − 1. 2 A significant difference between the domination game and the independent domination
game is that upon completion of the domination game, the set of played vertices is a dominating set although not necessarily a minimal dominating set, while upon completion of the independent
domination game, the set of played vertices is always a maximal independent set. Thus, the independent domination game numbers of a graph G are always squeezed between the independent domination
number i(G) of G and the independence number α(G) of G, which is the maximum cardinality of an independent set in G. We state this formally as follows. Theorem 3 If G is a graph of order n, then i(G)
≤ Id (G) ≤ α(G)
i(G) ≤ Is (G) ≤ α(G).
A graph G is well-covered if all of the maximal independent set in G have the same cardinality. The problem of determining which graphs have the property that every maximal independent set of
vertices is also a maximum independent set was proposed in 1970 by Plummer [48] and has subsequently been extensively studied in the literature. As observed earlier, upon completion of the
independent domination game, the set of played vertices is always a maximal independent set. Hence, any sequence of legal moves by Diminisher and Sweller (regardless of strategy) in the independent
domination game played in a well-covered graph of order n will always lead to the game ending in α(G) moves. Thus as a consequence of Theorem 3, we have the following interesting connection between
the independent domination game and the class of well-covered graphs. Theorem 4 If G is a well-covered graph, then Id (G) = Is (G) = α(G). The game domination number and the game total domination
number are related as follows. Theorem 5 ([31]) If G is a graph on at least two vertices, then γ g (G) ≤ 2γ tg (G) − 1. Proof. By Theorem 1, we have γ g (G) ≤ 2γ (G) − 1. Since every total dominating
set is by definition a dominating set of G, the inequality γ (G) ≤ γ t (G) holds. By Theorem 2, we have γ t (G) ≤ γ tg (G). These observations imply that γ g (G) ≤ 2γ (G) − 1 ≤ 2γ t (G) − 1 ≤ 2γ tg
(G) − 1. 2 As observed in [31], to see that Theorem 5 is close to being optimal consider the following examples. For any integer k ≥ 2, let Gk be the graph obtained from the complete graph on k
vertices by attaching k leaves to each of its vertices. As shown in [31], γ tg (Gk ) = k + 1 and γ g (Gk ) = 2k − 1, and so γ g (Gk ) = 2γ tg (Gk ) − 3. Thus we have the following result. Theorem 6
([31]) If n ≥ 2 is an integer and Gn denotes the class of all isolate-free graphs G of order n, then
M. A. Henning
sup n
γg (G) = 2, γtg (G)
where the supremum is taken over all graphs G ∈ Gn . The game total domination number can be bounded by the domination number as follows. Theorem 7 ([31]) If G is a graph such that γ (G) ≥ 2, then γ
(G) ≤ γ tg (G) ≤ 3γ (G) − 2. Proof. The lower bound follows immediately from the inequality chain γ (G) ≤ γ t (G) ≤ γ tg (G). To prove the upper bound, let D be an arbitrary γ -set of G. Dominator
adopts the following simple strategy in the total domination game. He selects vertices in D sequentially whenever such a move is legal. Once Dominator has played all allowable vertices in D, we note
that at most 2|D|− 1 = 2γ (G) − 1 moves have been made. At this point of the game all vertices that have a neighbor in the set D are totally dominated. There are two possible cases to consider. Case
1: No vertex in D is currently totally dominated. In this case, the set D is an independent set and both Dominator and Staller only played vertices from D. Thus, exactly |D| = γ (G) moves have been
made at this point in the game. The only remaining legal moves that can be played in the total domination game are those that totally dominate vertices in D. There are therefore at most |D|
additional moves that are played in order to complete the game, implying that the total number of moves played is at most 2|D| = 2γ (G) ≤ 3γ (G) − 2 noting that γ (G) ≥ 2. Case 2: At least one vertex
in D is currently totally dominated. In this case, the only legal moves remaining in the total domination game are those that totally dominate vertices in D, if any, are not yet totally dominated.
This implies that at most |D|− 1 additional moves are required to complete the game. Therefore, the total number of moves played is at most (2|D|− 1) + (|D|− 1) = 3γ (G) − 2. 2 As shown in [31], both
the lower and upper bounds in Theorem 7 are tight. We present here only an example illustrating the lower bound. For k ≥ 3 and ≥ 1, let G = Fk, be obtained from a complete bipartite graph K2,k by
selecting an arbitrary vertex v of degree 2 in K2,k , and appending to it vertex-disjoint paths of length 2. We note that γ (G) = + 2, and hence γ tg (G) ≥ γ (G) = + 2. Let u and w be the neighbors
of v that belong to the complete bipartite graph K2,k . Suppose now that Dominator plays as his first move in the total domination game the vertex v. The only possible legal moves in the remainder of
the game are the + 2 neighbors of the vertex v in G. However, exactly one of u and w can be played in the game, while every support vertex of v (of degree 2) in G must be played. Thus, exactly + 2
vertices are played in the game, namely the vertex v, exactly one of u and w, and all support vertices of G. This strategy of Dominator implies that γ tg (G) ≤ + 2. As observed earlier, γ tg (G) ≥ +
2. Consequently γ (G) = γ tg (G) = + 2.
An Introduction to Game Domination in Graphs
4 Paths and Cycles Determining exact values of the domination game parameters for even relatively simple classes of graphs is relatively complex. The exact values of the game domination number, the
game total domination number, and the game independent domination number for paths and cycles are known. In 2017 Košmrlj [43] determined the formulas for the game domination number for paths and
cycles, and also gave optimal strategies for both players. Theorem 8 ([43]) If n ≥ 3, then γg (Cn ) =
⎧n ⎨ 2 − 1;
n ≡ 3 (mod 4) ,
otherwise ,
γg (Cn )
⎧ n−1 ⎪ ⎪ ⎨ 2 − 1;
n ≡ 2 (mod 4) ,
⎪ ⎪ ⎩ n−1 ;
otherwise .
Theorem 9 ([43]) If n ≥ 1, then γg (Pn ) =
⎧n ⎨ 2 − 1;
n ≡ 3 (mod 4) ,
otherwise ,
and γg (Pn ) =
n 2
For small n, the values of the game domination numbers for a cycle Cn and a path Pn are shown in Table 1. The game total domination numbers for cycles and paths were determined in 2016 by Dorbec and
Henning [18]. Theorem 10 ([18]) If n ≥ 3, then γtg (Cn ) =
⎧ 2n+1 ⎨ 3 − 1; ⎩
2n+1 3 ;
n ≡ 4 (mod 6), otherwise,
Table 1 The game domination numbers for small cycles and paths n γ g (Cn ) γg (Cn )
n γ g (Pn ) γg (Pn )
M. A. Henning
and γtg (Cn )
⎧ 2n ⎨ 3 − 1; ⎩
2n 3 ;
n ≡ 2 (mod 6), otherwise.
Theorem 11 If n ≥ 1, then γtg (Pn ) =
and γtg (Pn ) =
2n 3
⎧ 2n ⎨ 3 ; ⎩
2n 3 ;
n ≡ 5 (mod 6), otherwise,
For small n, the values of the game total domination numbers for a cycle Cn and a path Pn are shown in Table 2. We remark that these results for paths and cycles show that for some families of graphs
Dominator has an advantage (paths) and for some families of graphs Staller has an advantage (cycles). And for still other families neither player has an advantage by going first. In 2002 Philips and
Slater [47] determined the game independent domination numbers of paths and cycles. Theorem 12 ([47]) The following holds. 3n+6 (a) For n ≥ 1, Id (Pn ) = 3n+4 7 and Is (Pn ) = 7 . 3n+2 (b) For n ≥ 3,
Id (Cn ) = 3n+3 7 and Is (Cn ) = 7 .
We note that the game independent domination numbers for a path immediately provide the value for a cycle, since the first move in a cycle Cn on n ≥ 3 vertices produces a path Pn−3 on n − 3 vertices.
Thus, Id (Cn ) = 1 + Is (Pn−3 ) and Is (Cn ) = 1 + Id (Pn−3 ). For small n, the values of the game independent domination numbers for a cycle Cn and a path Pn are shown in Table 3. Table 2 The game
total domination numbers for small cycles and paths n γ tg (Cn ) γtg (Cn )
n γ tg (Pn ) γtg (Pn )
Table 3 The game independent domination numbers for small cycles and paths n Id (Cn ) Is (Cn )
n Id (Pn ) Is (Pn )
An Introduction to Game Domination in Graphs
5 Continuation and Total Continuation Principles A partially dominated graph is a graph together with a declaration that some vertices are already dominated and need not be dominated in the rest of
the game. More formally, if G is a graph and S ⊆ V (G), then a partially dominated graph G|S is a graph together with a declaration that the vertices from S are already dominated. We use γ g (G|S)
(resp. γg (G|S)) to denote the number of moves remaining in the game on G|S under optimal play when Dominator (resp. Staller) has the next move. In 2013 Kinnersley, West, and Zamani in [38] presented
the following key lemma, named the Continuation Principle. Lemma 13 (Continuation Principle) If G is a graph and A, B ⊆ V (G) with B ⊆ A, then γ g (G|A) ≤ γ g (G|B) and γg (G|A) ≤ γg (G|B). As a
consequence of the Continuation Principle whenever x and y are legal moves for Dominator in the domination game and N[x] ⊆ N[y], then Dominator will play y instead of x, while Staller will play x
instead of y. As a further consequence, we have the fundamental property of the domination game that the number of moves in the D-game and the S-game when played optimally can differ by at most 1.
Theorem 14 If G is a graph, then |γg (G) − γg (G)| ≤ 1. There are graphs H1 , H2 , and H3 such that γg (H1 ) = γg (H1 ), γg (H2 ) = + 1, and γg (H3 ) = γg (H3 ) − 1. For example, by Theorems 8 and 9
the following holds where k ≥ 1 is an arbitrary integer. If H1 = Cn where n = 4k, then γg (H1 ) = γg (H1 ) = 2k. If H2 = Cn where n = 4k + 2, then γtg (H2 ) = γtg (H2 ) + 1 = 2k + 1. If H3 = Pn where
n = 4k + 3, then γtg (H3 ) = γtg (H3 ) − 1 = 2k + 1. A partially totally dominated graph is a graph together with a declaration that some vertices are already totally dominated and need not be
totally dominated in the rest of the game. If G is a graph and S ⊆ V (G), then a partially dominated graph G|S is a graph together with a declaration that the vertices from S are already totally
dominated. We use γ tg (G|S) (resp. γtg (G|S)) to denote the number of moves remaining in the total domination game on G|S under optimal play when Dominator (resp. Staller) has the next move. The
proof of the Continuation Principle can be modified to work for several variants of the domination game. In their introductory paper on the total domination game, the authors in [31] showed that the
Continuation Principle also holds for the total version of the game. γg (H2 )
Lemma 15 (Total Continuation Principle) If G is a graph and A, B ⊆ V (G) with B ⊆ A, then γ tg (G|A) ≤ γ tg (G|B) and γtg (G|A) ≤ γtg (G|B). Proof. Two games will be played in parallel, Game 1 on the
partially totally dominated graph G|A and Game 2 on the partially totally dominated graph G|B. The first of these will be the real game, while Game 2 will only be imagined by Dominator. In Game 1,
Staller will play optimally while in Game 2, Dominator will play optimally. In Game 2, Dominator will copy each move of Staller played in
M. A. Henning
Game 1, imagine that Staller played this move in Game 2, and then reply with an optimal move in Game 2. If this move is legal in Game 1, Dominator plays it in Game 1 as well. Otherwise, if the game
is not yet over, Dominator plays any other legal move in Game 1. We prove next the following claim. Claim 1 In each stage of the games, the set of vertices that are totally dominated in Game 2 is a
subset of the set of vertices that are totally dominated in Game 1. Proof. We proceed by induction. Since B ⊆ A, this is true at the start of the games. Suppose now that Staller has (optimally)
selected a vertex u in Game 1. Applying the induction assumption, the vertex u is a legal move in Game 2 because a new vertex v that was totally dominated by u in Game 1 is not yet dominated in Game
2. According to his strategy, Dominator copies the move of Staller by playing the vertex u in Game 2, and then replies with an optimal move in the imagined Game 2. If this move is legal in Game 1,
Dominator plays it in Game 1 as well. Otherwise, if the game is not yet over, Dominator plays any other legal move in Game 1. In either case the set of vertices that are dominated in Game 2 is a
subset of the set of vertices that are dominated in Game 1. By induction, this proves the desired claim. (2) Claim 1 implies that Game 1 finishes no later than Game 2. Suppose that m2 moves are
played Game 2. Since Dominator was playing optimally in Game 2, we note that m2 ≤ γ tg (G|B). Since Staller was playing optimally in Game 1 and Dominator has a strategy to finish Game 1 in m2 moves,
we infer that γ tg (G|A) ≤ m2 . Therefore, γ tg (G|A) ≤ m2 ≤ γ tg (G|B). Hence if Dominator is the first to play, then the desired result follows. In our earlier argument we made no assumption who
starts first. Thus in both cases, Game 1 will finish no later than Game 2. Hence the conclusion holds for γtg as well; that is, γtg (G|A) ≤ γtg (G|B). 2 As a consequence of the Total Continuation
Principle whenever x and y are legal moves for Dominator and N(x) ⊆ N(y), then Dominator will play y instead of x, while Staller will play x instead of y. The following fundamental property of the
total domination game that the number of moves in the D-game and the S-game when played optimally can differ by at most 1 follows readily from the Total Continuation Principle. We present a proof of
Theorem 16 along analogous lines to a proof of the result in Theorem 14. Theorem 16 If G is a graph with no isolated vertex, then |γtg (G) − γtg (G)| ≤ 1. Proof. Consider the D-game and let v be the
first move of Dominator. Let A = N(v) and consider the partially totally dominated graph G|A. Further let B = ∅ and note that G|B = G. By our choice of the vertex v as an optimal first move of
Dominator, we have γtg (G) = 1 + γtg (G|A). By the Total Continuation Principle, γtg (G|A) ≤ γtg (G|B) = γtg (G). Therefore, γtg (G) ≤ γtg (G|A) + 1 ≤ γtg (G) + 1. Next we consider the S-game and let
v be the first move of Staller. As before, let A = N(v) and B = ∅, and consider the partially totally dominated graph G|A. By our choice of the vertex v as an optimal first move of Staller, we have
γtg (G) = 1 + γtg (G|A). By the Total Continuation Principle, γ tg (G|A) ≤ γ tg (G|B) = γ tg (G), implying that γtg (G) ≤ γtg (G|A) + 1 ≤ γtg (G) + 1. 2
An Introduction to Game Domination in Graphs
There are graphs G1 , G2 , and G3 such that γtg (G1 ) = γtg (G1 ), γtg (G2 ) = + 1, and γtg (G3 ) = γtg (G3 ) − 1. For example, by Theorems 10 and 11 the following holds where k ≥ 1 is an arbitrary
integer. If G1 = Cn where n = 6k + 3, then γtg (G1 ) = γtg (G1 ) = 4k + 2. If G2 = Cn where n = 6k + 1, then γtg (G2 ) = γtg (G2 ) + 1 = 4k + 1. If G3 = Pn where n = 6k + 5, then γtg (G3 ) = γtg (G3
) − 1 = 4k + 3. If the Continuation Principle holds for some variant of the domination game, then the number of moves in the D-game and the S-game when played optimally on such a game can differ by
at most 1. Conversely, if for some variant of the domination game the number of moves in the D-game and the S-game when played optimally differ by more than 1, then the Continuation Principle does
not hold for such a game. There are several variants of the domination games for which the Continuation Principle does not hold. One such variant is the independent domination game. Indeed in the
independent domination game, the number of moves in the D-game and the S-game when played optimally can often differ by an arbitrarily large constant. As a simple example, let G be a star K1,k where
k is arbitrary large. In the D-game, the first vertex played by Diminisher is the central vertex (of degree k) and the game immediately ends. However, in the S-game, the first vertex played by
Sweller is a leaf, thereby forcing all k leaves to be played in the independent domination game. Thus in this example, Id (G) = 1 and Is (G) = k. Using the Continuation Principle, in 2013 Kinnersley,
West, and Zamani in [38] showed that the D-game in a partially dominated forest with no isolated vertex can never exceed its S-game. γtg (G2 )
Theorem 17 ([38]) If F is a partially dominated forest with no isolated vertex, then γg (F ) ≤ γg (F ). Using the Total Continuation Principle, in 2017 Henning and Rall [29] showed that the D-game in
a partially total dominated forest with no isolated vertex can never exceed its S-game. Theorem 18 ([29]) If F is a partially totally dominated forest with no isolated vertex, then γtg (F ) ≤ γtg (F
6 Upper Bounds and Conjectured Upper Bounds In this section, we present selected upper bounds and conjectured upper bounds on the game domination number and the game total domination number.
M. A. Henning
6.1 Domination Game Bounds In 2013 Kinnersley, West, and Zamani in [38] were the first to prove a general upper bound on the game domination number of an isolate-free graph in terms of its order. 7
Theorem 19 ([38]) If G is an isolate-free graph G of order n, then γg (G) ≤ 10 n.
The upper bound of Theorem 19 was subsequently improved by Bujtás [12] and Henning and Kinnersley [26] using completely different proof techniques. The ingenious approach adopted by Bujtás [12]
colors the vertices of the graph with three colors that reflect three different types of vertices and associates a weight with each vertex, and analyses the weight decrease resulting from each played
vertex as the game unfolds. The proof method in [26] proves a strong inductive statement in a partially dominated graph with a set of vertices predominated, and from this they deduce the desired
upper bound on the game domination number. As a consequence of these results we have the following improved upper bound on the game domination number of an isolate-free graph in terms of its order.
Theorem 20 ([12, 26]) If G is an isolate-free graph of order n, then γg (G) ≤
2 n 3
γg (G) ≤
2 n. 3
Much of the interest in the domination game was generated by the so-called 35 Conjecture posed in 2013 by Kinnersley, West, and Zamani [38]. There are two 3 5 -Conjectures: one for isolate-free
forests, and one for general isolate-free graphs. We state both conjectures. Conjecture 1 ([38]) If G is an isolate-free forest of order n, then γg (G) ≤ 35 n. Conjecture 2 ([38]) If G is an
isolate-free graph of order n, then γg (G) ≤ 35 n. Conjecture 1 for isolate-free forests is referred to as the 35 -Forest Conjecture, and Conjecture 2 for general isolate-free graphs as the 35 -Graph
Conjecture. It is not known whether the 35 -Forest Conjecture implies the 35 -Graph Conjecture. If the above two 35 -Conjectures are true, then the upper bound is tight. The simplest example is to
take G∼ =kP5 where k ≥ 1 is an arbitrary integer. The graph G has order n = 5k. By Theorem 9, we have γg (P5 ) = γg (P5 ) = 3. The optimal strategy of Staller is whenever Dominator plays on a
component of G, Staller plays on that component if at least one vertex in that component has not yet been dominated and adopts an optimal strategy on the component. If, however, Dominator previous
move played on a component of G results in all vertices of that component dominated, then Staller plays in a component with at least one vertex not yet dominated and adopts an optimal strategy on the
component. In this way, Staller can guarantee that three vertices are played from each component. This shows that
An Introduction to Game Domination in Graphs
γg (G) = 3k = 35 n. In Section 7, we show that there exist forests G of arbitrarily large order n satisfying γg (G) = γg (G) = 35 n. In 2016, Henning and Kinnersley [26] proved the 35 -Graph
Conjecture for the class of graphs of minimum degree at least 2. Theorem 21 ([26]) If G is a graph of order n with δ(G) ≥ 2, then γg (G) ≤
3n 5
γg (G) ≤
3n − 1 . 5
Bujtás [12] established the following improved upper bound on the game domination number for the class of graphs of minimum degree at least 3. Theorem 22 ([12]) If G is a graph of order n with δ(G) ≥
3, then γg (G) ≤
34 n 61
γg (G) ≤
34n − 27 . 61
More generally, Bujtás [12] proved the following remarkable result for graphs with large minimum degree. Theorem 23 ([12]) If G is a graph of order n with minimum degree δ(G) = δ ≥ 4, then 15δ 4 −
28δ 3 − 129δ 2 + 354δ − 216 γg (G) ≤ n. 45δ 4 − 195δ 3 + 174δ 2 + 174δ − 216 As an immediate consequence of Theorem 23, we have the following upper bound on the game domination number in terms of its
order with given minimum degree. Corollary 24 ([12]) If G is a graph of order n with minimum degree δ(G), then the following holds. (a) If δ(G) = 4, then γg (G) ≤ (b) If δ(G) ≥ 5, then γg (G) ≤
37 72 n < 0.5139n. 2102 4377 n < 0.4803n.
The 35 -Graph Conjecture has yet to be settled in general for graphs that contain vertices of degree 1.
6.2 Total Domination Game Bounds We now shift our attention to upper bounds on the game total domination number. If G is a graph of order n that consists of a disjoint union of copies of K2 , then γ
tg (G) = n. Hence it is only of interest to consider upper bounds on the game total domination number of a graph in which every component has order at least 3. The
M. A. Henning
first general upper bound on the game total domination number was given in 2017 by Henning, Klavžar, and Rall [32]. Theorem 25 ([32]) If G is a graph of order n in which every component contains at
least three vertices, then γtg (G) ≤
4 n 5
γtg (G) ≤
4n + 2 . 5
In 2018 Bujtás [14] obtained a new improved upper bound on the game total domination number that improves the 45 -bound established in Theorem 25. Theorem 26 ([14]) If G is a graph of order n in
which every component contains at least three vertices, then γtg (G) ≤
11 n 14
γtg (G) ≤
11n + 6 . 14
Bujtás’s bound in Theorem 26 is the best general upper bound on the game total domination number to date. In 2016, Henning, Klavžar, and Rall [32] posed the game total domination 34 -Conjecture.
Conjecture 3 ([32]) If G is a graph of order n in which every component contains at least three vertices, then γtg (G) ≤ 34 n. As remarked in [32], if the game total domination 34 -Conjecture is
true, then the upper bound is best possible. The simplest example is to take G∼ =kP8 where k ≥ 1 is an arbitrary integer. The graph G has order n = 8k. By Theorem 11, we have γtg (P8 ) = γtg (P8 ) =
6. The optimal strategy of Staller is whenever Dominator starts playing on a component of G, Staller plays on that component and adopts her optimal strategy on the component. Since γ tg (P8 ) = 6,
which is even, Staller can continue this strategy until the completion of the game. This shows that γtg (G) = 6k = 34 n. In 2016 Bujtás, Henning, and Tuza [16] studied upper bounds on the game total
domination number over the class of graphs with minimum degree at least 2. For this purpose, they introduced a transversal game in hypergraphs, and establish a tight upper bound on the game
transversal number of a hypergraph with all edges of size at least 2 in terms of its order and size. As an application of this result, they established the following result which proves the game
total domination 34 8 Conjecture for the class of graphs of minimum degree at least 2, noting that 11 < 34 . Theorem 27 ([16]) If G is a graph of order n with δ(G) ≥ 2, then γtg (G)
12 n and Is (T ) > 12 n are constructed in [22]. Proposition 36 ([22]) There exist trees T of maximum degree 3 and of arbitrarily large order n such that Id (T ) ≥
1 +ε n 2
Is (T ) ≥
1 +ε n 2
for some small ε.
8 Computational Complexity The algorithmic complexity of determining the game domination number of a given graph was studied by Brešar, Dorbec, Klavžar, Košmrlj, and Renault [9]. For this purpose,
they considered the following two game domination problems.
D-GAME DOMINATION PROBLEM Input: Question:
A graph G, and an integer . Is γ g (G) ≤ ?
M. A. Henning
S-GAME DOMINATION PROBLEM Input: Question:
A graph G, and an integer . Is γg (G) ≤ ?
Brešar et al. [9] presented a reduction to the Game Domination Problem from the POS-CNF problem, which is known to be log-complete in PSPACE (see [51] for the complexity result on this problem).
Using this reduction and careful analysis, they show that the complexity of both the D-GAME DOMINATION PROBLEM and S-GAME DOMINATION PROBLEM is in the class of PSPACE-complete problems. Theorem 37
([9]) Both the D-Game Domination Problem and the S-Game Domination Problem are log-complete in PSPACE. Hence the decision version of the game domination problem is computationally harder than any
NP-complete problem, unless NP=PSPACE. Klavžar, Košmrlj, and Schmidt [40] studied the D-GAME DOMINATION PROBLEM and S-GAME DOMINATION PROBLEM when the integer is fixed. In this case, when is not part
of the input, they were able to solve the game domination problems in polynomial time. Theorem 38 ([40]) If G is a graph of order n with maximum degree Δ and is a fixed integer, then the D-Game
Domination Problem and the S-Game Domination Problem can be solved in O( Δ · n ). The algorithmic complexity of determining the game total domination number of a given graph was studied by Brešar and
Henning [3] who considered the following game total domination problems.
D-GAME TOTAL DOMINATION PROBLEM Input: Question:
A graph G, and an integer . Is γ tg (G) ≤ ?
S-GAME TOTAL DOMINATION PROBLEM Input: Question:
A graph G, and an integer . Is γtg (G) ≤ ?
Analogously as in the Game Domination Problem, a reduction to the Game Total Domination Problem from the POS-CNF problem is presented, but using a
An Introduction to Game Domination in Graphs
different gadget graph. Using this reduction, they show that the complexity of both the D-GAME TOTAL DOMINATION PROBLEM and S-GAME TOTAL DOMINATION PROBLEM is in the class of PSPACE-complete
problems. Theorem 39 ([3]) Both the D-Game Total Domination Problem and the S-Game Total Domination Problem are log-complete in PSPACE.
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An Introduction to Game Domination in Graphs
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Domination and Spectral Graph Theory Carlos Hoppen, David P. Jacobs, and Vilmar Trevisan
1 Introduction From the early days of graph theory within Mathematics and Computer Science, matrices have played an important role as data structures to store graphs. Let G = (V, E) be a graph of
order n where V = {v1 , . . . , vn } and E = {e1 , . . . , em }. A graph may be naturally stored by recording adjacencies between vertices or by recording the incidence structure between vertices and
edges. Indeed, the adjacency matrix A = A(G) of G is the n × n symmetric matrix where the entry Aij = 1 if {vi , vj }∈ E, and Aij = 0 otherwise, while the incidence matrix B = B(G) of G is the n × m
matrix such that Bij = 1 if vi ∈ ej , and Bij = 0 otherwise. Much more recently, an entire branch of graph theory was born of the interest in extracting properties of graphs from algebraic
information about matrices associated with them. For square matrices, the spectrum is one such piece of information. Given a square matrix M of order n, a number λ is an eigenvalue of M if Mx = λx
for some nonzero column vector x, which is called an eigenvector for λ. Equation (1) is satisfied if and only if λ is a root of the characteristic polynomial p(x) = det(M − xI ), which has degree n,
so that any n × n real matrix has n complex eigenvalues, although some can be repeated. The multiset of eigenvalues is called the spectrum of the matrix and an eigenvalue’s multiplicity is the number
of times it occurs in C. Hoppen · V. Trevisan () UFRGS, 91509–900, Porto Alegre, Brazil e-mail: [email protected]; [email protected] D. P. Jacobs Clemson University, 29634, Clemson, SC, USA e-mail:
[email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66,
C. Hoppen et al.
the spectrum. For symmetric matrices, it is well-known that the eigenvalues are real numbers and that the eigenvectors associated with them produce an orthogonal basis of Rn , that is, the inner
product of two eigenvectors is zero. Clearly, the adjacency matrix of a graph is such a symmetric matrix. Many other symmetric matrices have been introduced. Given a graph G with vertex set V = {v1 ,
. . . , vn }, let D = D(G) be the diagonal matrix with entry Dii = d(vi ), the degree of vi . Then the Laplacian matrix of G is the matrix L(G) = D − A and the signless Laplacian matrix is the matrix
Q(G) = D + A. We observe that Q(G) = BBT , where B is the incidence matrix defined above, and that L(G) may also be written in the form CCT , where C is the incidence matrix of any orientation of G.
This makes L and Q positive semidefinite, so that their eigenvalues are all non-negative. Other symmetric matrices associated with a graph G are its normalized Laplacian matrix L (G) = D −1/2 LD −1/2
and its distance matrix, D(G), for instance. The latter is the matrix indexed by the vertex set of G such that the entry ij is given by the distance between vi and vj . The branch of graph theory
that studies the properties of graphs through the eigenvalues of the matrices associated with them is known as spectral graph theory. Often these matrices are A(G), L(G), Q(G), or L (G) defined
above, but many other matrices have been considered in various contexts. Even if we restrict ourselves to the adjacency and the Laplacian matrices, eigenvalues and eigenvectors have been particularly
useful for embedding graphs in the plane [4], for graph partitioning and clustering [44], in the study of random walks on graphs [13] and in the geometric description of data sets [15], just to
mention a few examples. It has had a particularly relevant influence in Chemistry, where spectral parameters are widely used as molecular descriptors, to the point that some of the results are
referred to as part of chemical graph theory. See [22, 30] for more information. We should point out that, even though graph matrices are defined with respect to some labelling of the graph G, their
spectrum does not depend on the labelling, so that isomorphic graphs share the same spectrum with respect to any fixed matrix. Given the important role of the isomorphism problem in graph theory, and
the nice fact that eigenvalues and eigenvectors can be computed efficiently, it is quite natural that spectral approaches to the isomorphism problem have been a flourishing research theme in this
area. In an ideal world, we would be able to test graph isomorphism by simply computing the eigenvalues of both graphs. However, it is not true that two non-isomorphic graphs must have distinct
spectra, and a graph G may have a cospectral mate H, namely a graph that is not isomorphic to G, but has the same spectrum as G. Figure 1, extracted from [9], shows two graphs that are cospectral
with respect to both the adjacency and the normalized Laplacian matrices. More generally, the seminal work of Schwenk [49] showed that almost every tree T has a mate T for which A(T) and A(T ) share
the same spectrum, in the sense that among all non-isomorphic trees on at most n vertices, the fraction that has a cospectral mate tends to 1 as n tends to infinity. This property of trees was shown
to hold for other matrices, see, for instance, McKay [45] for the Laplacian matrix. In sharp contrast with this, Haemers [31] conjectures that most graphs do
Domination and Spectral Graph Theory
Fig. 1 Cospectral graphs G and H
not have cospectral mates with respect to the adjacency matrix. This is one of the main conjectures in this area. We refer the interested reader to [7, 53]. Why so many different matrices to study
graphs? A natural question related to the previous paragraph is whether any particular matrix would distinguish more graphs than other matrices. This has been one of the driving forces to proposing
new matrices. For instance, in 2009, Cvetkovi´c and Simi´c, in [16–18], established many properties of the signless Laplacian matrix, and argued that this matrix had less spectral uncertainty than
other matrices, in the sense that more graphs are determined by their signless Laplacian spectrum than by their adjacency and Laplacian spectra. So far no definitive results have been obtained [54].
Moreover, as Butler and Chung point out in [10], each matrix has its advantages and disadvantages. As one might expect, since graph properties are often hard to compute, it would be unexpected that
footprints in the spectrum of a matrix, which is computable in polynomial-time, could fully capture these properties. For example, a graph is bipartite if and only if the eigenvalues of A(G) are
symmetric about the origin. That is, for each eigenvalue λ in the spectrum, − λ is an eigenvalue of the same multiplicity [7, Prop. 3.4.1]. On the other hand the multiplicity of the smallest
eigenvalue of L(G), which is 0, reveals how many connected components are in the graph, and the multiplicity of 0 in Q(G) is the number of bipartite components in G [16]. The normalized Laplacian,
introduced by Butler and Chung, will not be used in this chapter, but is closely connected to random walks in graphs. The focus of our chapter is the matrices A(G), L(G), Q(G), and D(G). Sometimes
the relationship between the eigenvalue and graph parameter can be startling. To illustrate, consider the result by Delsarte and Hoffman [20], obtained in the 1970s, involving the independence number
α(G), the cardinality of a maximum independent set of vertices in G, of a regular graph G. The theorem relates α(G) to the least eigenvalue of its adjacency matrix. Theorem 1 Let n > d ≥ 1 be
integers. Let G = (V, E) be a d-regular graph on n vertices whose adjacency matrix has least eigenvalue λn . If S ⊆ V is an independent set of G, then |S| ≤
−λn · n. d − λn
Moreover, given an independent set S, let yS be the characteristic vector of S, that is, the entry corresponding to vi is equal to 1 if vi ∈ S, and is equal to 0 otherwise. Equality holds in (2) if
and only if yS − |S| n 1 is an eigenvector associated with λn .
C. Hoppen et al.
Since the size i(G) of a minimum independent dominating set in a graph G is the minimum size of a maximal independent set, Theorem 1 immediately implies i(G) ≤
−λn ·n d − λn
for any d-regular graph G. To give an idea of the type of argument used for proving bounds in this area, we sketch the proof for Theorem 1. Proof. Let G be a d-regular graph with vertex set V = {v1 ,
. . . , vn } whose adjacency matrix has least eigenvalue λn . Since G is d-regular, λ1 = d is the largest eigenvalue of G, and it is associated with the eigenvector 1. The maximality of λ1 relies on
the Perron–Frobenius Theorem (see Theorem 8.4.4 in [37]). Let x1 = √1n 1 and let x2 , . . . , xn be eigenvectors associated with the remaining eigenvalues λ2 ≥· · · ≥ λn , respectively, with the
property that {x1 , . . . , xn } is an orthonormal basis of Rn , that is, the basis is orthogonal and ||xi || = 1, for each i. Let S ⊂ V be an independent set and let xS be the vector whose entry
corresponding to vi is equal to √1n if vi ∈ S, and is equal to 0 otherwise. Let a1 , . . . , an ∈ R be such that xS = a1 x1 + · · · + an xn . In particular, we have ||xS || = 2
ai2 =
The inner product xS ·x1 satisfies a1 = xS · x1 = the quadratic form xS · AxS satisfies xS · AxS =
|S| . n
|S| n .
Since S is an independent set
1 2 Ai,j = |E(G[S])| = 0. n n vi ,vj ∈S
Now compute xS · AxS by expressing the rightmost xS in terms of the basis, making use of the linearity of A, and noticing that inner products distribute over vector sums: 0 = xS · AxS = xS ·
ai Axi =
xS · ai Axi .
Next replace each Axi with λi xi , and write the other xS in terms of the basis. By orthogonality, each xS ·xi = ai , and so the above equation becomes 0=
n i=1
λi ai xS · xi =
λi ai2 .
Using the fact that λn is negative, writing d = λ1 , and applying (3) we get
Domination and Spectral Graph Theory
0 ≥ λ1 a12 + λn
n i=2
d|S|2 |S| |S|2 λn . − ai2 = da12 + ||xS ||2 − a12 λn = + n n2 n2 (4)
Inequality (4) leads to |S| ≤
−nλn , d − λn
as required. This bound holds with equality if and only if the inequality in (4) holds with equality, that is, if and only if ai = 0 implies that i = 1 or λi = λn . This means the vector xS − a1 x1 =
xS − |S| n x1 is a linear combination a2 x2 + . . . + an xn , where, for each i ≥ 2, either ai = 0 or λi = λn . This implies that xS − a1 x1 is a linear combination of eigenvectors associated with λn
, and is therefore such an eigenvector. As a consequence, √ √ |S| |S| √ |S| 1= x1 yS − n xS − n x1 = n xS − n n n is an eigenvector associated with λn . This completes the proof.
In this chapter we will highlight some of the interesting and important results in spectral graph theory involving domination parameters that have appeared in the last 25 years. These results usually
involve the well-known domination number γ (G). However, we will also give some results involving the total domination number γ t (G), and the signed domination number γ s (G). Generally, there have
been two kinds of spectral results involving domination in the literature: results that compare a specific eigenvalue to γ (G), and results that compare the number of eigenvalues in an interval with
γ (G). A meta-problem in this direction is the following. Let fM be a spectral parameter associated with a graph matrix M and let G be a class of graphs. The problem is to determine the functions max
{fM (G) : |V (G)| = n, G ∈ G } and min{fM : |V (G)| = n, G ∈ G },
and characterize the n-vertex graphs that attain the extremal values. For instance, if fA (G) = λk (G) is the k-th largest eigenvalue of the adjacency matrix of a graph G and Gγ is the set of all
graphs with domination number γ , solving (5) would lead to upper and lower bounds on the value of λk (G) in terms of its domination number. Bounds of this type may be often turned into upper and/or
lower bounds on γ (G) in terms of λk (G).
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The remainder of the chapter is organized as follows. Results involving domination and the adjacency matrix are in Section 2. This section includes a 1994 upper bound on γ (G) by Rowlinson using the
largest multiplicity in A(G), and a beautiful inequality involving the index and γ (G). In Section 3 we give several bounds for the largest and second smallest Laplacian eigenvalue using domination.
We also exhibit an early eigenvector partition due to Brand and Seifter for constructing disjoint dominating sets, as well as results relating the number of Laplacian eigenvalues in an interval to γ
(G). In Section 4 we discuss the signless Laplacian matrix, and give bounds for the largest and second smallest eigenvalue using domination. Domination results involving the distance matrix are given
in Section 5. We conclude this chapter with a few open problems
2 Adjacency Matrix For a graph G, we say the spectrum of G is the multiset given by the eigenvalues of the adjacency matrix A(G). Since A(G) is a real symmetric matrix, its eigenvalues are real and
we enumerate them as λ 1 ≥ λ2 ≥ · · · ≥ λn . It is well-known that the spectrum of G determines some structure of G as, for example, the number of vertices, the number of edges, the number of
triangles, whether G is bipartite and regular, among other properties. The adjacency matrix is by far the most studied matrix in spectral graph theory. However, the literature relating the spectrum
of the adjacency matrix of a graph and domination seems to be rare. Perhaps the first result about domination and matrices of a graph is the well-known fact that if G has no isolated vertices, then γ
(G) ≤ r, where r is the rank of the adjacency matrix of G. This result is from 1982 and is due to Van Nuffelen [55]. The first paper we found relating graph spectra to domination dates to 1994 by
Rowlinson [48], and involves the notion of star partition of a graph G. A star partition of a graph G whose distinct eigenvalues are λ1 , . . . , λm is a partition V (G) = X1 ∪· · · ∪ Xm with the
following two properties: the cardinality of each Xi is equal to the multiplicity of λi as an eigenvalue; λi is not an eigenvalue of the graph G − Xi obtained from G by deleting all vertices in Xi ,
namely the subgraph of G induced by the complement Xi of G. It is known that every graph has a star partition. Rowlinson showed that if X1 ∪· · · ∪ Xm is a star partition of a graph G with no
isolated vertices, then Xi is a dominating set of G. Moreover, for such a graph G, Rowlinson showed that
Domination and Spectral Graph Theory
γ (G) ≤ n − k, where k is the largest multiplicity of an eigenvalue of A(G). We remark that this improves on Van Nuffelen’s bound, since the RHS of the latter may be viewed as n − k0 , where k0 is
the multiplicity of the eigenvalue 0.
2.1 Domination and Spectral Radius The largest eigenvalue of the adjacency matrix A(G) of G, namely λ1 , is called the index of G, while the spectral radius ρ(G) of G is the maximum of the modulus of
the eigenvalues of A(G). By the Perron–Frobenius theory of matrices it is known that ρ(G) = λ1 . If G has at least two vertices and is connected, ρ(G) is always positive, simple, and its associated
eigenvector may be chosen with positive entries. By its many applications (see, for example, the book by Stevanovi´c [51]), the spectral radius is likely to be the most studied spectral parameter of
graphs. Brualdi and Solheid [8] proposed the following general problem, which is a subproblem of (5) and became one of the classic problems of spectral graph theory: Given a set G of graphs, find min
{ρ(G) : G ∈ G } and max{ρ(G) : G ∈ G }, and characterize the graphs which achieve the minimum or maximum value.
In 2008, Stevanovi´c, Aouchiche, and Hansen [52] studied this problem for the class of graphs having domination number γ . They characterize the graphs with n vertices having domination number γ with
maximum spectral radius. The main result of the paper is the following. In the statements hereafter, given graphs G and H, and a positive integer m, G ∪ H denotes the disjoint union of G and H, mG
denotes the disjoint union of m copies of G, and G is the complement of G. Theorem 2 If G is a graph on n vertices with domination number γ , then ρ(G) ≤ n − γ . Equality holds if and only if G∼ =
Kn−γ +1 ∪ (γ − 1)K1 or, when n − γ is even, n−γ +2 ∼ G = 2 K2 ∪ (γ − 2)K1 . Of course we can restate the result above as an upper bound for domination number in terms of the spectral radius of G. In
order to explain their result for graphs with no isolated vertices, we need the following definition. The surjective split graph SSG(n, k;a1 , . . . , ak ), defined for positive integers n, k, a1 , .
. . , ak , 3 ≤ k ≤ n, satisfying a1 + · · · + ak = n − k, a1 ≥· · · ≥ ak , is a split graph on n vertices formed from a clique Kn−k vertices and an independent set I with k vertices, in such a way
that the ith vertex of I is adjacent to ai vertices of K, and that no two vertices of I have a common neighbor in K. It
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Fig. 2 SSG(8, 4;2, 2, 2, 2)
is easy to see that γ (SSG(n, k;a1 , . . . , ak )) = k. As an illustration see the graph of Figure 2. Theorem 3 If G is a graph on n vertices with no isolated vertices and domination number n2 ≥ γ ≥
3, then ρ(G) ≤ ρ(SSG(n, γ ; n − 2γ + 1, 1, 1, . . . , 1)), with equality if and only if G∼ =SSG(n, γ ;n − 2γ + 1, 1, 1, . . . , 1). Moreover the authors also characterize the graphs with no isolated
vertices and maximum spectral radius having γ ∈{1, 2}. Precisely, they show that if γ = 1, then ρ(G) ≤ ρ(Kn ), if γ = 2 and n is even, then ρ(G) ≤ ρ( n2 K2 ) and if γ = 2 and n is odd, then ρ(G) ≤ ρ
(( n−1 2 )K2 ∪ P3 ). In the paper [61], B-X. Zhu also deals with a Brualdi–Solheid problem, but (n) restricting the set of candidates to bipartite graphs. Let Bγ be the set of bipartite (n) graphs
with n vertices and domination number γ . The author finds the graph of Bγ having maximum spectral radius: Theorem 4 Let G ∈ Bγ(n) . If G has the maximum spectral radius, then (i) G∼ =K1,n−1 for γ =
1, (ii) G ∼ = K n−γ +2 , n−γ +2 ∪ (γ − 2)K1 for γ ≥ 2. 2
2.2 Domination and Energy The energy E (G) of a graph G is defined as the sum of the absolute values of all eigenvalues of the adjacency matrix of the graph. This concept, introduced by Gutman in
1977 [29], has connections with theoretical Chemistry. Indeed, for the vast majority of conjugated hydrocarbons, the energy E (G) of a graph that models such a molecule is precisely the value of the
total π -electron energy, calculated by the simple Hückel tight-binding molecular orbital (HMO). This allows one to apply
Domination and Spectral Graph Theory Fig. 3 Tree T(n, γ )
253 γ−1
n − 2γ + 1
the energy E (G) to chemical and physical properties of organic molecules. Clearly, the graph theoretical definition is not restricted to molecular graphs. In the paper [35], He, Wu, and Yu present
sharp lower bounds for the energy of trees involving the domination number, determining also all extreme trees which attain these lower bounds. To state their result, we need the following
definition. For two given natural numbers n > 2γ > 2, the wounded spider is the tree T(n, γ ) obtained by subdividing exactly γ − 1 edges of the edges of the star K1,n−γ . It is easy to see that T(n,
γ ) has n vertices and domination number γ (Figure 3). (n) Consider the class Tγ of all trees having n vertices and domination γ . The authors show that T(n, γ ) is the unique tree with minimum
energy among all (n) elements of Tγ . By computing E (T (n, γ )), they show that if a tree T has n vertices and domination number γ , then ) ( E (T ) ≥ 2γ − 4 + 2 n − γ + 1 + 2 n − 2γ + 1. In 2011,
Xu and Feng [58] gave a shorter proof of the result about the minimum energy of T(n, γ ) over Tγ(n) . Moreover the paper characterizes the trees in Tγ(n) where n = kγ with maximal energy for k = 2,
3, n4 , n3 , n2 . In 2012, J. Zhu [62] shows that the tree with the second minimal energy is B(n, γ ) given in Figure 4.
2.3 Other Results In the paper [61] that was mentioned above in relation with a Brualdi and Soldheid problem, B-X Zhu characterizes the unique graph whose least eigenvalue achieves
C. Hoppen et al. γ−3
Fig. 4 Tree B(n, γ )
n − 2γ + 1
the minimum among all graphs with n vertices and domination number γ . The precise statement is the following. Theorem 5 Let G be a graph whose least eigenvalue λn is minimum among all graphs with n
vertices and domination number γ . Then (i) G ∼ = K1 ∨ K n−1 , n−1 for γ = 1 and n ≥ 6. 2 2 (ii) G ∼ = K n−γ +2 , n−γ +2 ∪ (γ − 2)K1 for γ ≥ 2. 2
A dominating set S ⊂ V is an efficient dominating set if each vertex of G is dominated by precisely one vertex of S or, equivalently, if the minimum length of a path between any two vertices of S is
at least three. Not every graph has an efficient dominating set, for example, the cycle C4 . A subset S ⊂ V (G) is a (k, τ )-regular set in G if it induces a k-regular subgraph in G and every vertex
outside S has exactly τ neighbors in S. An efficient dominating set can also be defined as follows: a set S of vertices of a graph G is an efficient dominating set if G[S] is a regular graph of
degree 0 (i.e., S is an independent set) and every vertex of G outside S has precisely one neighbor in S. Thus an efficient dominating set can be viewed as a (0, 1)-regular set. The efficient
dominating set problem is the problem of determining whether a given graph has an efficient dominating set and finding such a set if it exists. In the paper [11], Cardoso, Lozin, Luz, and Pacheco,
using spectral results on (k, τ )-regular sets, as well as the theory of star complements, present a simplexlike algorithm for detecting a (0, 1)-regular set in an arbitrary graph. This particular
algorithm can be used to find an efficient dominating set in any given graph or to conclude that such a set does not exist. The algorithm is not polynomial-time in general, however, the authors show
that if − 1 is not an eigenvalue of the adjacency matrix of the graph, it works in polynomial-time.
Domination and Spectral Graph Theory
3 Laplacian Matrix For a graph G of order n, its Laplacian eigenvalues always lie in the interval [0, n]. We number them 0 = μn ≤ μn−1 ≤ · · · ≤ μ1 . The multiplicity of 0 is the number of connected
components in G. There is a beautiful relationship between the Laplacian eigenvalues of G and of its complement G. The eigenvalues of G are the following: 0 ≤ n − μ1 ≤ n − μ2 ≤ · · · ≤ n − μn−1 . In
any the average Laplacian eigenvalue is the average vertex degree. That graph, is, n1 μi = n1 di (vi ). It is not surprising, then, that Laplacian eigenvalues reveal information about other
properties of a graph.
3.1 Largest Eigenvalue The largest Laplacian eigenvalue μ1 is called the Laplacian spectral radius. We will now give several bounds for μ1 using domination parameters. From what we can tell, the
earliest result relating Laplacian eigenvalues to the domination number appears in the 1996 paper [6] by Brand and Seifter where they gave an upper bound for the Laplacian spectral radius. Theorem 6
Let G be a connected graph order n. If γ (G) ≥ 3, then
γ (G) − 2 μ1 < n − . 2
If γ (G) = 1, then μ1 = n. If γ (G) = 2, then μ1 ≤ n, and no better bound exists. Obviously inequality (6) tells us something about μ1 if we know γ (G). But conversely, since Laplacian eigenvalues
are always in [0, n] their result implies that if μ1 is close to n then γ (G) is small. This is interesting because, unlike γ (G), it is easy to compute μ1 . In 2015 in [57], the upper bound for μ1
in (6) was improved by Xing and Zhou who showed μ1 ≤ n − γ (G) + 2,
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when 2 ≤ γ (G) ≤ n − 1, and characterized the structure of extremal graphs for fixed n and γ (G), without assuming connectivity. This is also a strict improvement over (6) when γ (G) ≥ 4. Let us now
give some lower bounds on μ1 based on domination parameters. In [46, Thr. 3] Nikiforov obtained the following lower bound on μ1 , and also characterized when equality occurs. A small domination
number would imply a large μ1 . n Theorem 7 If G is a graph containing an edge, then γ (G) ≤ μ1 .
Proof. Let H = (V, F) be a spanning subgraph consisting of a minimal set of edges γ such that γ (H) = γ (G). It is easy to see that H = ∪i=1 Si , a disjoint union of γ (G) stars, whose centers form a
minimum dominating set of G. The largest star S must n have order at least γ (G) . Since H is a subgraph of G, μ1 (G) ≥ μ1 (H ). Since H is a disjoint union, μ1 (H) will be the maximum eigenvalue of
the components. It is well-known that the largest Laplacian eigenvalue of a star S is precisely its order for ≥ 2. This happens for the largest S since G is not edgeless. Therefore
n . μ1 (H ) = μ1 (S ) = ≥ γ (G) Combining the last two lines completes the proof.
The vast majority of papers which relate domination to spectral properties of graphs contain results on the domination number γ (G). The paper [50] by Shi, Kang, and Wu contains a lower bound of μ1
using the signed domination number γ s . A function f : V →{−1, 1} is called signed dominating if the sum of the values over any closed neighborhood is positive. Recall that the closed neighborhood N
[v] of v is the set of neighbors of v together with v. The signed domination number γ s is the minimum weight over all signed dominating functions [21]. Theorem 8 ([50]) Let G be a connected graph of
order n. Then 4n ≤ μ1 , γs (G) + n with equality holding if and only if G = K3 .
Domination and Spectral Graph Theory
3.2 Second Smallest Eigenvalue The second smallest Laplacian eigenvalue μn−1 plays an important role in the structure of a graph. It is called the algebraic connectivity of the graph. The graph is
connected if and only if μn−1 > 0. Much of our understanding of this eigenvalue is due to Fiedler [24]. It plays an important role in isoperimetric parameters through the so-called Cheeger
inequalities (see [13]). As it turns out, the eigenvectors associated with the algebraic connectivity and with other small eigenvalues of the Laplacian matrix are widely used in graph partitioning
[44]. We will now give some upper bounds on μn−1 using γ and γ t . An upper bound for the algebraic connectivity first appeared in 2005 in the paper by Lu, Liu, and Tian [43], who showed the
following. Theorem 9 If G is a connected graph, then μn−1 ≤
n (n−2γ (G)+1) . n−γ (G)
This is equivalent to μn−1 ≤ n −
n (γ (G) − 1). n − γ (G)
Without assuming connectivity, in 2007 Nikiforov [46] showed that if n ≥ 2, then . μn−1 ≤
ifγ (G) = 1
n − γ (G)
ifγ (G) ≥ 2
and also characterized when equality occurs. As Har [33] showed in 2014 if one assumes no isolates, both results can be improved. Interestingly, Har cites the paper [43] but does not cite [46].
Theorem 10 If G has no isolates then μn−1 ≤ n − 2(γ (G) − 1). In the isolate-free case, clearly this improves upon (9). This theorem is also an improvement over inequality (8) since, when G has no
isolates, one has γ (G) ≤ n2 and therefore n−γn(G) ≤ 2. Upper bounds for the second Laplacian eigenvalue are also given in the 2010 paper [3] by Aouchiche, Hansen, and Stevanovi´c using the
domination number. Here the authors assume the graph is connected. The paper [50] also relates the total domination number γ t to μn−1 . A vertex set S is a total dominating set if every vertex in
the graph is adjacent to some member of S, or if for every v ∈ V , N(v) ∩ S = ∅. In a graph G without isolates, the total domination number γ t (G) is the minimum size of a total dominating set. The
authors of [50] give two upper bounds on μn−1 using γ t . One of them is the following. Theorem 11 Let G be connected graph having n ≥ 3 vertices, G = Kn . Then μn−1 ≤ n − γt (G),
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with equality holding if and only if G consists of a forest of K2 ’s or K2 ’s and isolates. This bound is clearly better than (9) above since γ (G) ≤ γ t (G).
3.3 Disjoint Dominating Sets A classic result of Ore [47, Thr. 13.1.5] states that in any graph without isolated vertices, if D is a minimal dominating set, then there exists another minimal
dominating set disjoint from it. Interestingly, Brand and Seifter’s paper has a simple construction [6, Prop. 3.5] for also obtaining disjoint dominating sets. Their method uses a clever partition of
an eigenvector of L(G). However, it is not the usual partition formed by taking negative and non-negative entries. The method requires choosing an eigenvalue larger than Δ(G), the maximum degree. One
might ask if this is always possible. The following result by Grone and Merris [27, Cor. 2] guarantees it. Theorem 12 If G has at least one edge, then μ1 ≥ Δ(G) + 1. 0 , D 0 , and D 0 denote the
vertices Let x be an eigenvector for G. Then let D− + 0 whose entries in x are negative, positive, and zero, respectively. For i ≥ 0 recursively define i+1 D+ (i+1)
D0 i+1 D− D0i+1
= = = =
i i D+ ∪ {v ∈ D0i | {v, w} ∈ E(G) for some w ∈ D− } i+1 i D0 − D+ (i+1)
i+1 i D− ∪ {v ∈ D0 | {v, w} ∈ E(G) for some w ∈ D+ } i+1 i D0 − D+ .
m and D = D m . The construction stops when D0m = ∅. Now define D+ = D+ − − By construction, D− dominates the new vertices in D+ , and D+ dominates the new vertices in D− . Figure 5 shows the
direction of the way vertices move from D0 in the partition constructed above.
D+ Positive and 0
adjacent to D−
Fig. 5 Constructing disjoint domination sets
D− adjacent to D+
Negative and 0
Domination and Spectral Graph Theory
We are now ready to state the result in [6]. Theorem 13 Let G = (V, E) be a connected graph, and μ be an eigenvalue of L(G), where μ > Δ(G). Let x be an eigenvector for μ. Then D+ and D− are disjoint
dominating sets. Proof. Clearly D+ ∪ D− forms a partition of V . Since x is an eigenvector we have L(G)x = μx, where L(G) = D − A(G). Considering the row corresponding to v, one has d(v)xv − xw = μxv
. (10) {v,w}∈E
0 . Then the right side of (10) is positive. Since μ > Δ(G) ≥ d(v), Suppose v ∈ D+ 0. there must be a negative term xw in the summation, and v is dominated by w ∈ D− 0 0 Hence D+ is dominated by D− .
0 . A similar argument shows that the On the other hand, suppose v ∈ D− summation must contain a positive element. Therefore v is dominated by some 0 . This shows that D 0 is dominated by D 0 . If D
0 = ∅, then both D 0 w ∈ D+ − + − 0 0 and D+ are dominating sets. Now suppose there is a v ∈ D00 . By connectivity, at some point it will enter i or D i . Assume it enters D i . This occurs because
it is adjacent to some either D+ − + i−1 i−1 w ∈ D− . Thus v is dominated by D− . The other case is similar.
3.4 Laplacian Distribution Here we are interested in results involving the number of Laplacian eigenvalues in an interval. If G is a graph and I is an interval, we let mG (I) denote the number of
Laplacian eigenvalues of G in I, counting multiplicities. In [28] the authors showed that in connected graphs G, mG [0, 1) ≤ α (G), where
α (G) is the matching number of G. In other words, the number of Laplacian eigenvalues less than 1 is at most the matching number of G. In graphs without isolates it is known that α (G) + β (G) = n,
where β (G) denotes the graph’s edge cover number. This implies that mG [1, n] ≥ β (G). The following theorem in 2016 by Hedetniemi, Jacobs, and Trevisan [36] is an improvement on the result in [28]
since γ (G) ≤ α (G) for any graph. Also the theorem holds for any graph regardless of its connectivity. Theorem 14 For any graph G, mG [0, 1) ≤ γ (G). Corollary 1 For any graph G, mG [1, n] ≥ n − γ
(G). Proof. Since there are n eigenvalues in [0, n] one has mG [1, n] = n − mG [0, 1).
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Fig. 6 Tree with 24 = mT [0, 1) < γ (T) = 25
For some simple classes of graphs one can have equality in Theorem 14. Theorem 15 If Pn is the path on n vertices then mPn [0, 1) = γ (Pn ) = n3 . Proof. In ascending order, the eigenvalues Pn are 2
− 2 cos ( iπ n ), for i = 0, . . . , 1 n − 1 (see [7, p. 9]). This implies that 0 ≤ μ < 1 if and only if 2 < cos ( iπ n ) ≤ 1. Therefore, i < n3 . The largest such i is n3 − 1, so there are exactly
n3 such numbers, mPn [0, 1) = n3 . Since γ (Pn ) = n3 , the theorem follows. Using similar arguments one can show: Theorem 16 If Cn is the cycle on n vertices, then mCn [0, 1) = 2 n6 − 1. Moreover,
mCn [0, 1) =
if n ≡ 1, 2, 3 mod 6 γ (Cn ) γ (Cn ) − 1 if n ≡ 0, 4, 5 mod 6.
There exist trees for which the inequality is strict, and Figure 6 depicts such a tree. To see this, we apply the algorithm in [5] which counts the number of Laplacian eigenvalues in any interval for
trees and obtain mT [0, 1) = 24. On the other hand, a minimum dominating set for the tree can be obtained by taking the 24 support vertices together with the root. If we think of Theorem 14 as a
lower bound of γ (G), the following result by Cardoso, Jacobs, and Trevisan [12] gives an upper bound. Theorem 17 If G has minimum degree 1, then γ (G) ≤ mG [2, n]. It should be noted that the result
in Theorem 17 was obtained by Zhou, Zhou, and Du for trees in [60, Cor. 3.2]. As in the case of Theorem 14, the ratio can become arbitrarily large. However, for certain classes, the approximation
ratio is small. The following results can be found in [12]. Theorem 18 If T is a tree, 1 ≤
mT [2,n] γ (T )
≤ 2.
Domination and Spectral Graph Theory
A connected graph of order n having n − 1 + c edges is called c-cyclic. A generalization of Theorem 18 is the following. Theorem 19 If G is a c-cyclic graph where c ≥ 1, then 1 ≤
mG [2,n] γ (G)
≤ c + 1.
3.5 A Spectral Nordhaus–Gaddum Result A Nordhaus–Gaddum inequality is a bound on the sum or product of a graph parameter for G and its complement G. A result of Jaeger and Payan [41] states that γ
(G) + γ (G) ≤ n + 1.
In [14] Cockayne and Hedetniemi proved the following. Theorem 20 For any graph G, γ (G) + γ (G) ≤ n + 1 with equality if and only if G = Kn or G = Kn . Using the above results we can obtain a
spectral Nordhaus–Gaddum inequality. Theorem 21 For any graph G, mG [0, 1) + mG [0, 1) ≤ n + 1 with equality if and only if G = Kn or G = Kn . Proof. From Theorem 14 and (11) we get mG [0, 1) + mG
[0, 1) ≤ γ (G) + γ (G) ≤ n + 1
for any G, establishing the inequality. Now assume G = Kn or G = Kn . Since mKn [0, 1) = 1 and mKn [0, 1) = n, we have equality. Conversely assume that mG [0, 1) + mG [0, 1) = n + 1. Then (12)
implies that γ (G) + γ (G) = n + 1. Applying Theorem 20 it follows that G = Kn or G = Kn . This completes the proof. To conclude the section, we mention that, more generally, computing the number of
eigenvalues of a matrix M associated with a graph G that lie in a given real interval I is a problem that has been intensively studied in the last few years. An algorithm is said to locate
eigenvalues for a graph class C if, for any graph G ∈ C and any real interval I, it finds the number of eigenvalues of G in the interval I. In recent years, efficient algorithms have been developed
for the location of eigenvalues of the adjacency matrix in trees [38], threshold graphs [39] (also called nested split graphs), chain graphs [1] and cographs [40], for instance. Several of these
algorithms have been adapted to the Laplacian matrix. There are also algorithms for general graphs that are very efficient when the graph admits a decomposition with “low complexity” (with respect to
measures such as the clique-width, see for instance [25]).
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4 Signless Laplacian Matrix For a given graph G = (V, E) of order n = |V | and size m = |E|, the signless Laplacian spectrum of G is the multiset given by the eigenvalues of the matrix Q(G) = D(G) +
A(G). As Q is real symmetric and positive semidefinite, its eigenvalues are real and non-negative, and we order them as q1 ≥ q2 ≥ · · · ≥ qn . In 2009, Cvetkovi´c and Simi´c, in the beautiful series
of papers [16–18], introduced many properties of the signless Laplacian matrix for graphs and, in particular, it is known that the multiplicity of 0 as a Q-eigenvalue is the number bipartite
components of G.
4.1 Bounds for the Index The largest eigenvalue of the signless Laplacian matrix of G is called the signless Laplacian index of G. The first record we found in the literature relating the spectrum of
the signless Laplacian matrix of a graph G and its domination number is from 2010 and due to Hansen and Lucas [32]. They considered relations between signless Laplacian eigenvalues and several graph
parameters. In our context, the relevant result is the following. Theorem 22 Let G be a connected graph on n ≥ 4 vertices with signless Laplacian index q1 and domination number γ . Then 1. q1 + γ ≤
2n − 1, 2. n ≤ q1 · γ . Equality is attained in (1) if and only G is the complete graph Kn . The case of general graphs has been addressed by Xing and Zhou [57], who were able to characterize all
graphs for which the bound is tight. Theorem 23 Let G be a graph on n ≥ 4 vertices with signless Laplacian index q1 and domination number γ . Then q1 (G) ≤ 2(n − γ ), with equality if and only if G∼
=Kn−γ +1 ∪ (γ − 1)K1 or when γ ≥ 2 and n − γ is n−γ +2 ∼ K2 ∪ (γ − 2)K1 . even, G = 2
Domination and Spectral Graph Theory
4.2 Bounds for the Smallest Signless Laplacian Eigenvalue He and Zhou in 2014 [34] presented a sharp upper bound for the smallest signless Laplacian eigenvalue of a graph involving its domination
number. They also determined extremal graphs which attain this bound. In order to state their result we need the following definition. Let H be a graph on n vertices v1 , v2 , . . . , vn . The corona
G ◦ K1 of a graph G is the graph obtained from G by attaching a leaf to every vertex of G. Theorem 24 Let G be a connected graph with even order n = 2k ≥ 6, least signless Laplacian eigenvalue qn and
domination number γ ≥ 3. Then qn ≤ 2k − 2γ +
(k − 2)2 + 4 , 2
with equality if and only if G # Kk ◦ K1 . For n odd, we consider the family F of graphs given in Figure 7. The result of [34] is the following. Theorem 25 Let G be a connected graph with odd order n
= 2k + 1 ≥ 7, smallest signless Laplacian eigenvalue qn , minimum degree δ and domination number γ ≥ 3. If G satisfies one of the following conditions: 1. 2. 3. 4.
δ is even and G ∈ F (see Figure 7), δ = 1, n ≥ 13, δ = 3, n ≥ 17, δ = 5, n ≥ 23, then qn ≤ 2k − 2γ +
( (k − 1)2 + 4 . 2
We notice that this result does not tell anything about graphs having odd minimum degree δ > 5. This case is not discussed in [34]. It is well-known that the smallest signless Laplacian eigenvalue qn
(G) = 0 if and only if G has a bipartite component. Hence it is natural to study lower bounds for
Fig. 7 Family F of graphs
C. Hoppen et al.
k n−1−g−k
g−1 2 Fig. 8 Graph Unk (g)
the qn (G) when a connected graph G is not bipartite. In 2014, Fan and Tan [23] presented a lower bound for the least eigenvalue of the signless Laplacian of G in terms of the domination number. For
a more precise statement of his result, we use the following notation. Denote by Unk (g) the unicyclic graph of order n, which is obtained from an odd cycle Cg (g < n) and a star S1,k by identifying
the end vertices of a path P to one vertex of the cycle and the center of the star , where = n + 1 − g − k (see Figure 8). It is easy to see that if k ≥ 2, then γ (Unk (g)) ≤ γ (Unk−1 (g)) ≤ · · · ≤
γ (Un1 (g)) := γn,g . For fixed n and odd g ∈ [3, n − 1], for each γ ∈ [ g3 , γn,g ], there exists one or more graphs Unk (g) with domination number γ . The unique one with minimum k γ among those
graphs is denoted by Wn (g). Theorem 26 Let G be a connected non-bipartite graph of order n with domination number γ ≤ n+1 3 . Then γ
qn (G) ≤ qn (Wn (3)), γ
with equality if and only if G = Wn (3). In an independent work in 2014 [59], Yu, Guo, Zhang, and Wu studied the same problem. With the above terminology, they determined exactly the structure γ of
Wn (3). The precise statement is as follows. Theorem 27 Among all the non-bipartite graphs with both order n ≥ 4 and domination number γ ≤ n+1 3 , we have (i) If n ∈{3γ − 1, 3γ , 3γ + 1}, then Wn1
(3) is the unique graph with minimal qn up to isomorphism;
Domination and Spectral Graph Theory n−γ
(ii) If n ≥ 3γ + 2, then Wn isomorphism.
(3) is the unique graph with minimal qn up to
4.3 k-Domination and Bounds for Q-Eigenvalues In the work by Liu and Lu [42], there are bounds for q2 (G) and qn (G) based on the k-domination number. For an integer k ≥ 1, a k-dominating set in G is
a subset X of V (G) such that each element of V (G) X is adjacent to at least k vertices of X. (This is sometimes called a k-fold-dominating set to distinguish the situation where every vertex in the
dominating set dominates all vertices at distance up to k.) The least cardinality of a k-dominating set is the k-domination number of G, denoted by γ k (G). The relevant results are as follows. For a
graph G, a partition V (G) = V1 ∪ V2 is called (r, s)-local-regular if all vertices in V1 have r neighbors in V2 and all vertices in V2 have s neighbors in V1 . Theorem 28 Let G be a graph of order n
≥ 2 with maximum degree Δ, minimum degree δ, and average degree d. Let X be a minimum k-dominating set such that |E(G[X])| is as large as possible. If δ ≥ k and |N(x) ∩ X|≥ k − 1 for each vertex x ∈
X, then q2 (G) ≤ 2δ −
n(n − (1 + 1/k)|X| + 1) , n − |X|
with equality only if G is (k, 1)-local-regular graph. Moreover, if k = 1, then equality holds if and only if G∼ =K2,2 . Theorem 29 Let G be a graph of order n ≥ 2 with maximum degree Δ. Then qn (G)
≤ 2 −
nk , γk (G)
where the equality holds only if G is a (k(n − γ k (G))/γ k (G), k)-local-regular graph. Moreover, if G is connected and k = 1, then equality holds if and only if G∼ =Kn .
5 Distance Matrices Let G be a connected graph with vertex set V (G) and edge set E(G). For u, v ∈ V (G), the distance between u and v in G is the length of a shortest path connecting them, denoted
by d(u, v). The distance matrix of G is defined as
C. Hoppen et al.
D(G) = (d(u, v))u,v∈V (G) . The eigenvalues of D(G) are called the distance eigenvalues of G. As D(G) is real and symmetric, the distance eigenvalues of G are real and ordered as ∂1 ≥ ∂2 ≥ · · · ≥ ∂n
. The distance spectral radius of G, ∂ 1 , is the largest distance eigenvalue of G. If |V (G)|≥ 2 and G is connected, then the matrix D(G) is irreducible and the Perron– Frobenius theorem implies
that ∂ 1 is positive, simple and there is a unique positive unit eigenvector x(G) corresponding to ∂ 1 , which is called the distance Perron vector of G. A remarkable property of the distance matrix
given by Graham and Pollack [26] is a formula of the determinant of the distance matrix of a tree depending only on the order n. The determinant is given by det(D) = (−1)n−1 (n − 1)2n−2 , implying
that any nontrivial tree has a single, positive distance eigenvalue and n − 1 negative eigenvalues. The work by Wang and Zhou [56] relates the spectral radius of a tree with its domination number. As
we shall describe, the authors determine the unique tree of given domination number with minimum distance spectral radius and the unique tree of given domination number with maximum distance spectral
radius. Let A(n, m) be the tree obtained from the star Sn−m+1 by attaching a new leaf to each of m − 1 chosen leaves of Sn−m+1 , where 1 ≤ m ≤ n2 . It is easy to see that γ (A(n, m)) = m. Theorem 30
Let T be a tree on n vertices with domination number γ , where 1 ≤ γ ≤ n2 . Then ∂1 (T ) ≥ ∂1 (A(n, γ )), with equality if and only if T ∼ =A(n, γ ). Let D(n, a, b) be the tree obtained from the path
Pn−a−b by attaching a and b leaves to the two end vertices, respectively, where a ≥ b ≥ 1 and a + b ≤ n − 1. Theorem 31 Let T be a tree on n vertices with domination number γ , where 1 ≤ γ < n3 .
Then ∂1 (T ) ≤ ∂1
n − 3γ + 2 n − 3γ + 2 D n, , , 2 2
n−3γ +2 n−3γ +2 with equality if and only if T ∼ = D n, 2 , 2 .
Domination and Spectral Graph Theory
The transmission Du of a vertex u is the sum of the distances from u to all other vertices of in G, that is, Du =
d(u, v).
If DL is the diagonal matrix of the vertex transmissions, whose i-th entry is Dui , the distance Laplacian matrix of a graph G is the matrix D L = D L − D, whose eigenvalues are going to be denoted
by ∂1L ≥ ∂2L ≥ · · · ≥ ∂nL . Among the properties given in [2] of the distance Laplacian matrix D L is that the spectrum of an n-vertex connected graph G with diameter at most 2 is given by ∂1L (G) =
2n − μn−1 ≥ ∂2L (G) = 2n − μn−2 ≥ · · · ≥ ∂n−1 (G) = 2n − μ1 > ∂nL (G) = 0, where the μi are the Laplacian eigenvalues. In [19], the authors relate the distance Laplacian spectral radius ∂1L of a
graph G and its domination number. The main results are summarized in the following theorem. Theorem 32 Let G be a connected graph of order n ≥ 2 with domination number γ . Then the following hold:
(i) ∂1L (G) ≥ n + γ − 1 with equality if and only if G∼ =Kn . (ii) If the diameter of G is d, then ∂1L (G) ≥ n + γ + d − 2 with equality holding if and only if G∼ =Kn or, in the case n = 2p, G ∼ =
K2, 2, . . . , 2 . / 01 2 p
L (G) = n, then γ ≤ 2. (iii) If ∂n−1
6 Final Remarks and Open Problems In writing this chapter, our aim was to collect some of the main results in spectral graph theory that involve domination parameters. As we have seen, many such
results provide bounds on domination parameters in terms of eigenvalues or eigenvalue-based parameters. It also became clear that the connection between
C. Hoppen et al.
graph spectra and domination parameters is still not well understood. In this section, we propose directions for further investigation. In the last decade, there have been considerable advances in
generalized Brualdi– Solheid problems involving domination. Recall that these are problems where, for a set G of graphs and a spectral parameter fM associated with a matrix M, one is asked to find
min{fM (G) : G ∈ G } and max{fM (G) : G ∈ G }, and to characterize the graphs which achieve the minimum or maximum value. Two prime examples are Theorems 2 and 3 in Section 2, which completely
determine, respectively, the n-vertex graphs with domination number γ and the n-vertex graphs with domination number γ and no isolated vertices that maximize the spectral radius of the adjacency
matrix. The maximization part has also been solved for the signless Laplacian matrix, as described in Section 4. However, the minimization part of the problem is often not studied, and we leave it
here as an open question. (n)
Problem 1 Let γ be a positive integer and Gγ = {G : |V (G)| = n, γ (G) = γ }. (n) Find the graphs G ∈ Gγ with minimum spectral radius. The Laplacian matrix is an example where both the maximization
and minimization part of the Brualdi–Solheid problem have been solved, but, for many other matrices, both parts are still open. This happens for the distance matrices of Section 5. More generally, it
is natural to consider the Brualdi–Solheid problem for other matrices and domination parameters. Problem 2 Let γ be a positive integer, let γ ∗ (G) be a domination parameter associated with a graph G
and let M be a symmetric matrix associated with a graph ∗(n) ∗(n) G. Define Gγ = {G : |V (G)| = n, γ ∗ (G) = γ }. Find the graphs G ∈ Gγ with maximum and minimum spectral radius (with respect to the
matrix M). Related to this problem, the result of Xing and Zhou in Section 3.1 describes the structure of extremal graphs with respect to the index of the Laplacian matrix. It turns out that these
graphs contain isolated vertices for all γ ≥ 3. It would be interesting to consider the case in which there are no isolates or where the graph is assumed to be connected. (n)
Problem 3 Let γ be a positive integer and let Cγ be the set of n-vertex connected (n) graphs. Find the graphs G ∈ Cγ with maximum and minimum index with respect to the Laplacian matrix. This problem
would also be interesting with respect to the signless Laplacian matrix, as the upper bound in Theorem 22 is tight if and only if γ = 1. As we mentioned in Section 2.3, there has been interest in
characterizing graphs with a given domination number such that the least eigenvalue is maximum or minimum. Another result in this direction, now in terms of the signless Laplacian
Domination and Spectral Graph Theory
matrix, is Theorem 27, which solves the problem of finding the smallest least Qeigenvalue for graphs of order n ≥ 4 and domination number γ ≤ n+1 3 . Problem 4 Determine which graph(s) among all
non-bipartite graphs on n vertices n and domination number n+1 3 < γ ≤ 2 has minimal least Q-eigenvalue. In the same vein, there has been substantial interest in ordering the elements of a graph
class according to the value of some parameter. So, in addition to finding the graph(s) that achieve the maximum or minimum value, one would be interested in graphs with the second largest value and
so on. In more restricted settings, it may even be possible to completely order the elements in terms of this parameter. This may be quite hard if we consider all graphs with domination number γ . A
particular question in this direction is stated in terms of the energy of trees with fixed domination number. (n)
Problem 5 Let γ be a positive integer and Tγ = {T : T tree, |V (T )| = n, γ (G) = (n) γ }. Order the elements of Tγ according to their energy. As mentioned in Section 2.2, the trees with least energy
and with second smallest (n) energy in Tγ are already known, and the tree with maximum energy is known in some cases. The distribution of the eigenvalues of a graph, or of the elements of a graph
class, is a topic of great interest in spectral graph theory. In Section 3.4, we have described results relating the domination number with the number of eigenvalues in a given real interval. For
instance, Theorem 14 determines that mG [0, 1) ≤ γ (G), where mG [0, 1) is the number of Laplacian eigenvalues of G in [0, 1). Even though we may have equality for some simple graph classes, the
ratio between mG [0, 1) and γ (G) can be arbitrarily large [12]. An open question is when Theorem 14 achieves equality. Problem 6 Characterize graphs G for which mG [0, 1) = γ (G). Another natural
question is whether the ratio mγG(G) [0,1) is bounded for some particular graph class. The following is a particular problem in this direction. Problem 7 Is the ratio
γ (T ) mT [0,1)
bounded for trees T?
To conclude this section, we also include a problem that is purely spectral, but has algorithmic consequences for efficient domination in light of the result of Cardoso, Lozin, Luz, and Pacheco [11]
described in Section 2.3. Problem 8 Describe the family of graphs G such that λ = −1 is not an eigenvalue of A(G).
C. Hoppen et al.
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Varieties of Roman Domination M. Chellali, N. Jafari Rad, S. M. Sheikholeslami, and L. Volkmann
1 Introduction For a graph G = (V, E), a function f : V →{0, 1, 2} is a Roman dominating function, or just an RDF, if every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which
f (v) = 2. The weight of a function f is ω(f ) = v ∈ V f (v). The Roman domination number γ R (G) of a graph G is the minimum weight of an RDF of G. Roman domination was introduced in 2004 by
Cockayne, Dreyer, and Hedetniemi [35] and is now well studied with over 200 papers published on it and its variations. For more on Roman domination, we refer the reader to the chapter written by the
authors on Roman domination in [30]. In it, we covered the core results on Roman domination. In this chapter, we continue that survey to include variations of Roman domination. As of the time of this
writing, there are at least twenty known Roman domination-related parameters that we review nine of them, each in a separate section. The remainder is surveyed in [31].
M. Chellali () LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria N. J. Rad Department of Mathematics, Shahed University, Tehran, Iran S. M. Sheikholeslami
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran e-mail: [email protected] L. Volkmann Lehrstuhl II für Mathematik, RWTH Aachen, 52056, Aachen, Germany e-mail: [email
protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://
M. Chellali et al.
For a positive integer k and a function f : V →{0, 1, . . . , k}, the weight of f is w(f ) = v ∈ V f (v), and for S ⊆ V we define f (S) = v ∈ S f (v). So w(f ) = f (V ). For every i ∈ {0, 1, . . . ,
k}, let Vi be the set of vertices assigned the value i under a function f. Note that there is a 1-to-1 correspondence between the functions f : V →{0, 1, . . . , k} and the ordered partitions (V0 ,
V1 , . . . , Vk ) of V , so we will write f = (V0 , V1 , . . . , Vk ). For a graph G, we denote by γ (G) the domination number, γ t (G) the total domination number, and i(G) the independent
domination number.
2 Weak Roman Domination In 2003, Henning and Hedetniemi [47] considered a less restrictive version of Roman domination, which they called weak Roman domination,but still guaranteeing the defense of
the Roman Empire from a single attack. Let f = (V0 , V1 , V2 ) be a function on a graph G = (V, E). A vertex v with f (v) = 0 is said to be undefended with respect to f if it is not adjacent to a
vertex u with f (u) > 0. A function f is called a weak Roman dominating function (WRDF) if each vertex v with f (v) = 0 is adjacent to a vertex u with f (u) > 0, such that the function f = (V0 , V1 ,
V2 )
defined by f (v) = 1, f (u) = f (u) − 1, and f (w) = f (w) for all w ∈ V {v, u}, has no undefended vertex. The weak Roman domination number γ r (G) is the minimum weight of a WRDF in G. It should be
noted that few papers have been published on weak Roman domination. Few exact values on the weak Roman domination number have been established. For cycles and paths, Henning and Hedetniemi [47]
obtained the following. Proposition 2.1 ([47]) For every n ≥ 4, γr (Cn ) = γr (Pn ) = 3n 7 . Roushini Leely Pushpam and Malini Mai [76] extended the exact value on paths to 2-by-n grid graphs G{2,n}
(Figure 1).
Fig. 1 The construction for G{2,n} , where n = 5k + i, 0 ≤ i ≤ 4. Filled-in circles denote vertices in V1
Varieties of Roman Domination
Proposition 2.2 ([76]) For any 2-by-n grid graph G{2,n} , ⎧ ⎨ 4n if n ≡ 0 (mod 5), 5 γr (G{2,n} ) = 4n ⎩ + 1 otherwise. 5
Independently, Valveny and Rodríguez-Velázquez [81] and Zhu and Shao [84] showed that for any connected nontrivial graph G of order n, γr (G) ≤ 2n 3 . Moreover, Zhu and Shao [84] characterized the
connected graphs achieving equality. Theorem 2.3 ([84]) If G is a connected graph of order n, then γr (G) = 2n 3 if and only if every vertex with degree at least 2 is adjacent to exactly two leaf
2.1 Relationships with γ R and γ Since for every WRDF f = (V0 , V1 , V2 ), V V0 is a dominating set in G and every Roman dominating function on G is a WRDF, we have the following inequality chain
which was observed in [47]. Theorem 2.4 ([47]) For every graph G, γ (G) ≤ γr (G) ≤ γR (G) ≤ 2γ (G).
Moreover, they provided a characterization of graphs G for which γ (G) = γ r (G) and the forests G for which γ r (G) = 2γ (G) Theorem 2.5 ([47]) For any graph G, γ (G) = γ r (G) if and only if there
exists a γ (G)-set S such that (i) pn(v, S) induces a clique for every v ∈ S; (ii) for every vertex u ∈ V (G) S that is not a private neighbor of any vertex of S, there exists a vertex v ∈ S such
that v dominates u and pn(v, S) ∪{u} induces a clique. For the purpose of characterizing the class of trees T for which γ (T) = γ r (T), Roushini Leely Pushpam and Malini Mai [76] defined a family F
of trees T satisfying the following four conditions (Figure 2). • No vertex of T is a strong support. • If u ∈ V (T) is a non-support, which is adjacent to a support, then N(u) contains exactly one
vertex which is neither a support nor adjacent to a support, and all other members of N(u) are either supports or adjacent to supports. • For any vertex u of degree at least two, there exists at
least one leaf v such that dT (u, v) ≤ 3. • Two vertices which are neither supports nor adjacent to supports are not adjacent.
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Fig. 2 A tree T ∈ E
Theorem 2.6 ([76]) For any tree T, γ (T) = γ r (T) if and only if T ∈ F. Split graphs G such that γ (G) = γ r (G) were also characterized in [76] as follows. Theorem 2.7 ([76]) For any split graph G,
γ (G) = γ r (G) if and only if G is the corona of K n2 for some even integer n. Clearly, if G is a graph with γ r (G) = γ R (G), then every γ R (G)-function is also a γ r (G)-function. However, not
every γ r (G)-function is a γ R (G)-function even when γ r (G) = γ R (G). For example, the double star S(2, 2) has three γ r (S(2, 2))-functions but only two γ r (S(2, 2))-functions are γ R (S(2, 2))
-functions. Hence, we say that γ r (G) and γ R (G) are strongly equal, denoted by γ r (G) ≡ γ R (G), if every γ r (G)function is a γ R (G)-function. It is worth mentioning that Haynes and Slater in
[44] were the first to introduce strong equality between two parameters. Alvarado, Dantas, and Rautenbach [10] characterized the trees T with γ r (T) ≡ γ R (T). Nevertheless, a characterization of
trees T with γ r (T) = γ R (T) was given in [84]. It was also shown in [10] that the problem of deciding whether γ r (G) = γ R (G) for a given graph G is NP-hard. Chellali et al. [27] gave an upper
bound on the weak Roman domination number of connected claw-free graphs G in terms of their total domination number γ t (G). Theorem 2.8 ([27]) Let G be a nontrivial, connected, claw-free graph.
Then, (i) γr (G) ≤ 32 γt (G); (ii) if further, G is {K1,3 + e}-free, then γ r (G) ≤ γ t (G).
2.2 Nordhaus–Gaddum Type Bounds Nordhaus–Gaddum type results on the weak Roman domination of a graph and its complement were provided by Valveny and Rodríguez-Velázquez in [81]. Theorem 2.9 ([81])
The following statements hold for any graph G of order n. (i) γr (G) + γr (G) ≤ n + 1. 2 (ii) γr (G)γr (G) ≤ (n+1) 4 . Furthermore, if G C5 is a connected graph with δ(G) ≥ 2 and Δ(G) ≤ n − 3, then
the following two statements hold.
Varieties of Roman Domination
(iii) γr (G) + γr (G) ≤ n − 1 if n is odd, and γr (G) + γr (G) ≤ n if n is even. 2 2 (iv) γr (G)γr (G) ≤ (n−1) if n is odd, and γr (G)γr (G) ≤ n4 if n is even. 4 The reader can find in [81] two
examples of graphs showing the sharpness of all inequalities in Theorem 2.9.
2.3 Algorithmic and Complexity Results In [47], it is shown that the decision problem for the weak Roman domination is NP-complete, even when restricted to bipartite or chordal graphs by describing a
polynomial transformation from the well-known NP-complete decision problem corresponding to the problem of computing the domination number γ (G). Furthermore, Liu, Peng, and Tang [66] designed a
linear-time algorithm for solving the weak Roman domination problem on block graphs. In [23], the authors have proven that the weak Roman domination problem can be solved in O∗ (2n ) time needing
exponential space, and have described an O∗ (2.2279n ) algorithm using polynomial space. Moreover, they proved that the problem can be solved in linear time on interval graphs.
3 Independent Roman Domination Independent Roman dominating functions were defined in [35] by Cockayne et al., but Adabi et al. [7] were the first to study these functions. A Roman dominating
function f = (V0 , V1 , V2 ) is an independent Roman dominating function (IRDF) if the set V1 ∪ V2 is independent. The independent Roman domination number iR (G) is the minimum weight of an IRDF on
G. Independent Roman domination has been studied in [7, 24, 38, 53, 57].
3.1 Bounds on iR It was observed in [7] that for every graph G of order n, iR (G) ≤ n, with equality if and only if G # pK2 ∪ Kq , for any positive integers p and q with n = 2p + q. Other bounds on
the independent Roman domination number have been established by Ebrahimi et al. [38], which are summarized by the following result. Proposition 3.1 ([38]) Let G be a graph of order n and girth g(G).
Then, (i) iR (G) ≤ n − Δ(G) + 1; (ii) iR (G) ≤ n − diam(G)−1 ; 3
2g(G)+2 (iii) ≤ iR (G) ≤ n − g(G)−2 . 3 3
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We note that it had been previously known that γ R (G) ≤ n − (G) + 1. For the class of trees T, Ebrahimi et al. [38] gave an upper bound on the independent Roman domination number in terms of the
order of T. Moreover, they provided a characterization of trees attaining this upper bound. Theorem 3.2 ([38]) For any tree T on n ≥ 3 vertices, iR (T) ≤ 4n/5, and equality holds if and only if there
exists a partition V (T) = X1 ∪ . . . ∪ Xk of V (T) such that each Xi induces a path P5 and the subgraph induced by the central vertices of these paths is connected. The first Nordhaus–Gaddum
inequality for the independent Roman domination number was given in [38]. Proposition 3.3 ([38]) For any graph G of order n ≥ 3, 5 ≤ iR (G) + iR (G) ≤ n + 3. Equality holds in the lower bound if and
only if G or G is K3 , or (δ(G), Δ(G)) or (δ(G), (G)) = (1, n − 1), and in the upper bound if and only if G or G is C5 or n2 K2 . Proposition 3.4 ([38]) If G is a connected graph of order n with diam
(G) ≥ 3, then 6 ≤ iR (G) + iR (G) ≤ n − δ(G) + 4.
3.2 Relationships Between iR and γ R The relationship between iR (G) and γ R (G) has been studied in [7, 24, 53, 57], where some interesting results bounding iR (G) in terms of γ R (G) are given. By
definition, it is obvious that for any graph G, γ R (G) ≤ iR (G). Adabi et al. [7] showed that for a graph G, γ R (G) = iR (G) if and only if there exists a γ R (G)-function f = (V0 , V1 , V2 ) such
that V2 is independent. They also showed that the two parameters γ R (G) and iR (G) are equal for any graph G with maximum degree at most three. Jafari Rad and Volkmann [57] showed that iR (G) = γ R
(G) for a large class of graphs G, by proving the following result. Theorem 3.5 ([57]) Let k ≥ 2 be an integer. If a graph G does not contain the star K1,k+1 as an induced subgraph, then iR (G) ≤ (k
− 1)γR (G) − 2(k − 2). Corollary 3.6 ([57]) If G is a claw-free graph, then γ R (G) = iR (G). In [7], Adabi et al. presented for any graph G with (G) ≥ 3 an upper bound for iR (G) in terms of γ R (G)
and (G), by showing that
Varieties of Roman Domination
iR (G) ≤ γR (G) +
(γR (G) − 2) ((G) − 3). 2
However, this bound has been improved by Jafari Rad [53] as follows. Let k ≥ 4, and let H be a bipartite graph with partite sets A and B each of cardinality k − 1 such that iR (H) ≥ k with the
condition that if iR (H) = k, then for every iR (H)-function f, either f (A) = 0 or f (B) = 0. Let Gk be the graph obtained from H by adding two new vertices x and y, and adding edges xy, xu for all
u ∈ A, and yv for all v ∈ B. Theorem 3.7 ([53]) For any connected graph G with 4 ≤ Δ(G) ≤ 6, iR (G) ≤ (G)+1 γR (G), with equality if and only if G is a double star S( Δ, Δ) or G = GΔ . 4 Theorem 3.8
([53]) For any graph G with Δ(G) ≥ 7,
18 iR (G) ≤ ((G) − )γR (G) − 1. 5 The proof of Theorem 3.7 has allowed Jafari Rad to deduce also the following result. Corollary 3.9 ([53]) If G is a graph in which the vertices of degree at least 4
form an independent set, then iR (G) = γ R (G). In [57], Jafari Rad and Volkmann defined Roman domination perfect graphs which is a concept closely related to domination perfect graphs introduced by
Sumner [80] in 1990. A graph G is called Roman domination perfect if γ R (H) = iR (H) for any induced subgraph H of G. They showed that a graph is Roman domination perfect if it does not contain
eight forbidden induced subgraphs (the same as those given in [39]). In particular, chordal graphs that do not contain a double star S(2, 2) as an induced subgraph are Roman domination perfect. We
close this subsection by mentioning that a constructive characterization of trees with strong equality between the Roman domination and independent Roman domination numbers was given by Chellali and
Jafari Rad in [24].
3.3 Relationships Between iR and i It has been noticed in [7] that for any graph G, i(G) ≤ iR (G) ≤ 2i(G). Adabi et al. [7] were interested in the characterization of all graphs G with iR (G) = i(G)
+ k, for 0 ≤ k ≤ i(G), and they obtained the following.
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Theorem 3.10 ([7]) Let G be any graph of order n. Then, (i) iR (G) = i(G) if and only if G = K n ; (ii) iR (G) = i(G) + 1 if and only if G has a vertex of degree n − i(G); (iii) for every integer k ∈
{2, . . . , i(G)}, iR (G) = i(G) + k if and only if the following holds. – For any integer s with 1 ≤ s ≤ k −31, 3 independent set Ut of there is no cardinality t such that 1 ≤ t ≤ s and 3 v∈Ut N [v]
3 = n − i(G) − s + 2t. – There is an independent set 3 3 Wl of cardinality l for some integer l ∈{1, . . . , 3 3 k} such that 3 v∈Wl N [v]3 = n − i(G) − k + 2l. The lower bound i(G) has been improved
by Chellali et al. [28] for all nontrivial connected graphs G. Let G be the family of connected graphs G of order n such that γ (G) = n/(1 + (G)). Recall that sets that are both dominating sets and
packings are called efficient dominating sets and have been defined by Bange et al. [16]. Note that every graph of G admits an efficient dominating set in which every vertex has maximum degree. Let F
be the family of graphs G such that G is the cycle C4 or the corona of any connected H ∈ G. Theorem 3.11 ([28]) Let G be a nontrivial connected graph with maximum degree Δ. Then, iR (G) ≥ i(G) + γ
(G)/ Δ, with equality if and only if G ∈ F. We note that for the class of trees, Chellali and Jafari Rad [25] gave a constructive characterization of trees T with iR (T) = 2i(T), answering an open
question in [38]. It should be noted that to our knowledge, the relationship between iR (G) and the independence number α(G) has not been discussed before and could be an interesting subject for
future work.
3.4 Algorithmic and Complexity Results It was mentioned in [35] that Alice McRae has also shown that the decision problem corresponding to independent Roman domination is NP-complete, even when
restricted to bipartite graphs. However, this result was never published. Liu and Chang [65] have studied the complexity and the algorithmic aspect of the independent Roman domination problem in a
more general context. For real numbers b ≥ a > 0, an independent (a, b)-Roman dominating function is an (a, b)-Roman dominating function f such that the set of vertices assigned a nonzero value is
independent. Clearly, for b = 2 and a = 1, this is an independent Roman dominating function. Liu and Chang showed that for any fixed (a, b), the independent (a, b)-Roman domination problem is
NP-complete for bipartite and chordal graphs. Moreover, using the framework of linear programming and the strong elimination ordering as a tool, they provided a linear-time algorithm for the weighted
independent (a, b)-Roman domination problem with 2a ≥ b ≥ a > 0 on strongly chordal graphs.
Varieties of Roman Domination
4 Roman k-Domination Kämmerling and Volkmann [59] studied a generalization of the Roman dominating functions. A Roman k-dominating function (Rk-DF) on G is a function f : V (G)→{0, 1, 2} such that
every vertex u for which f (u) = 0 is adjacent to at least k vertices v1 , v2 , . . . , vk with f (vi ) = 2 for i = 1, 2, . . . , k. The minimum weight of a Roman k-dominating function on a graph G
is called the Roman k-domination number γ kR (G). An Rk-DF of minimum weight is called a γ kR (G)-function. Clearly, the Roman 1-domination number γ 1R corresponds to the Roman domination number γ R
. Note that if k ≥ (G) + 1, then γkR (G) = |V |. Hence, it is only interesting to consider graphs G when k ≤ (G). Roman k-domination has been further studied in [18, 19, 59, 64, 69] and elsewhere.
Let k be a positive integer. A subset S ⊆ V (G) is a k -dominating set of G if every vertex of V (G) − S is adjacent to at least k vertices of S. The k-domination number γ k (G) is the minimum
cardinality of a k-dominating set of G. Note that the 1-domination number γ 1 (G) is the classical domination number γ (G). Some properties of minimum Roman dominating functions given in [35] are
generalized through the following result. Proposition 4.1 ([59]) Let f = (V0 , V1 , V2 ) be any γ kR (G)-function of a graph G. Then, (i) (ii) (iii) (iv) (v)
The complete bipartite graph Kk,k+1 is not a subgraph of G[V1 ]. If w ∈ V1 , then |NG (w) ∩ V2 | ≤ k − 1. If A = {u1 , . . . , uk }⊆ V0 , then |V1 ∩ NG (u1 ) ∩ . . . ∩ NG (uk )| ≤ 2k. V2 is a minimum
k-dominating set of G[V0 ∪ V2 ]. Let H = G[V0 ∪ V2 ], and let v ∈ V2 . Then, there exists a vertex u1 ∈ NH (v) ∩ V0 such that u1 has exactly k − 1 neighbors in V2 −{v}. In addition, there exists
either a second vertex u2 ∈ NH (v) ∩ V0 such that u2 has exactly k − 1 neighbors in V2 −{v} or v has at most k − 1 neighbors in V2 −{v}. (vi) Let v ∈ V2 such that degG[V2 ] (v) = k − 1 and v has
precisely one neighbor in V0 , say w, with the property that w has exactly k − 1 neighbors in V2 −{v}. If S1 ⊆ V1 is a set such that each vertex of S1 has precisely k − 1 neighbors in V2 −{v}, then
NG (w) ∩ S1 = ∅. (vii) Let S2 ⊆ V2 be the set of vertices of degree at least k in G[V2 ], and let C = {x ∈ V0 : |NG (x) ∩ V2 | ≥ k + 1}. Then % |V2 | + |S2 | |V0 | ≥ max |V2 | + + |C| . 2
4.1 Bounds on γ kR and Relationships with γ k Proposition 4.2 ([59]) For any graph G,
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γk (G) ≤ γkR (G) ≤ 2γk (G). Equality holds in the upper bound if and only if G has a γ kR (G)-function f = (V0 , V1 , V2 ) with V1 = ∅. Proposition 4.3 ([59]) If G is a graph of order n, then the
following conditions are equivalent: (i) γ kR (G) = γ k (G); (ii) γ k (G) = n; (iii) Δ(G) ≤ k − 1. An improvement of the upper bound in Proposition 4.2 was given by Bouchou et al. in [18] for the
class of graphs with at most one cycle. Moreover, the authors characterized extremal graphs attaining the new upper bound. Theorem 4.4 ([18]) Let T be a tree of order n ≥ 3 with (T ) ≥ k ≥ 2. Then, γ
kR (T) ≤ 2γ k (T) − k + 1, with equality if and only if (i) k = 2 and T is the subdivision graph of another tree, or (ii) k = n − 1 and T is a star. Let K1,p + e denote the graph obtained from a star
K1,p by adding an edge between two leaves of K1,p . Let F be the graph obtained from a path P5 whose vertices are labeled in order 1, 2, 3, 4, 5 by adding a new vertex x and edges x2 and x4. Theorem
4.5 ([18]) Let G be a unicyclic graph and (G) ≥ k ≥ 3. Then, γkR (G) ≤ 2γk (G) − k + 1, with equality if and only if either k ∈{3, 4, n − 1} and G = K1,k + e, or k = 3 and G = F. Theorem 4.6 ([59])
Let G be a graph of order n. Then, 2kn ; k + (G) γkR (G) ≥ min{n, γk (G) + k}; if n ≤ 2k, then γ kR (G) = n; if n ≥ 2k + 1, then γ kR (G) ≥ 2k; if n ≥ 2k + 1 and γ k (G) = k, then γ kR (G) = γ k (G)
+ k = 2k.
(i) γkR (G) ≥ (ii) (iii) (iv) (v)
Proposition 4.7 ([59]) If G is a graph of order n with at most one cycle and k ≥ 2 or a cactus graph with k ≥ 3, then γ kR (G) = n.
Varieties of Roman Domination
Jafari Rad [55] established a probabilistic upper bound on the Roman kdomination number of a graph improving slightly the one obtained by Hansberg and Volkmann [43]. Theorem 4.8 ([55]) Let G be a
graph of order n, with minimum degree δ ≥ 1 and δ+1 maximum degree Δ, and let k be a positive integer. If ln(δ+1) ≥ 2k, then γkR (G) ≤
k−1 δi 2n k ln(δ + 1) − ln(2) + δ+1 i!(δ + 1)k−1
k ln(δ + 1) − ln(2) 2n 1+ δ+1
1+ .
A slight improvement of Theorem 4.8 is given in [51] by using the well-known Brooks’ Theorem for vertex coloring. Kämmerling and Volkmann proved a lower bound on γkR (G) + γkR (G), and characterized
the extremal graphs attaining this lower bound. However, a counterexample of the characterization was given by Mojdeh and Moghaddam [69] and the result becomes as follows. Theorem 4.9 ([59, 69]) If G
is a graph of order n, then γkR (G) + γkR (G) ≥ min{2n, 4k + 1} and the equality holds if and only if one of the following holds: (i) n ≤ 2k; (ii) n = 2k + 1, and either γ k (G) = k or γk (G) = k;
(iii) k = 1, n ≥ 4 and G or G has a vertex of degree n − 1 and its complement has a vertex of degree n − 2.
4.2 Relationships Between γ kR and γ R Lower bounds on the Roman k-domination in terms of the Roman domination number have been obtained by Bouchou, Blidia, and Chellali in [19]. Proposition 4.10
([19]) Let k ≥ 2 be an integer and G a graph of order n such that γ kR (G) < n. Then, (i) γ kR (G) ≥ γ R (G) + 2k − 4; (ii) if further G is C4 -free, then γ kR (G) ≥ γ R (G) + 2k − 2. Corollary 4.11
([19]) If G is a C4 -free graph of order n with γ 2R (G) < n, then γ 2R (G) ≥ γ R (G) + 2. The next result improves the lower bound of Proposition 4.10-(i) when k = 2. Proposition 4.12 ([19]) If G is
a graph of order n with γ 2R (G) < n, then γ2R (G) ≥ γR (G) + 1.
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For k = 3, a characterization of cubic graphs G attaining the lower bound of Proposition 4.10-(i) has been given in [19] as follows. Theorem 4.13 ([19]) Let G be a connected cubic graph of order n.
Then γ 3R (G) = γ R (G) + 2 if and only if G = K4 , K3,3 or G is the complement graph of C6 . For k = 2, the following results have also been obtained in [19]. Theorem 4.14 ([19]) Let G be a
connected graph of order n with at most one cycle. Then γ 2R (G) = γ R (G) + 1 if and only if G ∈{P3 , C3 , P4 , C4 , P5 , C5 }. A caterpillar is a tree T with the property that the removal of its
leaves results in a path u1 , u2 , . . . , us which is called the spine of T. A sequence of nonnegative integers (t1 , t2 , . . . , ts ), where ti is the number of leaves adjacent to ui for s ≥ 1 is
associated with T that can be denoted by C(t1 , t2 , . . . , ts ). Let H = {H1 , H2 , H3 , H4 , H5 } as illustrated in Figure 3, and let C be the family of nine caterpillars illustrated in Figure 4.
Theorem 4.15 ([19]) Let T be a tree of order n. Then, γ 2R (T) = γ R (T) + 2 if and only if T ∈ {P6 , P7 , P8 } ∪ H ∪ C. For {K1,3 , K1,3 + e}-free graphs G, the authors of [19] provided a full
characteri zation of graphs G such that γ kR (G) = γ R (G) + t, where t ∈ {2k − 3, 2k − 2, n3 }. Let Rn denote the complete graph of an even order n minus a perfect matching. Clearly, Rn is an (n −
2)-regular graph of an even order n. Theorem 4.16 ([19]) Let G be a connected & {K1,3 , K'1,3 + e}-free graph of order n and k a positive integer with 2 ≤ k ≤ min (G) , n2 . Then, (i) γ kR (G) = γ R
(G) + 2k − 3 if and only if G ∈{P3 , C3 , P4 , C4 , P5 , C5 } and k = 2 or G = Rn and k is even or n = 2k; (ii) γ kR (G) = γ R (G) + 2k − 2 if and only if G ∈{P6 , C6 , P7 , C7 , P8 , C8 } and k = 2,
G = Kn , G = Rn and k is odd with n ≥ 2k + 1, or G = Kp + Rq with p ≥ 1, q is an even integer and n≥ 2k; (iii) γ2R (G) = γR (G) + n3 if and only if G = Pn or Cn for n ≥ 9.
Fig. 3 Five trees Hi with γ 2R (Hi ) = γ R (Hi ) + 2
Varieties of Roman Domination
C(1, 2)
C(2, 0, 1)
C(2, 2)
C(1, 1, 2)
C(1, 1, 1, 1)
C(1, 1, 1)
C(1, 1, 0, 1)
C(1, 0, 1, 0, 1)
Fig. 4 Nine caterpillars C with γ 2R (C) = γ R (C) + 2
4.3 k-Roman Graphs Generalizing the definition of Roman graphs given in [35], Kämmerling and Volkmann defined a graph G to be a k-Roman graph if γ kR (G) = 2γ k (G). k-Roman graphs have been studied
mainly in [18], where the following necessary condition for graphs to be k-Roman is given. Theorem 4.17 ([18]) If G is a k-Roman graph with k ≥ 2, then every vertex of G is adjacent to at most k − 1
leaves. For the case k = , -Roman graphs were characterized as follows. Theorem 4.18 ([18]) A graph G is Δ-Roman if and only if G is a bipartite regular graph. For k ≥ 2, it was shown in [18] that no
tree is k-Roman, and for k ≥ 3, no cactus is k-Roman. For 2-Roman unicyclic graphs, the following is obtained. Theorem 4.19 ([18]) A unicyclic graph G is a 2-Roman graph if and only if G is the
subdivided graph of another unicyclic graph (possibly with a cycle on two vertices).
4.4 Algorithmic and Complexity Results It is shown in [64] that the decision problem corresponding to the problem of computing γ kR (G) is NP-complete even when restricted to bipartite graphs and
chordal graphs. Moreover, for k = 2 the decision problem remains NP-complete for planar graphs. As of this writing, a linear algorithm for computing the Roman kdomination number for any tree has not
yet been designed.
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5 Roman {2}-Domination In 2016, Chellali, Haynes, Hedetniemi, and McRae [29] defined a new variant of Roman dominating functions which they called Roman {2} -dominating functions. A Roman {2}
-dominating function f : V →{0, 1, 2} has the property that for every vertex v ∈ V with f (v) = 0, f (N(v)) ≥ 2, that is, either there is a vertex u ∈ N(v), with f (u) = 2, or at least two vertices
x, y ∈ N(v) with f (x) = f (y) = 1. In terms of the Roman Empire, this defense strategy requires that every location with no legion has a neighboring location with two legions, or at least two
neighboring locations with one legion each. The minimum weight of a Roman {2}-dominating function of G is the Roman {2}-domination number, denoted γ {R2} (G). It should be noted that Roman {2}
-dominating functions are closely related to {2} -dominating functions defined in [37] as functions f : V →{0, 1, 2} having the property that for every vertex u ∈ V , f (N[u]) ≥ 2. Observe that a
Roman {2}-dominating function f relaxes the restriction that for every vertex u ∈ V , f (N[u]) = v ∈ N[u] f (v) ≥ 2 to only requiring that this property holds for every vertex assigned 0 under f.
Also, for a Roman {2}dominating function f, it is possible that f (N[v]) = 1 for some vertex with f (v) = 1. Moreover, it is worth noting that Roman {2}-domination was also studied in 2017 [48] and
2019 [62], where it was called Italian domination. The following property of γ {R2} (G)-functions can be found in [29]. Proposition 5.1 ([29]) For every graph G, there exists a γ {R2} (G)-function f
= (V0 , V1 , V2 ) such that either V2 = ∅ or every vertex of V2 has at least three private neighbors in V0 with respect to the set V2 . In 2018, the independent version of Roman {2}-domination was
initiated by Rahmouni and Chellali [74]. A Roman {2}-dominating function f = (V0 , V1 , V2 ) of G is an independent Roman {2}-dominating function (IR2DF) if the set V1 ∪ V2 is independent. The
independent Roman {2}-domination number i{R2} (G) is the minimum weight of an IR2DF on G. One of the main results of [74] relates iR and i{R2} . Theorem 5.2 ([74]) For every connected graph G of
order n, iR (G)−i{R2} (G) ≤ n4 . Note that the bound of Theorem 5.2 is sharp for cycles C4 and C8 .
5.1 Bounds on γ {R2} and Relationships with γ , γ 2 , γ r , and γ R We begin by giving a lower bound and an upper bound on the Roman {2}domination number established in [29] and [62], respectively.
Theorem 5.3 ([29]) If G is a connected graph of order n and maximum degree Δ, then γ{R2} (G) ≥ 2n/ ( + 2). Theorem 5.4 ([62]) For all connected graphs G with n ≥ 3 vertices,
Varieties of Roman Domination
γ{R2} (G) ≤
3n . 4
Proof. It suffices to show that γ{R2} (T ) ≤ 34 n for an arbitrary spanning tree T of G. We use an induction on the order n of T. Clearly, the result holds for n ∈{3, 4}, establishing the base case.
Next, choose an edge e of T and let T1 and T2 be the components of T − e. If both T1 and T2 have at least three vertices, then the result follows from induction. So we must only consider the case
when there is no such edge e, and thus T has diameter at most four. Clearly, if the diameter of T is two or three, then γ{R2} (T ) ≤ 34 n. Hence, we assume that the diameter of T is four. If T = P5 ,
then γ{R2} (T ) = 3 ≤ 34 n. Thus, let T = P5 . Then, T has a vertex of degree at least three and order n ≥ 6. Let v be a vertex in T of minimum eccentricity. Then, v has degree at least three, and
every neighbor of v has degree at most two (otherwise, the deletion of some edge incident with v provides two components each of order at least three). In this case, let f (v) = 2, f (u) = 1 for each
leaf u of T that is not adjacent to v, and f (w) = 0 for each other vertex. This shows that γ{R2} (T ) ≤ 34 n. A characterization of connected extremal graphs attaining the upper bound in Theorem 5.4
was provided by Haynes, Henning, and Volkmann [46]. Let F be an arbitrary connected graph of order nF , and let G be the graph of order n = 4nF obtained from F by adding to each vertex v of F three
new vertices u, w and x and the edges uv, vw, and wx. It can be seen that in any minimum Roman {2}-dominating function f on such a graph G, f ({u, v, w, x}) ≥ 3. Let G be the family of all such
graphs G. Theorem 5.5 ([46]) Let G be a connected graph of order n ≥ 3. Then, γ{R2} (G) = 3 4 n if and only if G ∈ G. The parameters γ {R2} (G) and γ 2 (G) for arbitrary graphs G are related as
follows. Proposition 5.6 ([29]) For every graph G, γ (G) ≤ γ {R2} (G) ≤ γ 2 (G). It is shown in [62] that for any graph G, γ {R2} (G) = γ 2 (G) if and only if there is a γ {R2} (G)-function f = (V0 ,
V1 , V2 ) such that V2 = ∅. In particular, Klostermeyer and MacGillivray showed that for cactus graphs G with δ(G) = 2, γ {R2} (G) = γ 2 (G). Hajibaba and Jafari Rad [41] were interested in graphs G
such that γ (G) = γ {R2} (G) and they proved the following result. Theorem 5.7 ([41]) For any nontrivial graph G, γ (G) = γ {R2} (G) if and only if γ (G) = γ 2 (G). Moreover, Hajibaba and Jafari Rad
provided a constructive characterization of all connected graphs G with γ (G) = γ {R2} (G) which solves the question of characterizing all connected graphs G with γ (G) = γ 2 (G) posed by Hansberg
and Volkmann in [42]. On the other hand, Caro and Rodity [21] showed that γ2 (G) ≤ 23 n for every graph of order n with δ(G) ≥ 2, while Chen and Zhou [33] showed that γ2 (G) ≤ 12 n for every graph of
order n with δ(G) ≥ 3. As a consequence, Proposition 5.6 leads
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Fig. 5 Graph G with γ (G) < γ r (G) < γ {R2} (G) < γ R (G) < 2γ (G)
x w v
y z
c3 c4
s r
t u
to the following corollary that improves the upper bound of Theorem 5.4 for graphs G with δ(G) > 1. Corollary 5.8 ([62]) Let G be a graph of order n. Then, (i) if δ(G) ≥ 2, then γ{R2} (G) ≤ 23 n;
(ii) if δ(G) ≥ 3, then γ{R2} (G) ≤ 12 n. Extremal connected graphs attaining the upper bound in Corollary 5.8-(i) have been characterized by Haynes, Henning, and Volkmann [46]. For two graphs G and
H, the corona graph G ◦ H is the graph obtained from one copy of G and |V (G)| copies of H and joining the ith vertex of G to every vertex in ith copy of H. Let G≥2 = {G ◦ K2 | G is a connected
graph}. Theorem 5.9 ([46]) Let G be a connected graph of order n with δ(G) ≥ 2. Then, γ{R2} (G) = 23 n if and only if G ∈ G≥2 . For graphs with maximum degree ≤ 2, any γ {R2} (G)-function f = (V0 ,
V1 , V2 ) satisfying Proposition 5.1 must have V2 = ∅, and so V1 is a 2-dominating set of G. Because of this, γ{R2} (Pn ) = (n + 1)/2 and γ{R2} (Cn ) = n/2 . An interesting string of inequalities is
established in [29] relating the parameters γ , γ r , γ {R2} , and γ R . This string extends the inequality chain (1) given in Subsection 2.1 as follows. Theorem 5.10 ([29]) For every graph G, γ (G)
≤ γ r (G) ≤ γ {R2} (G) ≤ γ R (G) ≤ 2γ (G). Furthermore, the authors of [29] provided the graph G illustrated in Figure 5 showing the strictness of all inequalities in Theorem 5.10. Indeed, for such a
graph G, we have γ (G) = 6, γ r (G) = 8, γ {R2} (G) = 9, γ R (G) = 11, and 2γ (G) = 12. In 2019, Martínez and Yero [68] gave a constructive characterization of trees T with γ {R2} (T) = γ R (T). In
[62], Klostermeyer and MacGillivray have shown that γ {R2} (T) ≥ γ (T) + 1 for any nontrivial tree T, and in [48], Henning and Klostermeyer characterized all trees T with γ {R2} (T) = γ (T) + 1. For
positive integers r and s, let Fr,s be the tree obtained from a double star S(r, s) by subdividing every edge exactly once. Let F be the family of all such trees Fr,s ;
Varieties of Roman Domination
that is, F = {Fr,s | r, s ≥ 1}. Also, let T be the family of trees Tk,j of order k ≥ 2, where k ≥ 2j + 1 and j ≥ 0, obtained from a star by subdividing j edges exactly once. Theorem 5.11 ([48]) Let T
be a nontrivial tree. Then, γ {R2} (T) = γ (T) + 1 if and only if T ∈ F ∪ T . Henning and Klostermeyer [48] gave also a constructive characterization of all trees T with γ {R2} (T) = 2γ (T), which
are called Italian trees.
5.2 Nordhaus–Gaddum Type Bounds Nordhaus–Gaddum type results on the Roman {2}-domination of a graph and its complement have been established by Haynes, Henning, and Volkmann [46]. Theorem 5.12 ([46])
If G is a graph of order n ≥ 3, then 5 ≤ γ{R2} (G) + γ{R2} (G) ≤ n + 2. Further, if γ{R2} (G) ≤ γ{R2} (G), then γ{R2} (G) + γ{R2} (G) = 5 if and only if there exists a vertex in G of degree n − 1
with a neighbor of degree 1 in G or with two adjacent neighbors of degree 2 in G. The upper bound in Theorem 5.12 has been slightly improved for graphs with no small components. Theorem 5.13 ([46])
If G is a graph of order n ≥ 16 and having no component with fewer than three vertices, then γ{R2} (G) + γ{R2} (G) ≤ n − 1.
5.3 Algorithmic and Complexity Results In [29], it is shown that the decision problem corresponding to the problem of computing γ {R2} (G) is NP-complete even when restricted to bipartite graphs.
Furthermore, Chen and Lu [34] showed that the problem remains NP-complete even for split graphs, and designed a linear-time algorithm for computing the value i{R2} (T) for any tree T, answering an
open problem posed in [74]. Note that Rahmouni and Chellali showed that the decision problem corresponding to the problem of computing i{R2} (G) is NP-complete for bipartite graphs. Poureidi et al.
[71] showed that the decision problem for computing the independent Roman {2}-domination number is NP-complete even when restricted to chordal graphs. Furthermore, aiming to answer a problem in [26]
on a parameter, namely, independent 2-Rainbow domination number, they proposed a linear algorithm that in particular can compute the independent Roman {2}-domination number of a given tree, providing
an answer to the open problem posed in [74].
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6 Double Roman Domination In 2016, Beeler, Haynes, and Hedetniemi [17] defined a stronger version of Roman domination which they called double Roman domination. This new strategy, where three legions
can be deployed at a given location, offers a high level of defense ensuring that any attack can be defended by at least two legions. A double Roman dominating function (DRDF) on a graph G is a
function f = (V0 , V1 , V2 , V3 ) that satisfies the following conditions: (i) If f (v) = 0, then v must have one neighbor in V3 or at least two neighbors in V2 ; (ii) If f (v) = 1, then v must have
at least one neighbor in V2 ∪ V3 . The double Roman domination number γ dR (G) equals the minimum weight of a double Roman dominating function on G, and a DRDF of G with weight γ dR (G) is called a γ
dR -function of G. Double Roman domination has been studied in [2, 3, 5, 13, 40, 56, 82, 83] and elsewhere. The following property of γ dR -functions was given in [17].
Proposition 6.1 ([17]) For any double Roman dominating function f , there exists a
double Roman dominating function f of no greater weight than f for which no vertex is assigned the value 1. According to Proposition 6.1, double Roman dominating functions can be assumed to be of the
form f : V (G) →{0, 2, 3} such that if f (v) = 0, then either v has a neighbor w with f (w) = 3 or v has two neighbors x and y with f (x) = f (y) = 2. Therefore, in a very good sense, double Roman
domination is equivalent to Roman {2}-domination as follows. If S ⊆ V (G) is a solution to double Roman domination, then by subtracting one from each vertex assigned the value 2 or 3 we get a
solution to Roman {2}-domination. Conversely, if S is a solution to Roman {2}-domination, then by adding one to every vertex assigned the value 1 or 2 we get a solution to double Roman domination.
The exact values on the double Roman domination number for paths and cycles have been established in [2]. Proposition 6.2 ([2]) For n ≥ 1, γdR (Pn ) =
n if n ≡ 0 (mod 3) n + 1 if n ≡ 1 or 2 (mod 3).
Proposition 6.3 ([2]) For n ≥ 3, γdR (Cn ) =
n if n ≡ 0, 2, 3, 4 (mod 6) n + 1 if n ≡ 1, 5 (mod 6).
Varieties of Roman Domination
6.1 Bounds on γ dR and Relationships with γ , γ R , γ {R2} , and γ 2 Various results relating γ dR (G) to γ (G) and γ R (G) are presented by Beeler et al. [17]. We begin by the following result
involving γ dR (G) and γ R (G) for connected graphs G. Proposition 6.4 ([17]) For any nontrivial connected graph G, 1 + γR (G) ≤ γdR (G) ≤ 2γR (G) − 1. Recall that a wounded spider is the graph
obtained by subdividing at most t − 1 of the edges of a star K1,t , for t > 0. The following result due to Zhang et al. [83] provides a characterization of trees T with γ dR (T) = 2γ R (T) − 1.
Theorem 6.5 ([83]) Let T be a nontrivial tree. Then, γ dR (T) = 2γ R (T) − 1 if and only if T is a wounded spider. The domination and double Roman domination numbers are related as follows for
arbitrary graphs. Theorem 6.6 ([17]) For any graph G, 2γ (G) ≤ γ dR (G) ≤ 3γ (G). Both bounds of the previous theorem are sharp. Indeed, if G is a nontrivial star K1,n−1 , then γ dR (G) = 3γ (G) = 3,
and if G = K2,k , for k ≥ 2, then γ dR (G) = 2γ (G) = 4. Moreover, Beeler et al. called a graph G having γ dR (G) = 3γ (G) a double Roman graph. They raised the problem of characterizing double Roman
graphs, in particular the double Roman trees. These have been independently characterized constructively by Abdollahzadeh Ahangar et al. [3] and Henning and Jafari Rad [50]. For graphs G such that γ
dR (G) = 2γ (G), the following necessary and sufficient was given in [17]. Theorem 6.7 ([17]) For any graph G, γ dR (G) = 2γ (G) if and only if γ (G) = γ 2 (G). An improvement of the lower bound in
Theorem 6.6 was given in [2] as follows. Recall that γ {R2} (G) ≥ γ (G) holds for every graph G. Proposition 6.8 ([2]) For every connected graph G, γ dR (G) ≥ γ {R2} (G) + γ (G). Proof. Clearly, the
result is valid for graphs of order n ∈{1, 2}. Hence, let n ≥ 3. According to Theorem 6.1, let f = (V0 , V1 , V2 , V3 ) be a γ dR (G)-function with V1 = ∅. Note that V2 ∪ V3 dominates V0 and so γ (G)
≤ |V2 | + |V3 | . Let V0 be the set of vertices of V0 having at least one neighbor in V3 , and let V0
= V0 − V0 . Now define the function g on G as follows: g(x) = f (x) − 1 for all x ∈ V2 ∪ V3 and g(x) = f (x) for all x ∈ V0 . Clearly, g is a Roman {2}-dominating function on G, and so γ{R2} (G) ≤ |
V2 | + 2 |V3 | . Therefore, γdR (G) = 2 |V2 | + 3 |V3 | = |V2 | + |V3 | + |V2 | + 2 |V3 | ≥ γ (G) + γ{R2} (G).
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As a consequence, the authors of [2] obtained the following result for the class of trees. Proposition 6.9 ([2]) Let T be a nontrivial tree. Then, γ dR (T) ≥ 2γ (T) + 1, with equality if and only if
T is a wounded spider. We also note that a characterization of trees T with γ dR (T) = 2γ (T) + 2 was given in [5]. On the other hand, since γ {R2} (G) ≤ 2γ (G) holds for any graph G, Proposition 6.8
leads to γdR (G) ≥ 32 γ{R2} (G), proved by Hajibaba and Jafari Rad in [40] who gave further a constructive characterization of trees T with γdR (T ) = 32 γ{R2} (T ). Moreover, it was also shown in
[40] that for every graph G, γ dR (G) ≤ 2γ {R2} (G). This result improves an earlier upper bound from [2], where is proved that γ dR (G) ≤ 2γ 2 (G) for every graph G. Abdollahzadeh Ahangar et al. [2]
also gave a lower bound on the double Roman domination number of a graph G in terms of the order, maximum degree, and the domination number. Proposition 6.10 ([2]) For any graph G of order n with
maximum degree Δ, γdR (G) ≥
2n − 2 + γ (G).
This bound is sharp for even cycles and paths of order 3k. n It is well known that γ (G) ≥ +1 for every graph G of order n with maximum degree . As an immediate consequence, Proposition 6.10 leads to
the following corollary.
Corollary 6.11 ([82]) If G is a graph of order n and maximum degree Δ ≥ 1, then 3n γdR (G) ≥ +1 . An upper bound on the double Roman domination number of connected graphs in terms of their order was
obtained by Beeler et al. [17], who also characterized the graphs reaching this upper bound. Let H be the family of connected graphs G of order n that can be built from n/4 copies of P4 by adding a
connected subgraph on the set of centers of n4 P4 . Theorem 6.12 ([17]) If G is a connected graph of order n ≥ 3, then γdR (G) ≤ 54 n, with equality if and only if G ∈ H. The authors [17] observed
that every connected graph G having minimum degree at least two satisfies the inequality γdR (G) ≤ 6n 5 and posed the question whether this bound can be improved. This question has been settled by
Amjadi et al. [13] by proving that γdR (G) ≤ 8n 7 except when G is a cycle C5 . However, this bound has been improved by Khoeilar et al. [60] as follows, where an infinite family of graphs attaining
the new bound was also provided in [60]. Theorem 6.13 ([60]) Let G be a graph of order n, δ(G) ≥ 2 and with no component isomorphic to C5 or C7 . Then, γdR (G) ≤ 11n 10 .
Varieties of Roman Domination
Moreover, the authors in [17] also asked the question: which classes of graphs, or trees satisfy γ dR (G) ≤ n? This issue has been dealt in [2], where it is proved that for every graph G with minimum
degree at least three, γ dR (G) ≤ n. Using the probabilistic method, Jafari Rad and Rahbani obtained the following upper bound. Theorem 6.14 ([56]) For a graph G of order n with minimum degree δ, γdR
(G) ≤ 3n
ln 2(1 + δ) − ln 3 + 1 . 1+δ
6.2 Nordhaus–Gaddum Type Bounds The first Nordhaus–Gaddum inequalities for the double Roman domination number were given in [56]. Theorem 6.15 ([56]) If G is a graph of order n ≥ 2, then 7 ≤ γdR (G)
+ γdR (G) ≤ 2n + 3. Equality holds for the lower bound if and only if G or G is K2 , and equality holds for the upper bound if and only if G or G is a complete graph. Theorem 6.16 ([56]) If G is a
graph of order n ≥ 3, then γdR (G)+γdR (G) = 2n+2 if and only if G or G is C5 , P4 , or a complete graph minus an edge. Theorem 6.17 ([56]) Let G be a graph of order n. Then, (i) if n ≥ 240 and diam
(G) = diam(G) = 2, then γdR (G)γdR (G) < (ii) if n ≥ 4 and diam(G) ≥ 3, then γdR (G)γdR (G) < 15 2 n; (iii) if n ≥ 3 and δ(G) = 1, then γdR (G)γdR (G) ≤ 25 n. 4
15 2 n;
6.3 Algorithmic and Complexity Results In [2], it is shown that the decision problem corresponding to the problem of computing γ dR (G) is NP-complete even when restricted to bipartite and chordal
graphs. Furthermore, Zhang et al. [83] gave a linear-time algorithm to compute the value of γ dR (G) for any tree T, answering an open problem posed in [17]. Poureidi et al. [72] showed that the
decision problem associated to double Roman domination is NP-complete even when restricted to planar graphs. Then, they showed that the problem of deciding whether a given graph is double Roman is
NP-hard even when restricted to bipartite or chordal graphs. They also gave a linear algorithm that computes the double Roman domination number of a given unicyclic graph.
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7 Total Roman Domination A total Roman dominating function (TRDF) of a graph G with no isolated vertex is a Roman dominating function f = (V0 , V1 , V2 ) on G such that the subgraph induced by V1 ∪
V2 under f has no isolated vertex. The total Roman domination number γ tR (G) is the minimum weight of a TRDF on G. A TRDF with minimum weight γ tR (G) is called a γ tR (G)-function. The concept of
total Roman dominating function in graphs was introduced in 2016 by Liu and Chang [65], namely total (a, b)-Roman domination for any given real numbers b ≥ a > 0. Total Roman domination is studied in
[1, 4, 12] and elsewhere. The following observation giving some properties of γ tR (G)-functions can be found in [1]. Observation 7.1 ([1]) Let G be a connected graph of order at least 3 and let f =
(V0 , V1 , V2 ) be a γ tR (G)-function. Then, the following hold. (i) |V2 | ≤ |V0 | . (ii) If x is a leaf and y a support vertex in G, then x∈V2 and y∈V0 . (iii) If z has at least three leaf
neighbors, then f (z) = 2 and at most one leaf neighbor of z belongs to V1 . The exact values of γ tR for grids G{2,n} and G{3,n} are determined in [6]. ⎧ 4n ⎪ 3 if n ≡ 0 (mod 3) ⎪ 4n+2 ⎨ 3 if n ≡ 1
(mod 3) and Proposition 7.2 ([6]) For n ≥ 2, γtR (G{2,n} ) = 4n+4 ⎪ if n ≡ 2 (mod 3) ⎪ ⎩ 3 γ tR (G{3,n} ) = 2n. Since the total Roman domination number of any nontrivial connected graph G of order n
is at most n (simply assign a 1 to each vertex of G), it is interesting to characterize those graphs G for which γ tR (G) = n. This problem has been considered in [1], where it was shown the
following. Let G be the family of graphs that can be obtained from a C4 : v1 v2 v3 v4 v1 by adding k1 + k2 ≥ 1 vertex-disjoint paths P2 and joining v1 to the end of k1 such paths and joining v2 to
the end of k2 such paths (possibly, k1 = 0 or k2 = 0). Let H be the family of graphs that can be obtained from a double star by subdividing each pendant edge once and subdividing the non-pendant edge
r ≥ 0 times. Theorem 7.3 ([1]) Let G be a connected graph of order n ≥ 2. Then, γ tR (G) = n if and only if one of (i) G is a path or a cycle or (ii) G is a corona of some graph F or (iii) G is a
subdivided star or (iv) G ∈ G ∪ H.
Varieties of Roman Domination
7.1 Bounds and Relations with Some Domination Parameters In what follows, we present the bounds on γ tR and some relationships with γ , γ t , and γ R . It is shown in [1] that for any graph G with no
isolated vertex, 2γ (G) ≤ γ tR (G) ≤ 3γ (G). Amjadi et al. [15] provided a constructive characterization of trees T with γ tR (T) = 2γ (T) or γ tR (T) = 3γ (T). Necessary conditions for graphs G
attaining each bound are also established [1]. Theorem 7.4 ([1]) Let G be a graph with no isolated vertex. If γ tR (G) = 2γ (G), then γ (G) = γ t (G) or there exists a set S of vertices of G such
that the following hold. (i) G[S] = kK2 for some k ≥ 1. (ii) γ (G − S) = γ t (G − S). (iii) No neighbor of a vertex of S in G belongs to a γ t (G − S)-set. Theorem 7.5 ([1]) Let G be a graph with no
isolated vertex. If γ tR (G) = 3γ (G), then every γ (G)-set is a packing in G. As shown in [1], the converse of Theorem 7.5 is not true and can be seen by considering the graph Gk , for k ≥ 3,
obtained from a star K1,k by subdividing k − 1 edges twice and subdividing the remaining edge exactly once. Then, γ (Gk ) = k, while γ tR (Gk ) ≤ 2(k + 1) < 3k. The total domination and total Roman
domination numbers are related as follows for arbitrary graphs. Theorem 7.6 ([1]) If G is a graph with no isolated vertex, then γ t (G) ≤ γ tR (G) ≤ 2γ t (G). Further, the following holds. (i) γ t
(G) = γ tR (G) if and only if G is the disjoint union of copies of K2 . (ii) If γ tR (G) = 2γ t (G) and S is an arbitrary γ t (G)-set, then epn(v, S) = ∅ for all v ∈ S. According to Theorem 7.6, γ tR
(G) ≥ γ t (G) + 1 for every connected graph of order at least three. It is shown in [1] that connected graphs G of order n ≥ 3 satisfy γ tR (G) = γ t (G) + 1 if and only if (G) = n − 1. By analogy
with Roman graphs defined in [35], Abdollahzadeh Ahangar et al. [1] called a total Roman graph any graph G satisfying γ tR (G) = 2γ t (G). Moreover, they gave a necessary and sufficient condition for
a graph to be a total Roman graph. It should be noted that a constructive characterization of total Roman trees was given by Amjadi et al. [12]. Proposition 7.7 ([1]) Let G be a graph without
isolated vertices. Then G is a total Roman graph if and only if there exists a γ tR (G)-function f = (V0 , V1 , V2 ) such that V1 = ∅.
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Obviously, for every graph G with no isolated vertex, γ R (G) ≤ γ tR (G). An upper bound relating the total Roman domination number to the Roman domination number was given by Abdollahzadeh Ahangar
et al. [1] as follows. Theorem 7.8 ([1]) If G is a graph of order n with no isolated vertex, then γ tR (G) ≤ 2γ R (G) − 1. Further, γ tR (G) = 2γ R (G) − 1 if and only if Δ(G) = n − 1. An upper bound
on the total Roman domination number of connected graphs different from stars in terms of their order, maximum degree, and matching number was obtained by Abdollahzadeh Ahangar et al. [6] who
characterized in addition the graphs with girth at least 4 reaching this upper bound. Recall that the matching
number α (G) is the size of a largest matching in G. Theorem 7.9 ([6]) Let G be a graph of order n ≥ 4, without isolated vertices,
different from a star. Then, γ tR (G) ≤ n − Δ(G) + α (G). They also gave a lower bound on the double Roman domination number of a graph G in terms of the order and maximum degree. Theorem 7.10 ([6])
For any graph G of order n ≥ 3 with maximum degree Δ, γtR (G) ≥ 2n . This bound is sharp for stars and double stars S(p, p). The following Nordhaus–Gaddum inequalities for the total Roman domination
are given in [14]. Theorem 7.11 ([14]) If G and G are graphs of order n without isolated vertices, then γtR (G) + γtR (G) ≤ n + 5. Furthermore, this bound is sharp for a 5-cycle. Theorem 7.12 ([14])
If G and G are graphs of order n without isolated vertices, then γtR (G)γtR (G) ≤ 6n − 5 with equality if and only if G is 5-cycle. Campanelli and Kuziak [20] were interested in total Roman
domination in the lexicographic product of two graphs. Recall that the lexicographic product of two graphs G and H is defined as the graph G · H with vertex set V (G) × V (H) and edge
set E(G · H) = {(u, v)(u , v )|uu ∈ E(G) or ((u = u and vv ∈ E(H))}. Theorem 7.13 ([20]) If G = (V, E) is a connected graph with a γ tR (G)-function h = (V0 , V1 , V2 ), then for every graph H, γtR
(G) ≤ γtR (G · H ) ≤ γtR (G) + |V1 | .
Varieties of Roman Domination
Clearly, if V1 = ∅, then γ tR (G · H) = γ tR (G). In particular, if G is a total Roman graph, then γ tR (G · H) = γ tR (G) for any graph H. As an immediate consequence of Theorem 7.13, they obtained
the following Vizing’s-like result. Corollary 7.14 ([20]) If G and H are nontrivial connected graphs, then γ tR (G · H) ≤ γ tR (G)γ tR (H). The total Roman domination and total domination numbers are
related for the lexicographic product of graphs as follows. Proposition 7.15 ([20]) If H is a graph and G is a graph without isolated vertices, then γ tR (G · H) ≤ 2γ t (G). If one of the graphs G or
H is a path or a cycle, Campanelli and Kuziak [20] established the following. Theorem 7.16 ([20]) If H is a graph and G is a path or a cycle of order n, then ⎧ ⎨
n if n ≡ 0 (mod 4), n ≤ γtR (G · H ) ≤ n + 1 if n ≡ 1, 3 (mod 4), ⎩ n + 2 if n ≡ 2 (mod 4).
7.2 Algorithmic and Complexity Results It is shown in [65] that the decision problem corresponding to the problem of computing γ tR (G) is NP-complete for bipartite graphs. Moreover, as of this
writing, a linear algorithm for computing the total Roman domination number for any tree has not yet designed. Poureidi et al. [70] studied the complexity issue for problems of deciding whether for a
graph G, γ tR (G) = 2γ (G), γ tR (G) = 2γ t (G), or γ tR (G) = 3γ (G), and showed that the corresponding decision problems are NP-hard even when restricted to bipartite graphs.
8 Perfect Roman Domination As defined in Livingston and Stout in [67], a perfect dominating set (PDS) is a dominating set S of G for which each vertex in V − S is adjacent to exactly one vertex in S.
The minimum cardinality of a PDS of a graph G is the perfect domination number γ p (G). Note that every graph G has a PDS since V (G) is trivially such a set. Perfect domination has been studied by
several authors, and for more details on this concept, the reader is referred to the survey in [61]. The study of the perfect Roman domination was initiated by Henning, Klostermeyer, and MacGillivray
in [52]. A perfect Roman dominating function (PRDF) on a graph G is a Roman dominating function f = (V0 , V1 , V2 ) on G such that every
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vertex of V0 is adjacent to exactly one vertex assigned in V2 . The minimum weight p of a PRDF on G is the perfect Roman domination number γR (G). A PRDF on G p p with weight γR (G) is called a γR
8.1 Bounds Note that every graph G has a PRDF since f = (∅, V (G), ∅) is trivially such a p function, and thus γR (G) ≤ n for all G of order n. One can easily see that this bound is sharp if and only
if (G) ≤ 1. Henning et al. [52] focused on the class of trees by giving an upper bound on the perfect Roman domination number in terms of the order. In addition, they characterized extremal trees
attaining this upper bound. Let T be the family of all trees T whose vertex set can be partitioned into sets, each set inducing a path P5 on five vertices, such that the subgraph induced by the
central vertices of these P5 ’s is connected. p
Theorem 8.1 ([52]) If T is a tree of order n ≥ 3, then γR (T ) ≤ 45 n, with equality if and only if T ∈ T . The bound of Theorem 8.1 has been improved by Darkooti et al. [36] for trees T with (T) ≥
2s(T) − 2, where (T) and s(T) are the number of leaves and support vertices of T, respectively. p
Theorem 8.2 ([36]) For any tree T of order n ≥ 3, γR (T ) (T) + 2s(T) − 2)/5.
Moreover, the same authors also showed that the decision problem associated p with γR (G) is NP-complete for bipartite graphs. The question of whether the 45 n upper bound on the perfect Roman
domination number for trees remains valid for any connected graph G of order n ≥ 3 was posed in [52]. This issue was addressed by Henning and Klostermeyer [49] for regular graphs, where a positive
answer was given to some cases. Other than the case of p cycles Cn for which γR (Cn ) ≤ 45 n, it is shown the following for k ≥ 3. Theorem 8.3 ([49]) Let G be k-regular graph of order n. Then, p
(i) if k = 3, then γR (G) ≤ 34 n; 2 p +k+3 n; (ii) if k ≥ 4, then γR (G) ≤ kk2 +3k+1 p
(iii) if k ≥ 3 and G has girth at least 7, then γR (G) ≤
k 2 −k+4 k 2 +k+2
As an immediate consequence of Theorem 8.3, the following result is obtained. Corollary 8.4 If G is 4-regular or k-regular with k ∈{5, 6, 7} and with girth at least p 7, then γR (G) ≤ 45 n.
Varieties of Roman Domination
The relationship between the perfect Roman domination and perfect domination numbers was raised in [32] and [79]. Obviously, if S is a minimum PDS of a graph p G, then clearly (V − S, ∅, S) is a PRDF
of G, and thus γR (G) ≤ 2γ p (G). We say p p that a graph G is a perfect Roman graph if γR (G) = 2γ (G). An open problem is to characterize the perfect Roman graphs. A constructive characterization
of perfect Roman trees was given in [79], where 7 operations have been defined to build such p trees. Moreover, it is shown in [32] that γ p (G) may be larger or smaller than γR (G). p p Clearly, for
nontrivial stars G, we have γR (G) > γ (G). To see the other situation, consider the following example of graphs given in [32]. Let H be the graph obtained from a double star S(p, p), (p ≥ 3) with
central vertices u, v by subdividing the edge uv with vertex w, and adding 2k (k ≥ 3) new vertices, where k vertices are attached to both u and w and the remaining k vertices are attached to both v
and w. Then, p p γ p (H) = 2k + 3 while γR (H ) = 5 and so the difference γ p (H )−γR (H ) can be even very large. In addition, the following bound relating the perfect Roman domination and perfect
domination numbers for trees was proved in [32], and a constructive characterization of extremal trees was also given. p
Theorem 8.5 ([32]) For any tree T of order n ≥ 2, γR (T ) ≥ γ p (T ) + 1. Haynes and Henning [45] extended the concept of perfect Roman domination to Italian domination (equivalently Roman {2}
-domination) by defining a perfect Italian dominating function, abbreviated PIDF, on G as a function f : V (G) →{0, 1, 2} such that for every vertex u ∈ V with f (u) = 0, the total weight assigned by
f to the vertices of N(u) is 2, that is, all the neighbors of u are assigned the weight 0 by f except for exactly one vertex v for which f (v) = 2 or for exactly two vertices v and w for which f (v)
= f (w) = 1. The perfect Italian domination number of G, denoted p γI (G), is the minimum weight of a PIDF of G. It was shown in [45] that if G is a p tree on n ≥ 3 vertices, then γI (G) ≤ 45 n.
Haynes and Henning proposed in [45] the p problem of determining the best possible constants cG such that γI (G) ≤ cG × n for all graphs of order n when G is a planar or regular graph. This problem
has been dealt by Lauri and Mitillos in [63], by proving that cG = 1 when G is planar or split and cG = 2/3 when G is cubic.
8.2 Algorithmic and Complexity Results Lauri and Mitillos in [63] studied the complexity-theoretic questions for perfect Italian domination number. They proved that deciding whether a given graph G
admits a perfect Italian dominating function of weight at most k is NP-complete, even when G is restricted to the class of bipartite planar graphs. They also strengthen the result of Chellali et al.
[29] by showing that deciding whether G admits a Roman {2}-dominating function of weight at most k is NP-complete, even when G is both bipartite and planar.
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9 Strong Roman Domination In [11], Álvarez-Ruiz et al. defined a new version of Roman domination ensuring the defense of the Roman empire from multiple attacks, unlike with the strategy of Roman
domination which only guarantees the defense of the empire from a single attack. Indeed, if several simultaneous attacks occur on weak places (places in which no army is stationed), then a strong
place (in which two legions are deployed) may not be able to defend efficiently its neighbors. The new strategy suggested by Álvarez-Ruiz et al. [11] considers that a strong place should be able to
defend itself and at least half of its weak neighbors. For a given graph G of order n and maximum degree , let f : V (G) → {0, 1, . . . , 2 + 1}. Let Bi = {v ∈ V : f (v) = i} for i ∈{0, 1}, and B2 =
V (G) − (B0 ∪ B1 ). Then f is called a strong Roman dominating function (StRDF) v ∈ B0 has a neighbor w, such that w ∈ B2 and f (w) ≥ for G, if every 1 1 + 2 |N (w) ∩ B0 | . The strong Roman
domination number γ StR (G) is the minimum weight of an StRDF on G. Clearly, γ StR (G) = γ R (G) for all connected graphs G with (G) ≤ 2. In particular, if G is a path or a cycle of order n, then
γStR (G) = 2n 3 . Various bounds on the strong Roman domination number obtained in [11] are gathered in the following proposition. Proposition 9.1 ([11]) Let G be a graph of order n with maximum
degree Δ. Then, ! " (i) γR (G) ≤ γStR (G) ≤ 1 + 2 γ (G); (ii) γStR (G) ≤ n − 2 ; ; (iii) γStR (G) ≥ n+1 2
; (iv) γStR (G) ≤ n − 1+diam(G) 3
(v) if G has girth g(G) ≥ 3, then γStR (G) ≤ n − g(G) . 3 The following result obtained by using a probabilistic approach is a generalization in some sense of the one presented in [35]. Proposition
9.2 ([11]) Let G be a graph of order n, minimum degree δ, and maximum degree Δ, such that 2 < δ. Then, (1 + 2 )n δ+1 γStR (G) ≤ ln +1 . δ+1 1 + 2 For the class of trees T, an upper bound on γ StR (T)
in terms of the order is presented. Let H be a tree obtained from a star K1,3 by subdividing each edge exactly
Varieties of Roman Domination
once. Let Fp be the family of trees obtained from any tree T of order p by identifying each vertex of T with the central vertex of H so that the H’s are vertex disjoint. Theorem 9.3 ([11]) If T is a
tree of order n, then γStR (T ) ≤ 67 n, with equality if and only if T ∈ Fp . Álvarez-Ruiz et al. have wondered if the 67 n upper bound on the strong Roman domination number for trees remains valid
for any connected graph G of order n ≥ 3. We close this section by mentioning some algorithmic and complexity results. It is shown in [11] that the decision problem corresponding to the problem of
computing γ StR (G) is NP-complete for planar graphs. A problem of constructing a polynomial algorithm for computing the value of γ StR (T) for any tree T is proposed in [11]. This problem is
answered by Poureidi et al. [73] who gave a linear algorithm that computes the strong Roman domination number of trees.
10 Edge Roman Domination The study of the edge version of Roman domination was initiated by Roushini Leely Pushpam and Malini Mai in 2009 [75]. An edge Roman dominating function (ERDF) of a graph G
is a function f : E(G) →{0, 1, 2} such that every edge e
with f (e) = 0 is adjacent to some edge e with f (e ) = 2. The weight of an ERDF f is w(f ) = e ∈ E(G) f (e), and the edge Roman domination number of G, denoted by γR (G), is the minimum weight of an
ERDF of G. An ERDF f : E(G) →{0, 1, 2} can be represented by the ordered partition (E0 , E1 , E2 ) of E(G), where Ei = {e ∈ E(G) : f (e) = i} for i ∈{0, 1, 2}. It is worth mentioning that the edge
Roman domination number of G equals the Roman domination number of its line graph. Edge Roman domination has been studied in [8, 9, 22, 54, 75, 77] and elsewhere. It should be noted that as far as we
know, no paper has dealt with either the complexity or the algorithmic aspect of the edge Roman domination problem. Several properties of edge Roman dominating functions can be obtained analogously
to Roman dominating functions given in [35]. Here are some summarized by the following result. Proposition 10.1 ([75]) Let f = (E0 , E1 , E2 ) be a minimum ERDF of an isolate-free graph G, such that
|E2 | is maximum. Then, (i) (ii) (iii) (iv)
E1 is independent. The edges of E0 dominate the edges of E1 . Each edge of E0 is adjacent to at most one edge of E1 . Let e ∈ G[E2 ] have exactly two private edges e1 and e2 in E0 with respect to E2
. Then there do not exist edges h1 , h2 ∈ E1 such that (h1 , e1 , e, e2 , h2 ) is the edge sequence of a path P6 .
Also, it is obvious that for every graph G, γ (G) ≤ γR (G) ≤ 2γ (G), where γ (G) is the edge domination number of G. A characterization of trees with edge
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Roman domination number twice the edge domination number was given by Jafari Rad in [54].
and γR (Cn ) = 2n In [75], it is shown that γR (Pn ) = 2n 3 3 . Akbari et al. [8] extended the exact value on paths to 2-by-n and 3-by-n grid graphs and proved that 4n
for n ≥ 2, γR (G{2,n} ) = 3 and γR (G{3,n} ) = 2n. The problem of determining the edge Roman domination number for every m-by-n grid graph G{m,n} remains open. For bounds on the edge Roman domination
number, we begin with the class of trees T, where Akbari et al. [8] have bounded it in terms of the order and number of leaves of T. Theorem 10.2 ([8]) If T is a tree of order n with (T) leaves, then
2(n − (T ) + 1) 3
γR (T )
2(n − 1) . 3
For arbitrary graphs, two upper bounds on the edge Roman domination number have been established by Akbari et al. [8]. Theorem 10.3 ([8]) Let G be a graph of order n with maximum degree Δ. Then, 2
(i) γR (G) ≤ 2+1 n; (ii) if G has a perfect matching, then γR (G) ≤
2−1 2 n.
Furthermore, the authors in [8] conjectured that γR (G) ≤ +1 n . This conjecture has been disproved by Chang, Chen, and Liu [22] who gave the following counterexample. Let G(r, t) be the graph
obtained from t copies of Kr,r+1 by i y i+1 for 1 ≤ i ≤ t with y t+1 = y 1 , where the partite sets of adding edges yr+1 1 1 1 i }. Note that G(r, the i-th Kr,r+1 are Xi = {x1i , . . . , xri } and Yi
= {y1i , . . . , yr+1 t) has order n = (2r + 1)t and maximum degree = r + 1. It was shown then that 2−2
γR (G(r, t)) = 2rt = 2−1 n > +1 n when r ≥ 2 and t a multiple of r + 2. For the class of planar graphs, it is shown in [9] that if G is outerplanar, then γR (G) ≤ 45 n and if G is planar claw-free,
then γR (G) ≤ 67 n. Furthermore, the authors in [9] conjectured that γR (G) ≤ 67 n for any planar graph G of order n. This conjecture was proved by Chang, Chen, and Liu in [22]. Also, they obtained
several other bounds that we list below. Recall that a graph G is k -degenerate if for every subgraph H of G, δ(H) ≤ k. Theorem 10.4 ([22]) If G is a k-degenerate graph of n vertices, then γR (G) ≤
2k 2k+1 n.
Since any tree is 1-degenerate, the upper bound 2n 3 in Theorem 10.2 becomes a simple consequence of Theorem 10.4. Moreover, since every graph is -degenerate,
Varieties of Roman Domination
the upper bound in Theorem 10.3 is also a straightforward consequence of Theorem 10.4. Theorem 10.5 ([22]) If G is a connected graph of order n with maximum degree 2 Δ, then γR (G) ≤ 2−2 2−1 n + 2−1
. Theorem 10.6 ([22]) If G is a subcubic graph of order n which does not contain K3,3 as a component, then γR (G) ≤ 45 n. Theorem 10.7 ([22]) If G is a graph of order n containing no subgraph
isomorphic to a subdivision of K2,3 , then γR (G) ≤ 45 n.
11 Open Problems In this chapter, we have surveyed some results concerning nine Roman dominationrelated parameters. There is much scope for further research, here are some suggestions. 1. Determine
the weak Roman domination number for every m-by-n grid graph G{m,n} . 2. Since the problem of deciding whether γ r (G) = γ R (G) for a given graph G is NP-hard, it is quite interesting to
characterize other classes of graphs G, other than trees, such that γ r (G) = γ R (G). 3. Design an algorithm for computing the Roman k-domination number for any tree T. 4. [Álvarez-Ruiz et al. [11]]
Is it true that for every connected graph G of order n ≥ 3, γStR (T ) ≤ 67 n? 5. What are the algorithmic, complexity, and approximation properties of edge Roman domination? 6. Determine the edge
Roman domination number for every m-by-m grid graph G{m,m} . In [78], it was shown that the generalized Petersen graph P(n, 1) is a double Roman graph for any n≡ 2 (mod 4), while in [58], it is shown
that the generalized Petersen graph P(n, 2) is not double Roman for all n ≥ 3. 7. [Jiang et al. [58]] Find other generalized Petersen graphs that are double Roman.
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Varieties of Roman Domination
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weak Roman domination number. Inf. Proc. Lett. 138, 12–18 (2018)
Part II
Domination in Selected Graph Families
Domination and Total Domination in Hypergraphs Michael A. Henning and Anders Yeo
AMS Subject Classification: 05C65, 05C69
1 Introduction Domination in hypergraphs has not been studied in as much depth as domination in graphs. Although the theory of domination in graphs is well developed in the literature, the theory of
domination in hypergraphs is currently in its infant stages. In this chapter, we give a survey of selected results on domination and total domination in hypergraphs. We establish an essential
connection in hypergraphs between dominating sets and transversals, and between total dominating sets and total transversals. Hypergraphs are systems of sets, which are conceived as natural
extensions of graphs. More formally, a hypergraph H = (V (H), E(H)) is a finite set V (H) of elements, called vertices, together with a finite multiset E(H) of subsets of V (H), called hyperedges. If
the hypergraph H is clear from the context, we simply write V = V (H) and E = E(H). We shall use the notation nH = |V | (or n(H)) and mH = |E| (or m(H)), and sometimes simply n and m without
subscripts if
M. A. Henning Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: [email protected] A. Yeo () Department of Mathematics and Applied
Mathematics, University of Johannesburg, Johannesburg, South Africa Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark © The Author(s), under exclusive
license to Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://doi.org/10.1007/978-3-030-58892-2_11
M. A. Henning and A. Yeo
the hypergraph H is clear from the context, to denote the order and size of H, respectively. A vertex v ∈ V is incident with a hyperedge e ∈ E in H if v ∈ e, i.e., if the vertex v belongs to the edge
e. A hypergraph H is linear if every two distinct edges of H intersect in at most one vertex. The edge set E in a hypergraph H = (V, E) is often allowed to be a multiset in the literature, but in
this chapter we exclude multiple edges without loss of generality. We refer to the cardinality |e| of an edge e in H as its size. In the problems studied here, we assume that every edge has size at
least 2. Throughout this chapter, we simply refer to a hyperedge as an edge. An isolated edge in H is an edge in H that does not intersect any other edge in H. A k-edge in H is an edge of size k. The
rank of a hypergraph H is the maximum size of an edge on H. Thus, H is of rank k if |e|≤ k holds for each edge e ∈ E in H. The hypergraph H is k-uniform if every edge of H is a k-edge. Every (simple)
graph is a 2-uniform hypergraph. Thus, graphs are special hypergraphs. For i ≥ 2, we denote the number of edges in H of size i by ei (H). The degree of a vertex v in H, denoted by dH (v) or d(v) if H
is clear from the context, is the number of edges of H, which contain v. A vertex of degree k is called a degree- k vertex. The minimum degree among the vertices of H is denoted by δ(H) and the
maximum degree by (H). A subhypergraph of a hypergraph H = (V, E) is a hypergraph H = (V , E ) satisfying V ⊆ V and E ⊆ E (and where E ⊆ V by definition of a hypergraph H ). Given a hypergraph H =
(V, E) and a nonempty set X ⊆ V , the subhypergraph of H induced by X is the hypergraph H = (V , E ) where E = {e ∈ E : e ⊆ X} and where V ⊆ X is the set of vertices of H contained in at least one
edge e ⊆ X. A 2-section graph, H2 , of a hypergraph H is defined as a graph H2 with the same vertex set as H and in which two vertices are adjacent in the graph H2 if and only if they belong to a
common edge in H. Two vertices x and y in a hypergraph H are adjacent if there is an edge e of H such that {x, y}⊆ e. The neighborhood of a vertex v in H, denoted by NH (v), is the set of all
vertices different from v that are adjacent to v, while the closed neighborhood of v is the set NH [v] = NH (v) ∪{v}. We call a vertex in NH (v) a neighbor of v in H. For a subset X ⊆ V (H), the
neighborhood of X is the set NH (X) = ∪x ∈ X NH (x), while the closed neighborhood of X is the set NH [X] = NH (X) ∪ X. Two vertices x and y are connected in the hypergraph H if there is a sequence x
= v0 , v1 , v2 . . . , vk = y of vertices of H in which vi−1 is adjacent to vi for i ∈ [k]. A connected hypergraph is a hypergraph in which every pair of vertices are connected. A component of H is a
maximal connected subhypergraph of H. Thus, no edge in H contains vertices from different components. A subset T ⊆ V of vertices in a hypergraph H is a transversal (also called vertex cover or
hitting set in many papers) if for every e ∈ E, T ∩ e=∅, that is, every edge has a vertex in T. A transversal of cardinality τ (H) is called a τ -transversal of H. A total transversal in H is a
transversal T in H with the additional property that every vertex in T has at least one neighbor in T. The total transversal number τ t (H) of H is the minimum size of a total transversal in H. A
total transversal in H of cardinality τ t (H) is called a τ t -total transversal of H. Transversals in hypergraphs are well-studied in the literature (see, for example [6, 11, 15, 23, 27–30, 38,
Domination and Total Domination in Hypergraphs
For a subset X ⊂ V of vertices in H, we define H − X to be the hypergraph obtained from H by deleting the vertices in X and all edges incident with X, and deleting all isolated vertices, if any, from
the resulting hypergraph. We note that if T is a transversal in H − X, then T ∪ X is a transversal in H. If X = {x}, then we write H − X simply as H − x. A dominating set in a hypergraph H is a
subset of vertices D ⊆ V such that for every vertex v ∈ V D, there exists an edge e ∈ E for which v ∈ e and e ∩ D=∅. Equivalently, a set D is a dominating set in H if every vertex outside the set D
has a neighbor in D. The domination number γ (H) is the minimum cardinality of a dominating set in H. A dominating set of H of cardinality γ (H) we call a γ -set of H. A total dominating set,
abbreviated TD-set, in H is a subset of vertices D ⊆ V (H) such that every vertex in H is adjacent with a vertex in D. Equivalently, a set D is a total dominating set in H if D is a dominating set in
H with the additional property that for every vertex v ∈ D there exists an edge e ∈ E(H) for which v ∈ e and e ∩ (D {v})=∅. The total domination number γ t (H) is the minimum cardinality of a TD-set
in H. We remark that a set is a (total) dominating set in H if and only if it is a (total) dominating set in the 2-section graph H2 . Domination in hypergraphs was introduced in 2007 by Acharya [2]
and studied in [4, 8, 22, 34, 35] and elsewhere, and also in the Ph.D. thesis of Bibin Jose [33]. Total domination in hypergraphs was introduced by Bujtás, Henning, Tuza, and Yeo in 2014 [9] and
studied further by Henning and Yeo in 2015 [32]. We use the standard notation [k] = {1, . . . , k} and [k]0 = {0, 1, . . . , k}.
2 Domination in Hypergraphs In this section, we focus on domination in hypergraphs. We proceed as follows. In Section 2.1, we consider disjoint dominating sets in hypergraphs. In Section 2.2 we
discuss an interplay between domination and transversals in hypergraphs. As a consequence of the results in Section 2.2, we establish in Section 2.3 upper bounds on the domination numbers of uniform
hypergraphs with minimum degree at least one. In Sections 2.4, 2.5, 2.6, and 2.7, we present results on hypergraphs with specified edge size having large domination numbers. In Section 2.8, we
discuss a general setting of upper bounds on the domination number in terms of its order and size and present a general way to formulate the problem. In Section 2.9, we address the problem of finding
the minimum order of a connected, k-uniform hypergraph with a given domination number. In Section 2.10, we present a Nordhaus–Gaddum-type result for the sum of domination parameters in hypergraphs
and their complements. Hypergraphs with equality of the domination and transversal numbers are discussed in Section 2.11. In Section 2.12, we discuss a relationship between domination and matching in
hypergraphs and present an upper bound on the domination number of a uniform hypergraph in terms of its matching number.
M. A. Henning and A. Yeo
2.1 Disjoint Dominating Sets The maximum number of vertex-disjoint dominating sets in a graph G is called the domatic number of G denoted by dom(G). The domatic number of a graph was introduced in
1975 by Cockayne and Hedetniemi [12]. The domatic number of a hypergraph H, denoted by dom(H), is defined analogously as the maximum number of vertex-disjoint dominating sets in H. A fundamental
result in domination theory in 1962 due to Ore [40] is the property that every graph without isolated vertices contains two disjoint dominating sets. Indeed as observed by Ore [40], if G = (V, E) is
a graph without isolated vertices, then the complement V D of every minimal dominating set D in G is a dominating set of G. Thus, every 2-uniform hypergraph H without isolated vertices contains two
vertex-disjoint dominating sets; that is, dom(H) ≥ 2. This property holds for k-uniform hypergraphs for all k ≥ 2. To see this, recall that an independent set in a hypergraph H = (V, E) is a set S ⊆
V such that no two vertices of S are adjacent; that is, every edge of H intersects S in at most one vertex, and therefore no two vertices of S belong to the same edge in H. Proposition 1 For k ≥ 2 an
integer, every k-uniform hypergraph H without isolated vertices satisfies dom(H) ≥ 2. Proof. Let S be a maximal independent set in the k-uniform hypergraph H = (V, E). Thus, |e ∩ S| = 1 for every
edge e ∈ E. By the maximality of the set S, every vertex in V S is adjacent to at least one vertex in S; that is, if v ∈ V S, then there exists at least one edge ev such that |ev ∩ S| = 1. In
particular, we note that the set S is a dominating set in H. Since H is without isolated vertices, every vertex in S is adjacent to at least one vertex outside S, implying that V S is a dominating
set of H. Thus, H has at least two vertex-disjoint dominating sets, namely S and V S. 2 We note that the result of Proposition 1 also holds for hypergraphs without isolated vertices and where every
edge has size at least k. However it is not true that for any given k ≥ 3, every k-uniform hypergraph without isolated vertices contains k vertex-disjoint dominating sets, as the following result
shows. Proposition 2 For k ≥ 3 an integer, there exist k-uniform hypergraphs H without isolated vertices satisfying dom(H) = 2. Proof. For a given integer k ≥ 3, we construct a k-uniform hypergraph
as follows. Let H = (V , E ) be a complete (k − 1)-uniform hypergraph on 3k − 5 vertices, that is, every (k − 1)-element subset of V forms an edge in H . We now construct a kuniform hypergraph H from
H by adding a new vertex v to each edge e ∈ E in H
and replacing the edge e with the edge e ∪{v} in such a way that all the new vertices are distinct and therefore have degree 1 in H. We show that dom(H) = 2. Suppose, to the contrary, that dom(H) ≥
3. Let D1 , D2 , D3 be three vertex-disjoint dominating sets of H. We may assume that (D1 , D2 , D3 ) is a partition of V , since vertices not in D1 ∪ D2 ∪ D3 can be assigned randomly to some set Di
where i ∈ [3]. We note that
Domination and Total Domination in Hypergraphs
3k − 5 = |V | =
|Di ∩ V (H )|.
Renaming the dominating sets D1 , D2 , D3 if necessary, we may assume that |D1 ∩ V (H )|≥ (3k − 5)/3, which implies that |D1 ∩ V (H )|≥ k − 1. Thus, there exists an edge e ∈ E of H such that e ⊆ D1 ∩
V (H ). Let v be the vertex added to the edge e when constructing H from H to produce the edge e = e ∪{v}. At least one of the two dominating sets D2 and D3 does not contain the vertex v. Such a
dominating set contains no vertex from the edge e and therefore does not dominate the vertex v, a contradiction. Hence, dom(H) ≤ 2. By Proposition 1, dom(H) ≥ 2. Consequently, dom(H) = 2. 2 We remark
that the hypergraph constructed in the proof of Proposition 2 is nonlinear. Even if we restrict our attention to linear hypergraphs, it is not true that for any given k ≥ 3, every k-uniform linear
hypergraph without isolated vertices contains k vertex-disjoint dominating sets. Proposition 3 For k ≥ 3 an integer, there exist k-uniform linear hypergraphs H without isolated vertices satisfying
dom(H) < k. Proof. For a given integer k ≥ 3, we construct a k-uniform, linear hypergraph as follows. Let Y = {y1 , y2 , . . . , yk+1 } be a set of k + 1 vertices. For each 2-element subset {yi , yj
} of Y where 1 ≤ i < j ≤ k + 1, let Xij be a set of k − 2 new vertices, and let X= Xij . 1≤i 0 if and only if b ≥ 0 and a > − bk . (b) For every H ∈ Hk−1 , for k ≥ 3, anH + (a + b)mH > 0 if and only
if b ≥ 0 and a > − bk . Proof. Suppose that anH + bmH > 0 for some k ≥ 2 and for every H ∈ Hk . !" The k-uniform complete hypergraph H of order nH = and size mH = k gives !" a + b k > 0, which
implies that b ≥ 0 as →∞. The k-uniform hypergraph with exactly one edge of order nH = k and mH = 1 gives a · k + b · 1 > 0, and so a > − bk . Hence, both conditions b ≥ 0 and a > − bk are necessary
in part (a). To prove the sufficiency part in (a), suppose that b ≥ 0 and − bk < a. We note that for every hypergraph H ∈ Hk we have by definition that δ(H) ≥ 1, and so mH = k1 v∈V (H ) dH (v) ≥ k1 ·
nH , or, equivalently, that nH ≤ kmH holds. If a ≤ 0, this implies that anH + bmH ≥ (ak + b)mH > 0. If a > 0, then trivially anH + bmH > 0 holds noting that b ≥ 0 by supposition. This proves part
(a). To prove part (b), suppose that k ≥ 3, b ≥ 0, and − bk < a. Similarly as before, nH ≤ (k − 1)mH holds for every hypergraph H ∈ Hk−1 with k ≥ 3. If a ≤ 0, this implies that anH + (a + b)mH ≥ (a(k
− 1) + (a + b))mH = (ak + b)mH > 0. If a > 0, then trivially anH + (a + b)mH > 0 holds noting that b ≥ 0 by supposition. This proves part (b). 2 We are now in a position to prove the following key
relationship between the transversal number and the domination number of uniform hypergraphs. Theorem 6 ([8]) For every integer k ≥ 3 and for a, b ∈ R satisfying b ≥ 0 and a > − bk , the following
equality holds: γ (H ) τ (H ) = sup . an + bm an + (a + b)mH H ∈Hk H ∈Hk−1 H H H sup
Domination and Total Domination in Hypergraphs
Proof. Let the parameters a, b, and k be fixed, where k ≥ 3 is an integer and a, b ∈ R satisfy b ≥ 0 and a > − bk . We define γ (H ) an H ∈Hk H + bmH
gk = sup
tk−1 =
H ∈Hk−1
τ (H ) . anH + (a + b)mH
By Lemma 5, we have anH +bmH > 0 for every H ∈ Hk and anH +(a+b)mH > 0 for every H ∈ Hk−1 . We show first that tk−1 ≤ gk . Let H be an arbitrary hypergraph in the family Hk−1 , and so H is a (k − 1)
-uniform hypergraph and δ(H) ≥ 1. Let H have vertex set V (H) = {v1 , . . . , vn } and edge set E(H) = {e1 , . . . , em }. Let H ∈ Hk be the k-uniform hypergraph constructed from H by adding to it m
new vertices u1 , . . . , um and extending each edge ei to the edge ei = ei ∪ {ui } for i ∈ [m]; that is, V (H ) = V (H) ∪{u1 , . . . , um } and E(H ) = {ei : i ∈ [m]}. We note that H has n = n + m
vertices and m = m edges. Every dominating set of H contains at least one vertex from every edge of H in order to dominate the newly added vertices of degree 1, and so τ (H ) ≤ γ (H ). By Observation
4, γ (H ) ≤ τ (H ). Consequently, γ (H ) = τ (H ) holds. We also note that every transversal of H remains a transversal of H , and so τ (H ) ≤ τ (H). Further we can always choose a τ -transversal T
of H to contain no added vertices of degree 1 since if T contains an added vertex ui for some i ∈ [m], then we can simply replace ui in T with a vertex in the edge ei . Thus, T is also a transversal
of H, and so τ (H) ≤|T | = τ (H ). Consequently, τ (H) = τ (H ). As observed earlier, τ (H ) = γ (H ), and so τ (H) = γ (H ). By the definition of gk , we therefore have τ (H )=γ (H ) ≤ gk (an +bm )=
gk (an+(a+b)m) and hence τ (H ) ≤ gk an + (a + b)m holds for every hypergraph H ∈ Hk−1 . Equivalently, tk−1 ≤ gk . To prove the converse relation tk−1 ≥ gk , let F be an arbitrary hypergraph in the
family Hk , and so F is a k-uniform hypergraph and δ(F) ≥ 1. Let F have order n and size m. Let F be a hypergraph obtained from F by successively deleting edges of F that do not contain any vertices
of degree 1 in the resulting hypergraph at each stage. We note that F is a k-uniform hypergraph with n = n vertices and m ≤ m edges. When F is transformed into F , isolated vertices cannot arise, and
so F ∈ Hk . Since removing edges cannot decrease the domination number, we note that γ (F) ≤ γ (F ). Moreover, every edge of F contains at least one vertex of degree 1 and hence τ (F ) = γ (F ).
Consequently, τ (F ) ≥ γ (F). Deleting exactly one vertex of degree 1 from each edge of F , we obtain a (k − 1)-uniform hypergraph F
of order n
= n − m and of size m
= m such that the transversal number remains unchanged. Thus, γ (F ) ≤ τ (F )=τ (F
) ≤ tk−1 (an
)=tk−1 (an +bm ) ≤ tk−1 (an+bm)
M. A. Henning and A. Yeo
and hence γ (F ) ≤ tk−1 an + (a + b)m holds for every hypergraph F ∈ Hk . Equivalently, gk ≤ tk−1 . As shown earlier, tk−1 ≤ gk . Consequently, gk = tk−1 . 2 We show next that the uniformity
condition in Theorem 6 can be relaxed. For this purpose, we shall need the following definition from [8]. Definition 2 For an integer k ≥ 2, let Hk+ ( Hk− ) denote the class of all hypergraphs H with
δ(H) ≥ 1, in which every edge is of size at least k ( at most k, respectively) . As shown in [8], every valid upper bound on the domination number of k-uniform hypergraphs of the form γ (H ) ≤ anH +
bmH can be extended to hypergraphs with a less strict condition on edge sizes. The exact formulation will depend, however, on the sign of a. For a ≥ 0, the result of Proposition 7 applies, while for
a ≤ 0, the result of Proposition 8 applies. We state these results without proof. Proposition 7 ([8]) For any two nonnegative reals a and b (with a + b > 0) and for every integer k ≥ 2, the following
equality holds: sup
H ∈Hk+
γ (H ) γ (H ) = sup . anH + bmH H ∈Hk anH + bmH
Proposition 8 ([8]) For every integer k ≥ 2 and for any two reals a ≤ 0 and b > 0, if a > − bk , then the following equality holds: sup
H ∈Hk−
γ (H ) γ (H ) = sup . anH + bmH H ∈Hk anH + bmH
2.3 Upper Bounds on the Domination Number As a consequence of Theorem 6 and known results on the transversal number of a hypergraph, we establish in this section upper bounds on the domination number
of a k-uniform hypergraph with minimum degree at least one. As remarked in [8], as a consequence of Theorem 6 the following two-way correspondence is obtained. • For k ≥ 3, if we have a general bound
on the transversal number of the form c2 τ (H ) ≤ c1 nH + c2 mH with − k−1 < c1 ≤ c2 for all (k − 1)-uniform hypergraphs H ∈ Hk−1 , then the inequality γ (H ) ≤ c1 nH + (c2 − c1 )mH on the domination
number necessarily holds for every k-uniform hypergraph H ∈ Hk . Moreover, if the former bound is sharp, then the latter one is sharp, as well.
Domination and Total Domination in Hypergraphs
• For k ≥ 3, similarly from every valid upper bound on the domination number of the form γ (H ) ≤ anH + bmH for all k-uniform hypergraphs H ∈ Hk with real numbers b ≥ 0, a > − bk , we can derive the
upper bound on the transversal number of the form τ (H ) ≤ anH +(a+b)mH for all (k − 1)-uniform hypergraphs H ∈ Hk−1 . Moreover, if the former bound is sharp, then the latter one is sharp, as well.
We next present the results on the transversal number. For k ≥ 2 an integer, the so-called Tuza constant ck is defined by ck = sup
τ (H ) , nH + mH
where the supremum ranges over all k-uniform hypergraphs H. Erd˝os and Tuza [16, p. 1180] showed that c2 = 13 . Chvátal and McDiarmid [11] and Tuza [43] independently established that c3 = 14 , while
Lai and Chang [38] showed that c4 = 29 . We summarize these results below. Theorem 9 The following hold: (a) ([16, p. 1180]) c2 = 13 . (b) ([11, 43]) c3 = 14 . (c) ([38]) c4 = 29 . The precise value
of ck has yet to be determined for any values of k, with k ≥ 5, some 30 years after the Tuza constants ck were first introduced and studied. Applying probabilistic arguments, Alon [6] determined the
asymptotic behavior of ck as k grows. ln(k) Theorem 10 (Alon [6]) ck = (1 + o(1)) as k →∞. k Chvátal and McDiarmid [11] established the following upper bound on the transversal number of a uniform
hypergraph in terms of its order and size. Theorem 11 ([11]) For k ≥ 2, if H is a k-uniform hypergraph, then n + τ (H ) ≤ H
k 2 3k 2
As a consequence of Alon’s result in Theorem 10 and the relation established in Theorem 6 (with b = 0), we have the following asymptotic equality. Theorem 12 ([8]) As k tends to infinity, ln(k) γ (H
) ln(k − 1) = (1 + o(1)) . = (1 + o(1)) nH k−1 k H ∈Hk sup
M. A. Henning and A. Yeo
Moreover, as an immediate consequence of the Chvátal–McDiarmid result in Theorem 11 and the relation given in Theorem 6, we obtain the following upper bound on the domination number of a uniform
hypergraph without isolated edges. Recall that for k ≥ 2, we denote by Hk the class of all k-uniform hypergraphs H with δ(H) ≥ 1. Theorem 13 ([8]) For k ≥ 3, if H ∈ Hk , then nH +
γ (H ) ≤
k−3 2
3(k−1) 2
and this bound is sharp. We remark that the k-uniformity condition in Theorem 13 can be relaxed to edge sizes at least k. We state this formally as follows. Theorem 14 ([25]) For k ≥ 3, if H is a
hypergraph with all edges of size at least k and with δ(H) ≥ 1, then nH +
γ (H ) ≤
k−3 2
3(k−1) 2
As an immediate consequence of Theorem 9 and the relation given in Theorem 6, we obtain the following upper bounds on the domination number of a hypergraph without isolated vertices and with edges
sizes at least k where k ∈{3, 4, 5}. Theorem 15 If H is a hypergraph with all edges of size at least k and with δ(H) ≥ 1, then the following hold: (a) If k = 3, then γ (H ) ≤ 13 nH . (b) If k = 4,
then γ (H ) ≤ 14 nH . (c) If k = 5, then γ (H ) ≤ 29 nH . We remark that the result of Theorem 15(a) and 15(b) can also be deduced from a result in [26], which states that for k ∈{3, 4}, if every
edge in a graph G without isolated vertices and of order n is contained in a clique Kk , then γ (G) ≤ k1 n.
2.4 Edge Size at Least Three In this section, we present a characterization, due to Henning and Löwenstein [22], of the hypergraphs that achieve equality in the upper bound for the domination number
given in Theorem 15(a). For this purpose, let H1 , H2 , . . . , H15 be the fifteen
Domination and Total Domination in Hypergraphs
Fig. 1 The hypergraphs H1 , H2 , . . . , H15
hypergraphs shown in Figure 1. Let Hunder , standing for underlying hypergraph, be a hypergraph every component of which is isomorphic to a hypergraph Hi for some i ∈ [15]. Each component of Hunder
we call a unit of Hunder . In each unit, we 2-color the vertices with the colors black and white as indicated in Figure 1, and we call the white vertices the link vertices of the unit and the black
vertices the nonlink vertices.
M. A. Henning and A. Yeo
We now add edges between the units as follows. Let H be a hypergraph obtained from Hunder by adding edges of size at least three, called link edges, in such a way that every added edge contains
vertices from at least two units and contains only link vertices. Possibly, H is disconnected or H = Hi for some i ∈ [15]. We call the hypergraph Hunder an underlying hypergraph of H, and we let U
(Hunder ) denote the set of all units in Hunder . Let F≥3 denote the family of all such hypergraphs H. We are now in a position to present the characterization due to Henning and Löwenstein [22] of
the hypergraphs, without isolated vertices and with all edges of size at least three, whose domination number is one-third their order. Theorem 16 ([22]) If H is a hypergraph with all edges of size
at least three and with δ(H) ≥ 1, then γ (H ) ≤ 13 nH with equality if and only if H ∈ F≥3 .
2.5 Edge Size at Least Four It remains an open problem (see Problem 7 in Chapter 4) to characterize the hypergraphs that achieve equality in the upper bound for the domination number given in Theorem
15(b). We remark that due to the interplay between domination and transversals in hypergraphs discussed in Section 2.2, a characterization of hypergraphs H ∈ H4 satisfying γ (H ) = 14 nH builds on a
characterization of the 3uniform hypergraphs H satisfying τ (H ) = 14 (nH + mH ). A characterization of the extremal connected hypergraphs that achieve equality in the Chvátal–McDiarmid Theorem 11
for k = 2 and for all k ≥ 4 is a relatively simple task. As shown in [24], there exist two such hypergraphs when k is even and one such hypergraph when k is odd. Surprisingly the case for k = 3 is
much more challenging. The infinite extremal connected hypergraphs in this case were characterized by Henning and Yeo [27]. However, it remains an open problem to deduce the hypergraphs that achieve
equality in Theorem 15(b) from the characterization in [27].
2.6 Edge Size at Least Five In this section, we present a characterization due to Henning and Löwenstein [25] of the hypergraphs that achieve equality in the upper bound of Theorem 15(c). For this
purpose, let H9 be the hypergraph shown in Figure 2. Let Hunder be a hypergraph Fig. 2 The hypergraph H9
Domination and Total Domination in Hypergraphs
every component of which is isomorphic to H9 . Let H be a hypergraph obtained from Hunder by adding edges of size at least five, called link edges, in such a way that every added edge contains only
vertices of degree 2 in Hunder . Possibly, H is disconnected or H = H9 . We call the hypergraph Hunder an underlying hypergraph of H. Let F≥5 denote the family of all such hypergraphs H. We are now
in a position to present the characterization due to Henning and Löwenstein [25] of the hypergraphs with no isolated edge and with all edges of size at least five whose domination number is
two-ninths their order. Theorem 17 ([25]) If H is a hypergraph with all edges of size at least five and with δ(H) ≥ 1, then γ (H ) ≤ 29 nH with equality if and only if H ∈ F≥5 .
2.7 A Characterization of Hypergraphs Achieving Equality in Theorem 14 In this section, we present a characterization due to Henning and Löwenstein [25] of the hypergraphs that achieve equality in
the upper bound for the domination number given in Theorem 14. For this purpose, we first define some special hypergraphs. For k ≥ 4, let Ek denote the k-uniform hypergraph on k vertices with exactly
one edge. The hypergraph E4 is illustrated in Figure 3. For k ≥ 4, the k-uniform hypergraph Tk is defined in [25] as follows. Let A, B, C, and D be vertexdisjoint sets of vertices with |A| = k/2, |B|
= |C| = k/2, and |D| = k/2 − k/2. In particular, if k is even, the set D = ∅, while if k is odd, the set D consist of a singleton vertex. Let Tk denote the k-uniform hypergraph with V (Tk ) = A ∪ B ∪
C ∪ D and with E(Tk ) = {e1 , e2 , e3 }, where V (e1 ) = A ∪ B, V (e2 ) = A ∪ C, and V (e3 ) = B ∪ C ∪ D. The hypergraphs T4 and T5 are illustrated in Figure 3. For odd k ≥ 5, the hypergraph Tk∗ is
defined in [25] as follows. Let A, B, and C be vertex-disjoint sets of vertices with |A| = |B| = (k + 1)/2 and |C| = (k − 1)/2. Let Tk∗ denote the hypergraph with V (Tk∗ ) = A ∪ B ∪ C and with E(Tk∗
) = {e1 , e2 , e3 }, where V (e1 ) = A ∪ B, V (e2 ) = A ∪ C, and V (e3 ) = B ∪ C. The hypergraph T5∗ is illustrated in Figure 3. We note that every edge in Tk∗ has size at least k.
Fig. 3 The hypergraphs E4 , T4 , T5 , and T5∗
M. A. Henning and A. Yeo
Fig. 4 The hypergraphs D5 , D6 , and D6∗
The expanded hypergraph, abbreviated expa(H), of a hypergraph H is defined in [25] as the hypergraph obtained from H by expanding every edge in H by adding to it one new vertex, where all added
vertices have degree 1 in expa(H). Thus for every edge e ∈ E(H), if ve denotes the new vertex added to e, where ve =vf for edges e=f in H, then expa(H) has edge set {e ∪{ve } : e ∈ E(H)} and vertex
set V (H ) ∪
{ve }.
e∈E(H )
For k ≥ 5, let Dk = expa(Tk−1 ). The hypergraphs D5 and D6 are illustrated in ∗ ). The Figure 4. We note that Dk is k-uniform. For even k ≥ 6, let Dk∗ = expa(Tk−1 ∗ ∗ hypergraph D6 is illustrated in
Figure 4. We note that every edge in Dk is of size at least k. The authors in [25] define a special family Dk of hypergraphs as follows. For odd k ≥ 5, they define Dk = {Ek , Dk }, and for even k ≥
6, they define Dk = {Ek , Dk , Dk∗ }. We are now in a position to present the characterization given in [25] of the hypergraphs that achieve equality in the upper bound for the domination number
given in Theorem 14. Theorem 18 ([25]) For k ≥ 3, if H is a hypergraph with all edges of size at least k and with δ(H) ≥ 1, then nH +
γ (H ) ≤
k−3 2
3(k−1) 2
with equality if and only if H ∈ Dk .
2.8 General Setting In the previous sections, we determined upper bounds of the form anH +bmH on the domination number of k-uniform hypergraphs. In particular, we proved in Lemma 5 that b ≥ 0 and a >
− bk must hold in every valid upper bound. In this setting,
Domination and Total Domination in Hypergraphs
we next present a general way to formulate the problem as described by Bujtás, Henning, and Tuza [8]. Problem 1 ([8]) Given an integer k ≥ 2, determine the shape of the surface Γ k (x, y, z), which
is the subset of Dk = {(x, y, z) | y ≥ 0 ∧ x > −y/k} ⊂ R3 defined by the rule γ (H ) . xn H ∈Hk H + ymH
z = sup
In other words, for k given, determine z = z(x, y) as a function of x and y. The hypergraph H with nH = k vertices and mH = 1 edge of size k has γ (H) = 1. Hence, we have the following simple general
lower bound first observed in [8]. Observation 19 ([8]) For every integer k ≥ 2 and reals y ≥ 0 and x > −y/k, z(x, y) ≥
1 . kx + y
Bujtás, Henning, and Tuza [8] gave a complete solution for Problem 1 for k ∈{2, 3}, showing that Observation 19 holds with equality in this case. Theorem 20 ([8]) For k ∈{2, 3}, the surface Γ k (x,
y, z) is determined by z(x, y) =
1 . kx + y
An equivalent formulation of Theorem 20 gives us the following general upper bound on the domination number of a k-uniform hypergraph for k ∈{2, 3}. Theorem 21 ([8]) For k = 2 and k = 3, the bound γ
(H ) ≤ anH + bmH is valid for every k-uniform hypergraph H, which does not contain isolated vertices, if and only if both ka + b ≥ 1 and b ≥ 0 hold. For example, to illustrate Theorem 21, let H be a
k-uniform hypergraph without isolated vertices. For k ∈{2, 3} and taking (a, b) = ( k1 , 0), we have that γ (H ) ≤ 1 k mH . For k = 2, this yields the classical result due to Ore [40], while for k =
3, this yields the result of Theorem 15(a). As remarked in [8], the general description of k (x, y, z) in Problem 1 appears to be a rather hard problem, already for k = 4. Indeed it remains an open
problem to give a complete solution for Problem 1 for any value of k ≥ 4. In particular, it remains an open problem to determine whether the result of Theorem 21 also holds for k = 4. We know that
the result of Theorem 21 does not hold for k = 5. We pose two open questions in these cases when k = 4 and k = 5 in Section 4.
M. A. Henning and A. Yeo
2.9 Hypergraphs with Given Domination Number In this section, we address the problem of finding the minimum number of vertices that a connected k-uniform hypergraph with high domination number must
contain. Since all isolated vertices always are contained in every dominating set, we can simply delete them and restrict our attention to hypergraphs without isolates. Let n(k, γ ) be the minimum
number of vertices that a k-uniform hypergraph with no isolated vertices must contain if its domination number is at least γ .
The Case γ = 1
We observe that the k-uniform hypergraph with exactly one edge, which we denoted by Ek in Section 2.7, has order k and is the smallest such hypergraph with γ = 1. We state this trivial case formally
as follows. Observation 22 For k ≥ 2, we have n(k, 1) = k.
The Case γ = 2
By a classical result due to Ore [40], if H is 2-uniform hypergraph without isolated vertices, then γ (H ) ≤ 12 nH . By Theorem 15 for k ∈{3, 4}, if H is a k-uniform hypergraph without isolated
vertices, then γ (H ) ≤ k1 nH . In particular, if k ∈{2, 3, 4} and γ (H) = 2, then nH ≥ 2k. If k = 2, then let F2 be the 4-cycle, and let F3 and F4 be the hypergraph shown in Figure 5(a) and 5(b),
respectively. For k ∈{2, 3, 4}, if H = Fk , then γ (H) = 2 and nH = 2k. Hence, we have the following result for small k. Observation 23 For k ∈{2, 3, 4}, we have n(k, 2) = 2k. The general case when γ
= 2 was studied by Erd˝os, Henning, and Swart [17], albeit in a graph theory setting. In order to state the result in [17], for each integer r ≥ 2, let Ir be the set of integers in the interval Ir =
[r2 − r + 2, r2 + r + 1]. We note further that if (r − 1)2 ≤ k ≤ (r + 1)2 , then either (r − 1)2 ≤ k ≤ r2 − r + 1, Fig. 5 The hypergraphs F3 and F4
Domination and Total Domination in Hypergraphs Table 1 Values of n(k, 2) for small k
k n(k,2)
in which case k ∈ Ir−1 , or r2 − r + 2 ≤ k ≤ r2 + r + 1, in which case k ∈ Ir , or r2 + r + 2 ≤ k ≤ (r + 1)2 , in which case k ∈ Ir+1 . Hence, each integer k ≥ 4 belongs to a unique interval Ir for
some r ≥ 2. Recall that [r] = {1, . . . , r} and [r]0 = {0, 1, . . . , r}. We note that if k ∈ Ir , then either k = r2 + 1 − i where i ∈ [r − 1]0 or k = r2 + 1 + i where i ∈ [r]. We are now in a
position to state the result in [17]. Theorem 24 ([17]) If k ≥ 4 is an arbitrary integer, then k belongs to a unique interval Ir for some r ≥ 2, and the following holds: n(k, 2) =
(r + 1)2 − i (r + 1)2 + 1 + i
if if
k = r 2 + 1 − i and i ∈ [r − 1]0 k = r 2 + 1 + i and i ∈ [r].
For example, in the special cases when k ∈ I2 = {4, 5, 6, 7} and k ∈ I3 = {8, 9, . . . , 13} we summarize the results of Theorem 24 in Table 1 above. For each integer k ≥ 4, we next present an
example of a k-uniform hypergraph H with γ (H) = 2 and of order nH = n(k, 2) given by the expression in Theorem 24. Let k ≥ 4 be fixed. By our earlier observations, the integer k belongs to a unique
interval Ir for some r ≥ 2, and either k = r2 + 1 − i where i ∈ [r − 1]0 or k = r2 + 1 + i where i ∈ [r]. We consider the two cases in turn. Case 1. k = r2 + 1 − i where i ∈ [r − 1]0 . In this case,
for j ∈ [r + 1], let Aj be a set of vertices defined as follows. If i = 0, let |Aj | = r for j ∈ [r + 1]. If i ∈ [r − 1], let |Aj | =
r −1 r
if if
j ∈ [i] j ∈ [r + 1] \ [i].
Further, let the sets Aj be pairwise disjoint sets, and let A=
B = {v1 , v2 , . . . , vr+1 },
j =1
where the vertices in B are r + 1 additional vertices that do not belong to the set A. Let v be an arbitrary vertex in Ar+1 . Let H be the hypergraph with vertex set V (H) = A ∪ B of order nH = |A| +
|B| = (r + 1)2 − i and with edge set E(H) = {e1 , e2 , . . . , er+1 } where the edge ej for j ∈ [r + 1] is defined as follows: ej =
(A ∪ {vj }) \ (Aj ∪ {v}) (A ∪ {vj }) \ Aj
if if
i ∈ [r − 1] and j ∈ [i] i = 0 or i ∈ [r − 1] and j ∈ [r + 1] \ [i].
M. A. Henning and A. Yeo
By construction, we note that |ej | = r2 + 1 − i = k for all j ∈ [r + 1], and so H is a k-uniform hypergraph. Since no vertex of H dominates the set B, we note that γ (H) ≥ 2. However, if ur ∈ Ar and
ur+1 ∈ Ar+1 {v}, then the set {ur , ur+1 } is an example of a dominating set of H, and so γ (H) ≤ 2. Consequently, H is a k-uniform hypergraph of order nH = (r +1)2 −i without isolated vertices
satisfying γ (H) = 2. Case 2. k = r2 + 1 + i and i ∈ [r]. In this case, for j ∈ [r + 1], let Aj be a set of vertices defined as follows: r + 1 if j ∈ [i + 1] |Aj | = r if i < r and j ∈ [r + 1] \ [i +
1]. Further, let the sets Aj be pairwise disjoint sets, and let A=
B = {v1 , v2 , . . . , vr+1 },
j =1
where the vertices in B are r + 1 additional vertices that do not belong to the set A. Let v be an arbitrary vertex in A1 . Let H be the hypergraph with vertex set V (H) = A ∪ B of order nH = |A| + |
B| = (r + 1)2 + 1 + i and with edge set E(H) = {e1 , e2 , . . . , er+1 } where the edge ej for j ∈ [r + 1] is defined as follows: ej =
(A ∪ {vj }) \ Aj (A ∪ {vj }) \ (Aj ∪ {v})
if if
j ∈ [i + 1] i < r and j ∈ [r + 1] \ [i + 1].
By construction, we note that |ej | = r2 + 1 + i = k for all j ∈ [r + 1], and so H is a k-uniform hypergraph. Since no vertex of H dominates the set B, we note that γ (H) ≥ 2. However, if u1 ∈ A1 {v}
and u2 ∈ A2 , then the set {u1 , u2 } is an example of a dominating set of H, and so γ (H) ≤ 2. Consequently, H is a k-uniform hypergraph of order nH = (r + 1)2 + 1 + i without isolated vertices
satisfying γ (H) = 2.
The Case γ ≥ 3
The case when γ ≥ 3 was addressed by Bujtás, Patkós, Tuza, and Vizer [10] in a more general setting. For an integer s ≥ 1, they define an s-dominating set of a hypergraph H as a dominating set D of H
with the property that every vertex outside the set D has at least s neighbors inside the set D; that is, |NH (v) ∩ D|≥ s for all vertices v ∈ V (H) D, where NH (v) denotes the open neighborhood of
v. The authors in [10] also define an s-tuple dominating set of H as a dominating set D of H with the property that every vertex of H either belongs to D and has at least s − 1 neighbors inside the
set D or belongs outside the set D and has at least s neighbors inside the set D; that is, |NH [v] ∩ D|≥ s for all vertices v ∈ V (H),
Domination and Total Domination in Hypergraphs
where NH [v] denotes the closed neighborhood of v. We note that dominating sets are precisely the 1-dominating sets and 1-tuple dominating sets. The s-domination number γ (H, s) of a hypergraph H is
the minimum size of an s-dominating set in H, and the s-tuple domination number γ × (H, s) of H is the minimum size of an s-tuple dominating set in H. By definition, we have γ (H, s) ≤ γ × (H, s).
For every pair γ and s of integers with γ ≥ s, let n(k, γ , s) denote the minimum number of vertices that a k-uniform hypergraph H with no isolated vertices must have if γ (H, s) ≥ γ holds, and let
n× (k, γ , s) denote the minimum number of vertices that a k-uniform hypergraph H with no isolated vertices must have if γ × (H, s) ≥ γ holds. As observed in [10], we have n× (k, γ , s) ≤ n(k, γ , s)
and n(k, γ ) = n× (k, γ , 1) = n(k, γ , 1). We are now in a position to state the result due to Bujtás, Patkós, Tuza, and Vizer [10]. Theorem 25 ([10]) For integers γ ≥ 2 and s ≥ 1 with γ > s, we
have k + k 1−1/(γ −s+1) ≤ n× (k, γ , s) ≤ n(k, γ , s) ≤ k + (4 + o(1))k 1−1/(γ −s+1) . In the special case in Theorem 25 when s = 1, we have the following result. Theorem 26 ([10]) For integers k ≥ 2
and γ ≥ 2, we have k+k
1− γ1
≤ n(k, γ ) ≤ k + (4 + o(1))k
1− γ1
2.10 Nordhaus–Gaddum-Type Results In this section, we present a Nordhaus–Gaddum-type result for the sum of domination parameters in hypergraphs and their complements. Given a hypergraph H = (V, E),
the complement H of H is the hypergraph H = (V , E) where E = {V \ e : e ∈ E}. In 2008, Hedetniemi et al. [20] defined the disjoint domination number γ γ (G) of a graph G as the minimum sum of the
cardinalities of two disjoint dominating sets in G. The disjoint domination number γ γ (H) of a hypergraph H is defined analogously as the minimum sum of the cardinalities of two disjoint dominating
set in H; that is, γ γ (H )= min{|D1 |+|D2 | : D1 , D2 are minimal dominating sets in H with D1 ∩ D2 =∅}.
Recall that a γ -set of a hypergraph H is a dominating set of H of cardinality γ (H). An inverse dominating set with respect to a given γ -set D of H is a dominating set D
of H such that D ⊆ V (H) D. The inverse domination number γ −1 (H) is defined as γ −1 (H ) = min{|D | : D is an inverse dominating set with respect to some γ -set D of H }.
M. A. Henning and A. Yeo
We remark that the inverse domination number of a graph was first defined in 1991 by Kulli and Sigarkanti [37]. Acharya [4] posed the problem of finding best possible lower and upper bounds for γ γ
(H ) + γ γ (H ). This problem was subsequently solved by Jose and Tuza [34], who proved an even stronger statement for the upper bound. Theorem 27 ([34]) For every integer n ≥ 4, if H is a hypergraph
of order n, then 4 ≤ γ γ (H ) + γ γ (H ) ≤ γ (H ) + γ −1 (H ) + γ (H ) + γ −1 (H ) ≤ max{8, n + 2}, and the bounds are tight. That the bounds of Theorem 27 are tight may be seen as follows. For the
lower bound, Jose and Tuza [34] constructed a hypergraph H = (V, E) with n vertices as follows. Let V = (V1 , V2 , V3 , V4 ) be a partition of the set V into four nonempty sets, and let E = {Vi ∪ Vj
: 1 ≤ i < j ≤ 4}. We note that H and H are isomorphic hypergraphs, and all the vertices are adjacent to each other in both H and H . Hence, γ γ (H ) = γ γ (H ) = 2. Therefore the lower bound is
attainable for all n ≥ 4. To prove tightness in the upper bound, suppose first that 4 ≤ n ≤ 6. In this case, Jose and Tuza [34] constructed a hypergraph H = (V, E) with n vertices by once again
taking a partition of the set V = (V1 , V2 , V3 , V4 ) into four nonempty sets, but now letting E = {V1 ∪ V2 , V2 ∪ V3 , V3 ∪ V4 }. The edge set of H = (V , E) is E = {V1 ∪ V2 , V1 ∪ V4 , V3 ∪ V4 }.
Thus, γ γ (H ) = γ γ (H ) = 4. Suppose next that n ≥ 7. In this case, Jose and Tuza [34] let H be a double star S((n − 2)/2, (n − 2)/2), which is a tree with two (adjacent) nonleaf vertices one of
which has (n − 2)/2 leaf neighbors and the other (n − 2)/2) leaf neighbors. Let x and y be the two nonleaf vertices, and let X and Y be the set of leaf neighbors of x and y, respectively. We note
that there are exactly four minimal dominating sets, namely D1 = {x}∪ Y , D2 = {y}∪ X, D3 = {x, y}, and D4 = X ∪ Y , forming two disjoint pairs, namely (D1 , D2 ) and (D3 , D4 ). Since both pairs
partition V (H), we note that γ γ (H) = n. Further, we note that γ γ (H ) = 2. Thus, γ γ (H ) + γ γ (H ) = n + 2.
2.11 Equality of Domination and Transversal Numbers Every transversal in a hypergraph H without isolated vertices is a dominating set in H, implying that γ (H) ≤ τ (H) is valid for every such
hypergraph H. Arumugam, Jose, Bujtás, and Tuza [7] investigated the hypergraphs H without isolated vertices satisfying γ (H) = τ (H). We first consider the special case when H is a 2-uniform
hypergraph, that is, when H is a graph. A stem, also called a support vertex in the literature, is a vertex in a graph G that is adjacent to a vertex of degree 1. Let Stem(G) denote
Domination and Total Domination in Hypergraphs
the set of stems in G. Let S(G) denote the graph obtained from the graph G by deleting all edges contained entirely in Stem(G). We note that the transformation S does not create isolated vertices,
unless G contains a component isomorphic to K2 . Arumugam et al. [7] provided the following characterization of connected graphs satisfying γ (G) = τ (G). Theorem 28 ([7]) For a connected graph G of
order at least 3, γ (G) = τ (G) holds if and only if there exists a bipartition (A, B) of S(G) such that Stem(G) ⊆ A, and moreover, for every pair u, v ∈ A Stem(G), if u and v have some common
neighbor, then they have at least two common neighbors of degree two. As a consequence of the characterization given in Theorem 28, Arumugam et al. [7] showed that graphs G without isolated vertices
and with γ (G) = τ (G) can be recognized in polynomial time. Theorem 29 ([7]) It can be decided in time ⎛ O⎝
⎞ (dG (v))2 ⎠
v∈V (G)
whether an arbitrary graph G without isolated vertices satisfies γ (G) = τ (G). In contrast, Arumugam et al. [7] proved that the corresponding problem is NPhard for hypergraphs, even if edges of size
greater than 3 are excluded. Theorem 30 ([7]) It is NP-hard to decide whether a generic input linear hypergraph H of rank 3 and of minimum degree 1 satisfies γ (H) = τ (H). As remarked in [7], “it is
very likely that the problem of testing γ (H) = τ (H) is harder than any problem in NP.” Arumugam et al. [7] also present several structural results on hypergraphs in which each subhypergraph H
without isolated vertices fulfills the equality γ (H ) = τ (H ). They also investigate hypergraphs for which the equality γ (H) = τ (H) holds hereditarily. That is, the property is required not only
for the hypergraph H itself but also for all of its subhypergraphs or for all of its induced subhypergraphs. For this purpose, they define the following notions: (P1) For a hypergraph H, the equality
γ = τ hereditarily holds if every subhypergraph H of H satisfies γ (H ) = τ (H ). (P2) For a hypergraph H, the equality γ = τ induced-hereditarily holds if every induced subhypergraph H of H
satisfies γ (H ) = τ (H ). By definition, property (P1) implies property (P2). Arumugam et al. [7] prove that in fact properties (P1) and (P2) are equivalent. Theorem 31 ([7]) For a hypergraph H, the
equality γ = τ holds hereditarily if and only if it holds induced-hereditarily.
M. A. Henning and A. Yeo
The following notions are also defined in [7]: (P3) A hypergraph H is minimal for γ < τ if γ (H) < τ (H), but for every proper subhypergraph H of H (without isolated vertices) the equality γ (H ) = τ
(H ) holds. (P4) A hypergraph H is induced-minimal for γ < τ if γ (H) < τ (H), but for every proper induced subhypergraph H of H (without isolated vertices) the equality γ (H ) = τ (H ) holds. By
definition, property (P3) implies property (P4). However, properties (P3) and (P4) are not equivalent since there exist hypergraphs, which are induced-minimal but not minimal for γ < τ . Arumugam et
al. [7] prove that for any given integer k ≥ 2, the number of hypergraphs of rank k that are minimal or induced-minimal for γ < τ is bounded. Theorem 32 ([7]) For every fixed k ≥ 2, the following
holds: (a) There exist only finitely many hypergraphs of rank k, which are minimal for γ < τ. (b) There exist only finitely many hypergraphs of rank k, which are induced-minimal for γ < τ .
2.12 The Relationship Between Domination and Matching In this section, we present an upper bound on the domination number of a uniform hypergraph in terms of its matching number. A matching in a
hypergraph is a set of disjoint edges. Thus, if M is a matching in a hypergraph H, then e ∩ f = ∅ for every pair of edges e and f in M. The matching number, α (H), of a hypergraph H is the maximum
size of a matching in H. In order to cover every edge of a hypergraph H, we note that τ (H) ≥ α (H) since a transversal of H must contain at least one vertex from every edge in a maximum matching in
H. If H is a k-uniform hypergraph, then the union of the edges of a maximum matching of H forms a transversal of H, implying that τ (H) ≤ kα (H). We state this formally as follows. Observation 33 If
H is a k-uniform hypergraph, then τ (H) ≤ kα (H). A hypergraph is k-partite if its vertex set can be partitioned into k sets such that every edge contains exactly one vertex from each of these
partite sets. In particular, a k-partite hypergraph is a k-uniform hypergraph. A long-standing open problem, known as Ryser’s conjecture, states that if H is a k-partite hypergraph, then τ (H) ≤ (k −
1)α (H). When k = 2, this is the classical theorem of König. When k = 3, Ryser’s conjecture was proven by Aharoni [5]. However, the conjecture remains open for k ≥ 4. Motivated by Ryser’s conjecture,
Kang, Li, Dong, and Shan [36] gave the following Ryser-like relation between the domination number and matching number
Domination and Total Domination in Hypergraphs
of a uniform hypergraph. The proof is based on the approach presented in the proof of Theorem 6 by Bujtás et al. [9]. Theorem 34 ([36]) For k ≥ 2, if H is a k-uniform hypergraph without isolated
vertices, then γ (H) ≤ (k − 1)α (H). Proof. Let H be the hypergraph obtained from H by successively deleting edges of H that do not contain any vertices of degree 1 in the resulting hypergraph at
each stage. We note that H is a k-uniform hypergraph with nH = nH vertices and mH ≤ mH edges. When H is transformed into H , isolated vertices cannot arise. Since removing edges cannot decrease the
domination number, we note that γ (H) ≤ γ (H ). We also note that removing edges cannot increase the matching number, and so α (H ) ≤ α (H). Moreover, every edge of H contains at least one vertex of
degree 1, and so τ (H ) = γ (H ). Consequently, τ (H ) ≥ γ (H). Deleting exactly one vertex of degree 1 from each edge of H , we obtain a (k − 1)-uniform hypergraph H
of order nH − mH and of size mH . When constructing H
from H , the transversal number and the matching number of H
remains unchanged, that is, τ (H
) = τ (H ) and α (H
) = α (H ). These observations, together with the result of Observation 33, imply that γ (H ) ≤ τ (H ) = τ (H
) ≤ (k − 1)α (H
) = (k − 1)α (H ) ≤ (k − 1)α (H ). This establishes the desired upper bound, namely γ (H) ≤ (k − 1)α (H). 2 To show that the upper bound is achievable, we recall the definition of a finite projective
plane. For q = pn ≥ 2, where some prime p and some integer n ≥ 1, a finite projective plane of order q, denoted by PG(q), consists of a set of q2 + q + 1 points and the same number of lines, having
the following properties: • • • • •
Every line contains q + 1 points. Every point lies in q + 1 lines. Any two distinct points lie on a unique line. Any two distinct lines intersect in exactly one point. There are four points such that
no line is incident with more than two of them.
A finite projective plane may be considered as a hypergraph, whose vertex set is the set of points and whose edge set is the set of lines of the plane. The hypergraph associated with a finite
projective plane PG(2, q) is a (q + 1)-uniform hypergraph on q2 + q + 1 vertices in which every two edges intersect in exactly one vertex. Kang, Li, Dong, and Shan [36] showed that when k − 2 is a
prime power, the upper bound in Theorem 34 on the domination number is tight. Suppose that k − 2 is a prime power and consider the hypergraph H associated with a finite projective plane PG(k − 2). By
our earlier observations, H is a (k − 1)uniform hypergraph in which every two edges intersect in exactly one vertex. We note that α (H) = 1. Let H be the k-uniform hypergraph obtained from H by
adding to it mH new vertices of degree 1, one vertex for each edge in H. Thus, each edge of H (of size k − 1) is extended to an edge of size k by adding to it one new
M. A. Henning and A. Yeo
vertex of degree 1. We note that H has nH = nH + mH vertices and mH = mH edges. Further we note that τ (H ) = τ (H) and α (H ) = α (H). Since every edge of H contains a vertex of degree 1, we have γ
(H ) = τ (H ) = τ (H). Since H is the hypergraph associated with a finite projective plane, the set of minimum transversals in H consists precisely of the set of edges of H. Thus, since H is a (k −
1)-uniform hypergraph, each τ -transversal in H is an edge of H, which has size k − 1, and so τ (H) = k − 1. Hence, γ (H ) = τ (H) = k − 1 = (k − 1) · 1 = (k − 1)α (H ). Thus, when k − 2 is a prime
power, the upper bound in Theorem 34 on the domination number is tight. Shan, Dong, Kang, and Li [41] observed that the inequality in Theorem 34 still holds for arbitrary hypergraphs of rank k. We
state this formally as follows. Corollary 35 ([36]) For k ≥ 2, if H is a hypergraph of rank k without isolated vertices, then γ (H) ≤ (k − 1)α (H). Shan, Dong, Kang, and Li [41] give a complete
characterization of the extremal hypergraphs H of rank 3 without isolated vertices satisfying γ (H) = 2α (H). As remarked in [36], for k ≥ 4 a constructive characterization of hypergraphs of rank k
without isolated vertices satisfying γ (H) = (k − 1)α (H) seems difficult to obtain, even in the special case when H is an intersecting hypergraph, that is, hypergraphs in which every two edges have
a nonempty intersection. We note that if H is an intersecting hypergraph, then α (H) = 1. Dong, Shan, Kang, and Li [14] present structural properties of intersecting hypergraphs H with no isolated
vertices and with rank k satisfying the equality γ (H) = k − 1. Their main result is that all linear intersecting hypergraphs H with no isolated vertices and with rank 4 such that γ (H) = 3 can be
constructed from the Fano plane, where a linear hypergraph is one in which every two edges intersect in at most one vertex. It remains, however, an open problem to characterize the nonlinear
intersecting hypergraphs H with no isolated vertices and with rank 4 satisfying γ (H) = 3. Li, Kang, Shan, and Dong [39] show that all the 5-uniform linear intersecting hypergraphs H with no isolated
vertices satisfying γ (H) = 4 are generated by the finite projective plane of order 3.
3 Total Domination in Hypergraphs In this section, we focus on total domination in hypergraphs. We proceed as follows. We first establish a relationship between the total transversal number and the
total domination number of uniform hypergraphs due to Bujtás, Henning, Tuza, and Yeo [9]. Using this interplay between total transversals and total domination in hypergraphs, we prove tight
asymptotic upper bounds on the total transversal number in terms of the number of vertices, the number of edges, and the edge size. We shall need the following definitions given in [9].
Domination and Total Domination in Hypergraphs
Definition 3 ([9]) For an integer k ≥ 2, let Hk be the class of all k-uniform hypergraphs containing no isolated vertices or isolated edges or multiple edges. Further, for k ≥ 3 let Hk∗ consist of
all hypergraphs in Hk that have no two edges intersecting in k − 1 vertices. We note that Hk∗ is a proper subclass of Hk . Definition 4 ([9]) For an integer k ≥ 2, let τt (H ) . n H ∈Hk H + mH
bk = sup
We remark that it is not known if the supremum in Definition 4 is a maximum. Bujtás, Henning, Tuza, and Yeo [9] proved the following upper bounds on the total domination number of a uniform
hypergraph in terms of its total transversal number, order, and size. Theorem 36 ([9]) For k ≥ 3, if H ∈ Hk , then % 2 nH . , bk−1 γt (H ) ≤ max k+1 In view of Theorem 36, it is of interest to
determine the values of bk for k ≥ 2. The value of bk for small k was determined in [9]. Theorem 37 ([9]) The following hold: (a) b2 =
and b3 = 13 .
(b) b4 ≤ 13 . (c) bk ≤
for all k ≥ 5.
Henning and Yeo [32] continued the study of total transversals in hypergraphs and proved the following result. Theorem 38 ([32]) The following hold: (a) b4 = 27 . (b) b6 ≤ 14 . (c) bk ≤
for all k ≥ 7.
By Theorems 37 and 38, we observe that bk−1 ≤
2 k+1
for k ∈ {3, 4, 5, 6, 7, 8}.
Hence as a consequence of Theorem 36 and Theorem 37, and the well-known fact (see, [13]) that if H ∈ H2 , then γt (H ) ≤ 23 nH , we have the following result.
M. A. Henning and A. Yeo
Theorem 39 ([9, 32]) For k ∈ [8] [1], if H ∈ Hk , then γt (H ) ≤
2 nH , k+1
and this bound is sharp. That the upper bound in Theorem 39 is sharp may be seen as follows. Let Fk ⊂ Hk be the subfamily of hypergraphs in Hk that can be obtained as follows. Let F be an arbitrary
hypergraph in the family Hk . For each vertex v in F, add k new vertices v1 , v2 , . . . , vk and two new k-edges ev = {v, v1 , . . . , vk−1 } and fv = {v1 , v2 , . . . , vk }. Let H ∈ Hk denote the
resulting k-uniform hypergraph. As observed in [9], every total dominating set of H must contain at least two vertices from ev ∪ fv for every vertex v ∈ V (F), implying that γt (H ) ≥ 2nF . The set
∪v ∈ V (F) {v, v1 }, where the union is taken over all vertices v ∈ V (F), forms a total dominating set of H, and so γt (H ) ≤ 2nF . Consequently, γt (H ) = 2nF . Since nH = (k + 1)nF , this yield
the following result from [9], implying that the upper bound in Theorem 39 is tight. 2 Observation 40 ([9]) For k ≥ 2, if H ∈ Fk , then γt (H ) = k+1 nH . Bujtás et al. [9] proved the following
result, which is a strengthening of the upper bound of Theorem 36 if we restrict the edges to intersect in at most k − 2 vertices. Theorem 41 ([9]) For k ≥ 4, if H ∈ Hk∗ , then % 2 γt (H ) ≤ max nH .
, bk−1 k+2 Corollary 42 ([9]) For k ≥ 4, if H ∈ Hk∗ , then γt (H ) ≤ 13 nH . Bujtás et al. [9] established the following tight asymptotic bound on bk for sufficiently large k. Theorem 43 ([9]) For
all k ≥ 2, bk = (1 + o(1))
ln(k) . k
2 Theorem 43 implies that the inequality bk−1 ≤ k+1 is not true when k is large enough. By definition, γ t (H) ≥ γ (H) for every hypergraph H without isolated vertices. Hence as a consequence of
Theorem 36, Theorem 43, and Corollary 12 (established in Section 2.3), we have the following result.
Theorem 44 ([9]) For all k ≥ 3, γt (H ) ln(k) . = (1 + o(1)) nH k H ∈Hk sup
We remark that Theorem 44 implies that Theorem 39 is not true for large k.
Domination and Total Domination in Hypergraphs
In view of Theorem 36, it is of interest to determine the value of bk for k ≥ 2. In Theorems 37 and 38, we have that b2 = 25 , b3 = 13 , and b4 = 27 . Further, b5 ≤ 27 , 2 for k ∈{3, 4, 5, 6, 7, 8}.
b6 ≤ 14 , and bk ≤ 29 for all k ≥ 7. Thus, bk−1 ≤ k+1
4 Conjectures and Open Problems This chapter presents an overview of research on domination and total domination in hypergraphs. Results not presented in this chapter can be found, for example, in
[1, 2, 18, 19, 21, 31]. We close this chapter with a conjecture and a list of open problems for future research. Problem 2 Characterize the hypergraphs H with all edges of size at least 4 and with δ
(H) ≥ 1 that achieve equality in the upper bound γ (H ) ≤ 14 nH given in Theorem 15. Problem 3 ([8]) Given an integer k ≥ 4, determine the shape of the surface Γ k (x, y, z), which is the subset of
Dk = {(x, y, z) | y ≥ 0 ∧ x > −y/k} ⊂ R3 defined by the rule γ (H ) . xn H ∈Hk H + ymH
z = sup
In other words, for k ≥ 4 given, determine z = z(x, y) as a function of x and y. Problem 4 Prove or disprove: The bound γ (H ) ≤ anH + bmH is valid for every 4-uniform hypergraph H without isolated
vertices, if and only if both 4a + b ≥ 1 and b ≥ 0 hold. Problem 5 Prove or disprove: The bound γ (H ) ≤ anH + bmH is valid for every 5-uniform hypergraph H without isolated vertices, if and only if
both 92 a + b ≥ 1 and b ≥ 0 hold. Problem 6 Determine the exact value of n(k, 3) for all k ≥ 5. From our earlier results, we note that for k ∈{2, 3, 4}, we have n(k, 3) = 3k. Conjecture 1 Prove
Ryser’s conjecture when k ≥ 4; that is, prove that if k ≥ 4 and H is a k-partite hypergraph, then τ (H) ≤ (k − 1)α (H). Problem 7 Close the gap between the upper and lower bounds of Theorem 26.
Problem 8 ([9]) Determine the exact value of bk for k ≥ 5. Problem 9 ([9]) Determine the smallest value of k for which bk−1 >
2 k+1 .
M. A. Henning and A. Yeo
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Domination and Total Domination in Hypergraphs
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Domination in Chessboards Jason T. Hedetniemi and Stephen T. Hedetniemi
1 Introduction In this chapter we consider chessboards, which are finite, uniform (or regular) tessellations of the plane into identical cells or squares. With square cells, we consider either the
more common n × n chessboards, with n rows and n columns, or the rectangular m × n chessboards. But we briefly consider these variations: (i) triangular, or diamond, shaped boards, (ii) sawtooth
square boards, (iii) torus (or circular) boards (where the leftmost squares are adjacent to the corresponding rightmost squares and the topmost squares are adjacent to the corresponding bottommost
squares), and (iv) three-dimensional square boards. For each type of board, we consider the following chess pieces: (i) queens, (ii) kings, (iii) rooks, (iv) bishops, and (v) knights, each of which
defines a graph, whose vertices correspond one-to-one with the squares of the board, and two squares are considered to be adjacent if and only if a piece of a given type can move from one square to
the other square in one (legal) move. Thus, we define: (i) the queens graph Qn , where two squares are adjacent if and only if they lie on a common row, column, or diagonal; (ii) the kings graph Kn ,
where two squares are adjacent if and only if they are next to each other on a common row, column, or diagonal; (iii) the rooks graph Rn , where two squares are adjacent if and only if they lie on a
common row or column;
J. T. Hedetniemi () Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458, USA S. T. Hedetniemi School of Computing, Clemson University, Clemson, SC, USA e-mail: [email protected] ©
The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. W. Haynes et al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://doi.org/10.1007/
J. T. Hedetniemi and S. T. Hedetniemi
(iv) the bishops graph Bn , where two squares are adjacent if and only if they lie on a common diagonal; and (v) the knights graph Nn , where two squares are adjacent if and only if you can move from
one square to the other by a two-step process of moving either one square in one direction followed by two squares in a perpendicular direction, or similarly, by moving two squares in one direction
followed by one square in a perpendicular direction. Before continuing we should comment that the notation Kn is used throughout graph theory to denote the complete graph of order n. Similarly, the
notation Qn is used throughout graph theory to denote the n-dimensional cube graph. However, in the context of chessboards, Kn denotes the n × n kings graph and Qn denotes the n × n queens graph. For
each chessboard graph, Qn , Kn , Rn , Bn , and Nn , we consider what is known about the values of the following seven graph theory parameters, the first six of which form what is known as the
Domination Chain of inequalities: ir(G) ≤ γ (G) ≤ i(G) ≤ α(G) ≤ (G) ≤ I R(G), and the seventh parameter is the total domination number, γ t (G). These parameters are defined in the Glossary in this
volume. But with respect to chessboard problems these can be defined as follows: (i) the lower irredundance number ir(G) equals the minimum number of pieces of one type that can be placed on a set S
of squares so that every piece attacks or occupies a square that is not attacked by any other piece in S and no additional piece can be added to S which preserves this property. (ii) the domination
number γ (G) equals the minimum number of pieces of one type that can be placed on a set S of squares so that all other squares are attacked by a piece in S. (iii) the independent domination number i
(G) equals the minimum number of pieces of one type that can be placed on a set S of squares so that all other squares are attacked by a square in S, and no two pieces in S attack each other. (iv)
the independence number α(G) equals the maximum number of pieces of one type that can be placed on a set S of squares, so that no two pieces in S attack each other. (v) the upper domination number
(G) equals the maximum number of pieces of one type that can be placed on a set S of squares so that all squares are attacked by a piece in S, and every piece attacks or occupies a square not
attacked by another piece in S. (vi) the upper irredundance number IR(G) equals the maximum number of pieces of one type that can be placed on a set S of squares so that every piece attacks or
occupies a square that is not attacked by any other piece in S. (vii) the total domination number γ t (G) equals the minimum number of pieces that can be placed on a set S of squares so that all
other squares are attacked by a piece in S and every piece in S is attacked by another piece in S.
Domination in Chessboards
The focus on these seven domination parameters is not to suggest that they are the only four types of domination worth studying with respect to chessboards. Indeed, more than 50 types of domination
have been defined and studied. But when applied to chessboards, these are the major types of domination that have been studied. We add in closing this section that several papers have studied queens
domination, when queens can only be placed on (i) the major diagonal, (ii) the column nearest the center column, or (iii) the border, or outermost, squares of the chessboard.
2 Historical Origins The independence number α(G), the independent domination number i(G), the domination number γ (G), and the total domination number γ t (G), although not defined at the time as
formal graph theory parameters, are all considered in the early mathematical studies of chessboard problems. This is documented by W. W. Rouse Ball in his book Mathematical Recreations and Problems
of Past and Present Times [81], published in 1892, and by P. J. Campbell in his 1977 paper entitled “Gauss and the eight queens problem: a study in miniature of the propagation of historical error”
[30]. In the book by Rouse Ball and Coxeter, Mathematical Recreations and Essays [79], we find the following passage on page 166: “One of the classical problems connected with the chessboard is the
determination of the number of ways in which eight queens can be placed on a chessboard (or, more generally, in which n queens can be placed on a board of n2 cells) so that no queen can take any
other. This was proposed originally by Franz Nauck in 1850.” Actually, Ball was in error here, as the problem was originally stated by a chess player, Max Bezzel [5] in 1848 (this is discussed in
Campbell’s paper, mentioned above). But Dr. Franz Nauck [73] is given credit, by Campbell, for being the first person to show that one can always place n non-attacking queens on an order n board.
Thus, Nauck can be given credit for showing that the vertex independence number of the queens graph Qn is n, that is, α(Qn ) = n. On page 119 [79], we find the following: “MAXIMUM PIECES PROBLEM. The
Eight Queens Problem suggests the somewhat analogous question of finding the maximum number of kings - or more generally of pieces of one type - which can be put on a board so that no one can take
any other, and the number of solutions possible in each case.” It is clear that Ball had in mind the general idea of independent sets in graphs (non-attacking chess pieces of one type) and
particularly of finding maximum independent sets, hence α(G). Ball notes the following values for 8 × 8 chessboards: α(Q8 ) = 8 α(R8 ) = 8
J. T. Hedetniemi and S. T. Hedetniemi
α(B8 ) = 14 α(K8 ) = 16 α(N8 ) = 32. On page 119 [79], Ball continues: “MINIMUM PIECES PROBLEM. Another problem of a somewhat similar character is the determination of the minimum number of kings -
or more generally pieces of one type - which can be put on a board so as to command or occupy all the cells.” It is clear that Ball had in mind the general idea of dominating sets in graphs, and
particularly of finding minimum dominating sets, hence γ (G). Ball notes the following values (for example, cf. Figure 1 which shows a minimum dominating set of 3 queens on Q6 ): γ (Q8 ) = 5, γ (Q7 )
= 4, γ (Q6 ) = 3, γ (Q5 ) = 3, γ (Q4 ) = 2 γ (B8 ) = 8 γ (R8 ) = 8 γ (K8 ) = 9 γ (N8 ) = 12. On page 120 [79], Ball continues: “Jaenisch [43] proposed also the problem of the determination of the
minimum number of queens which can be placed on a board of n2 cells so as to command all the unoccupied cells, subject to the restriction that no queen shall attack the cell occupied by any other
queen.” Thus, de Jaenisch should be given credit for the idea of the independent domination number i(G) in graphs. Ball gives the following values of i(Qn ) (for example, cf. Figure 1, which shows a
minimum independent dominating set of 4 queens on Q7 ):
Fig. 1 γ (Q6 ) = 3 and i(Q7 ) = 4
Domination in Chessboards
Fig. 2 A minimum total dominating set of 12 kings on K8
i(Q4 ) = 3, i(Q5 ) = 3, i(Q6 ) = 4, i(Q7 ) = 4, i(Q8 ) = 5. On p.120 [79], Ball continues: “A problem of the same nature would be the determination of the minimum number of queens (or other pieces)
which can be placed on a board so as to protect one another and command all the unoccupied cells.” Thus, Ball effectively defines the concept of the total domination number of a graph, something that
would not be formally defined graph theoretically until 1980 by Cockayne, Dawes, and Hedetniemi [38]. Ball notes the following values for 8 × 8 chessboards, where (i, j) indicates a piece placed on
the cell in row i and column j: γ t (Q8 ) = 5: (2,4),(3,4),(4,4),(5,4),(8,4) γ t (R8 ) = 8 γ t (B8 ) = 10: (2,4),(2,5),(3,4),(3,5),(4,4),(4,5),(6,4),(6,5),(7,4),(7,5) γ t (N8 ) = 14: (3,2),(3,3),
(3,6),(3,7),(4,3),(4,4),(4,5),(4,6),(6,3),(6,4),(6,5),(6,6), (7,3),(7,6) γ t (K8 ) = 16. It is interesting to note that Ball did not discuss the total domination numbers of kings graphs. The kings
total domination number of the 8 × 8 chessboard was given as 12 by Garnick and Nieuwejaar in 1995 [56]. Although the authors did not present a solution, Figure 2 shows a simple solution.
3 Early Chessboard Domination The literature on chessboard domination problems is far greater than can be reported in this chapter. We therefore make no claims of being comprehensive in covering
J. T. Hedetniemi and S. T. Hedetniemi
Fig. 3 Dudeney [46] p. 84
this literature. But certain publications stand out as being significant sources of information, either historically or in terms of key results, beginning with the book originally published in 1892
by W. W. Rouse Ball [81], which is cited in the previous section. It should also be pointed out that the well-known English author and mathematician, Henry Ernest Dudeney, created many puzzles around
the turn of the 20th century, which either directly or indirectly involve dominating sets of chess pieces on varying sizes of chessboards. Some of these puzzles can be seen in his book entitled The
Canterbury Puzzles and Other Curious Problems [46], which is freely available on the web. On page 84 of this book [46] is Puzzle 69. The Frogs and Tumblers. It shows eight frogs each sitting on a
different tumbler in an 8 × 8 array of tumblers (cf. Figure 3). When viewed as queens, this set forms a maximum independent set on the queens graph Q8 (cf. Figure 4). The puzzle is to move three
queens to different squares in order to form another maximum independent set of 8 queens. It also suggests that there is only one such solution, up to isomorphism. On page 108 of this book [46] is
Puzzle 92. The Four Porkers. It shows four pigs placed on the squares of a 6 × 6 chessboard, which when viewed as queens,
Domination in Chessboards
Fig. 4 A maximum independent set of 8 queens on Q8
Fig. 5 A minimum independent dominating set of 4 queens on Q6
form a set of 4 independent dominating queens of Q6 (cf. Figure 5). The puzzle is to determine the number of different sets of 4 queens which form an independent dominating set of Q6 (cf. Figure 5).
In 1910 Pauls proves the following well-known result. Theorem 1 (Pauls [76, 77]) For queens graphs Qn , (i) α(Q1 ) = α(Q2 ) = 1 (ii) α(Q3 ) = 2 (iii) for n ≥ 4, α(Qn ) = n. The basic idea for proving
this theorem for n ≥ 4 is indicated in Figure 6. Historically, the first true graph theory book, published by Dénes König in 1936 [67], is noteworthy in that essentially the (independent) domination
number was first formally defined as a graph theory concept, although not by this name; it was called a punktbasis, or point basis. In his book [67], König presents the following illustration that i
(Q8 ) = γ (Q8 ) = 5 (cf. Figure 7). In 1964 [96] the Yaglom brothers published the book Challenging Mathematical Problems with Elementary Solutions. Vol. I: Combinatorial Analysis and Probability
Theory, which contains 9 chessboard domination problems. Notable among their
J. T. Hedetniemi and S. T. Hedetniemi
Fig. 6 Ahrens’ maximum independent set of 10 queens on Q10
Fig. 7 A minimum (independent) dominating set of 5 queens on Q8
contributions were the following three theorems about the kings domination number of square and rectangular chessboards, and the kings independent domination and independence numbers of square
chessboards. 2 Theorem 2 (Yaglom and Yaglom [96]) For kings graphs, γ (Kn ) = n+2 3 .
Theorem 3 (Yaglom and Yaglom [96]) For rectangular kings graphs, γ (Km,n ) =
m+2 n+2 . 3 3
Theorem 4 (Yaglom and Yaglom [96]) For kings graphs, i(Kn ) = α(Kn ) = 2 n+1 2 .
Domination in Chessboards
Chessboard domination problems often appeared in the many columns written by Martin Gardner in Scientific American in the 1970s, cf. [53–55]. In 1995 [52] Fricke, Hedetniemi, Hedetniemi, McRae,
Wallis, Jacobon, Martin, and Weakley present the first comprehensive survey of chessboard domination results. This was followed in 1998 by a survey by Hedetniemi, Hedetniemi, and Reynolds [63]
containing even more chessboard domination results. In 2004 [90] Watkins, publishes the book Across the Board: The Mathematics of Chessboard Problems, in which Chapter Seven Domination, Chapter Eight
Queens Domination, Chapter Nine Domination on Other Surfaces, and Chapter Ten Independence discuss some of the results reviewed in this chapter. In 2008 [3] Bell and Stevens present a comprehensive
32-page survey of everything that is known about the n-queens problem, including the placement of non-attacking queens on a wide variety of chessboards, including n-dimensional boards, Möbius boards,
and modular boards. Finally, in 2018 [95] Weakley presents an excellent survey of research on the queens domination number in the last 25 years. We should also add that there is a huge amount of
chess information at http://www.kotesovec.cz.
4 Queens In this section we focus on chessboard domination using only queens. In subsequent sections we will focus on chessboard domination using the other pieces of bishops, knights, kings, and
rooks. In 1977 [68] Larson shows that for primes of the form n = 4k + 1, elegant solutions can be constructed for the n-queens problem using the following simple rule: place a queen on the center
square and then place other queens by making successive (2, 3) movements—two squares to the right and three squares upward, where the top and bottom edges of the board are identified, as well as the
right and left edges of the board. The resulting queen placement for n = 13 is shown in Figure 8. Larson shows that whenever u and v are positive integers and u2 + v2 is an odd prime p, then queens
located at successive (u, v) movements from a queen on the center square of the p × p chessboard give a solution to the p-queens problem, and such solutions exist whenever p is a prime of the form 4k
+ 1. In 1984 [88] Wagner and Geist discuss the results of a programming assignment given to students in a graduate computer science class: write a program to solve the following variant of the
8-queens problem. A crippled queen CQ is a chess queen that can move at most two squares at a time in any direction (vertical, horizontal, or diagonal). Find the maximum number α(CQn ) of CQs that
can be placed on an 8 × 8 chessboard so that no two CQs can attack one another. Find also the number of ways that this number of CQs can be so placed. If possible, generalize the program to compute α
(CQ8,n ) or α(CQm,n ).
J. T. Hedetniemi and S. T. Hedetniemi
Fig. 8 Thirteen non-attacking queens in a (2,3)-pattern on a 13 × 13 chessboard
Fig. 9 Thirteen independent crippled queens on an 8 × 8 chessboard
Figure 9 illustrates a crippled queens solution for the 8 × 8 board. Notice that in this solution every CQ is a knight’s move away from at least one other CQ. Notice also that a crippled queen is
very much like a super king, which can move two squares: one square in any direction, followed by a second square in any direction. In 1986 [35] Cockayne and Hedetniemi introduce a variation of the
standard queens domination problem, in which you seek to find the minimum number of queens which, when placed only on the main diagonal of an n × n chessboard, dominate all squares. Let diag(n)
denote this diagonal queens domination number; this is also denoted by γ diag (Qn ). The authors show that the diagonal queens domination problem is equivalent to the problem of finding a
midpoint-free, even-sum set of integers up to n, which, as well, is equivalent to that of finding a midpoint-free subset of [n/2]; this is a collection of integers up to n/2 not containing a
three-term arithmetic progression. A subset K ⊂ N is called midpoint-free if for all {i, j}⊆ K, (i + j)/2∈K, and K is called an even-sum subset if the sum of each pair of elements of K is even, i.e.,
its elements are either all odd or all even.
Domination in Chessboards Table 1 Values of diag(n)
351 n 7 8 11 15 20 24 25 30
diag(n) 4 5 7 11 15 18 18 22
Minimum diagonal dominating set {2, 4, 5, 6} {2, 4, 5, 6, 8} {1, 3, 5, 6, 7, 9, 11} N −{2, 4, 8, 10} N −{2, 4, 8, 10, 20} N −{2, 4, 8, 10, 20, 22} N −{1, 3, 7, 9, 19, 21, 25} N −{1, 3, 7, 9, 19, 21,
25, 27}
The authors show that the diagonal queens domination number is related to the number-theoretic function r3 (n), which equals the smallest number of integers in a subset of {1, 2, . . . , n} that must
contain three terms in arithmetic progression. Suppose that the squares of a chessboard are labeled (i, j), so that black and red squares have (i + j) even or odd, respectively. A subset K ⊂ N = {1,
2, . . . , n} is called a diagonal dominating set if queens placed in positions {(k, k) : k ∈ K} on the black major diagonal dominate the entire board. Theorem 5 (Cockayne, Hedetniemi [35]) A subset
K is a diagonal dominating set if and only if N − K is a midpoint-free, even-sum set. Proof. Let K be a diagonal dominating set and let {i, j}⊆ N − K. Then square (i, j) is not covered by a queen
along a row or column. Since only black squares are covered diagonally, square (i, j) must be black, which implies that (i + j) is even, i.e., N − K is an even-sum set. Since square (i, j) is
covered, by a queen at square (k, k), for some k ∈ K, we have i + j = 2k. Hence, (i + j)/2∈N − K and N − K are midpoint-free. Conversely, suppose N − K is a midpoint-free, even-sum set. Place queens
at {(k, k)|k ∈ K}. If (i, j) is a red square, i.e., i + j is odd, then by the even-sum property, either i or j is in K and (i, j) is covered by a queen along a row or column. If (i, j) is a black
square and is not covered by a row or a column, then (i, j) ⊆ N − K and i + 2j = 2l, for some l ∈ N. Since N − K is midpoint-free, l∈N − K. Therefore, l ∈ K and (i, j) are dominated by the queen at
position (l, l). This completes the proof. 2 Corollary 6 diag(n) = n − max{|K||K is a midpoint-free, even-sum subset of N}. In Table 1, notice that the complements of the indicated minimum diagonal
dominating sets are midpoint-free; for example, for n = 25, {1, 3, 7, 9, 19, 21, 25} is a midpoint-free set. See Figure 10 for a minimum diagonal queens dominating set on Q8 . In 1985 [40] Cockayne,
Gamble, and Shepherd consider another variation of the standard queens domination problem. Denote by col(n) the minimum number of queens on any single column that is required to dominate the n × n
chessboard. It is easy to see that a column nearest the center is as good as any other. The authors show that like diag(n), col(n) is also related to the number-theoretic function r3 (n), as follows.
J. T. Hedetniemi and S. T. Hedetniemi
Fig. 10 Diagonal minimum dominating set for Q8
Let A(n) = n − r3 (n/3). Let B(n) = n − maxk+l=n/2 {r3 (k/2) + r3 (l/2)}. Theorem 7 (Cockayne, Gamble, Shepherd [40]) For n ≥ 2, col(n) = min{A(n), B(n)}. Corollary 8 For any n, diag(n) ≤ col(n).
They also raised the following question. Question 9 For all n, is A(n) ≥ B(n)? In 1987 [78] Raghavan and Venkatesan prove the following bounds on the queens domination number. The proof of this upper
bound is essentially the same as the proof attributed to Welch below. Theorem 10 (Raghavan, Venkatesan [78]) For any n ≥ 1, 12 n ≤ γ (Qn ) ≤ 23 n + 2. This upper bound shows up in several subsequent
papers, and can be proved in several different ways. In 1988 [37] Cockayne and Spencer provide the following upper bound for the independent queens domination number. Theorem 11 (Cockayne, Spencer
[37]) For any n ≥ 1, i(Qn ) ≤ 0.705n + 2.305. In 1990 [34] Cockayne surveys results known at the time on domination and independent domination numbers of the queens graph, the diagonal queens
domination problem, domination by queens in a single column and domination, independent domination and total domination of the bishops graph. In this paper he presents the following basic result,
attributed to L. Welch in an undated private
Domination in Chessboards
communication to Cockayne. This construction is very similar to the same upper bound presented earlier by Raghavan and Venkatesan [78]. Theorem 12 (Welch [34]) For n = 3q + r, where 0 ≤ r ≤ 2, γ (Qn
) ≤ 2q + r. Proof Sketch. For the case n = 3q, divide the n × n chessboard into nine q × q boards, where the top three boards are numbered B1 , B2 , and B3 , the middle three boards are numbered B4 ,
B5 , and B6 , and the bottom three boards are numbered B7 , B8 , and B9 . Place q queens on the main (northwest down to southeast) diagonal of board B3 . Place q − 1 queens on the diagonal
immediately above the main diagonal of board B7 , and place the last queen in the bottom left corner of board B7 . It is easy to see that the q queens on the main diagonal of board B3 dominate all
squares in boards B1 , B2 , B3 , B6 , and B9 . Similarly, the q queens in board B7 dominate all squares in boards B4 , B7 , and B8 . It only remains to show that all squares in the middle board B5
are dominated by these 2q queens. If n = 3q + 1 or n = 3q + 2, it is easy to add one or two extra queens in squares (3q + 1, 3q + 1) and (3q + 2, 3q + 2). 2 It is worth noting that one can always add
just one queen to the pattern suggested in Welch’s proof to obtain an upper bound for the connected domination number of the queens graph. Cockayne also presents one of the most often quoted results
in queens domination theory, attributed to P. H. Spencer, when he was an undergraduate research student at the University of Victoria in the summer of 1984 (C.M. Mynhardt, Private communication,
2020). Theorem 13 (Spencer, 1984) For any n ≥ 1, γ (Qn ) ≥ (n − 1)/2. Proof. Since γ (Q1 ) = γ (Q2 ) = γ (Q3 ) = 1, we can assume that n ≥ 4. It is easy to see that the set of n − 2 queens placed on
the main diagonal on every square (i, i) except for (1, 1) and (3, 3) is a dominating set. Thus, for all n ≥ 4, γ (Qn ) ≤ n − 2, and therefore any minimum dominating set of queens on Qn will have at
least two rows and two columns with no queen on them. Assume that the columns of Qn are numbered 1, 2, . . . , n from left to right, and that the rows are similarly numbered from top to bottom. Let S
be a minimum dominating set of queens on Qn . Let a be the leftmost column, b the rightmost column, c the lowest row, and d the highest row not containing a queen. By symmetry, we may assume that δ 2
= d − c ≤ δ 1 = b − a. Consider the sets of squares Sa and Sb in columns a and b, respectively, which lie between rows c and c + δ 1 − 1 inclusively, and let S = Sa ∪ Sb . Thus, |Sa | = |Sb | = δ 1 −
1. Since δ 2 ≤ δ 1 , no diagonal intersects both Sa and Sb . Therefore, every queen diagonally dominates at most two squares of S, one in Sa and one in Sb . Furthermore, all queens situated above row
c or below row c + δ 1 − 1 do not dominate any squares of S by row or column. By the definition of c, there are at least c − 1 queens above row c and each row below d is occupied by at least one
queen, where d = c + δ 2 ≤ c + δ 1 . Therefore,
J. T. Hedetniemi and S. T. Hedetniemi
since all of the n − (c + δ 1 ) rows below row c + δ 1 are occupied, there are at least n − (c + δ 1 ) queens below row c + δ 1 − 1. It follows that at least (c − 1) + (n − c − δ 1 ) = n − δ 1 − 1
queens dominate at most 2 squares of S. The remaining queens, at most γ (Qn ) − (n − δ 1 − 1), may cover at most 4 squares of S. Since all of the 2δ 1 squares of S must be dominated, we must have 2(n
− δ1 − 1) + 4(γ (Qn ) − (n − δ1 − 1)) ≥ 2δ1 , which gives γ (Qn ) ≥ (n − 1)/2, as required.
Note, by the way, that the bound γ (Qn ) ≥ (n − 1)/2 means that for even values of n, γ (Qn ) ≥ n/2; this is noted in many papers dealing with queens domination numbers. Thus, whenever for even n you
can find a dominating set of n/2 queens, you know it is best possible, and this happens quite often. In 1991 Grinstead, Hahne, and Van Stone (see also Eisenstein, Grinstead, Hahne and Van Stone [47])
prove the following two theorems, which were the best known bounds at the time, the second of which is very much like the theorem of Welch and the theorem of Raghavan and Venkatesen above. Theorem 14
(Grinstead, Hahne, Van Stone [60]) For any n ≥ 1, γ (Qn ) ≤
14 n + O(1). 23
Theorem 15 (Grinstead, Hahne, Van Stone [60]) For any n ≥ 1, i(Qn ) ≤ 23 n + O(1). One basic pattern of independent dominating queens which achieves the upper bound in Theorem 15 is shown in Figure
11. In 1991 Weakley studies (Qn ) and t (Qn ) and proves the following two lower bounds. Theorem 16 (W.D. Weakley, Private communication, July 26, 1991) For n ≥ 5, (i) Γ (Qn ) ≥ 2n − 5. (ii) Γ t (Qn
) ≥ 2n − 5. He also shows that n = 6 is the smallest value for which (Qn ) > n. Weakley then studies the value of (Qm,n ) and t (Qm,n ) for rectangular m × n chessboards, and shows the following.
Theorem 17 (W.D. Weakley, Private communication, July 26, 1991) For any n ≥ 1, (i) Γ (Q2,n ) = IR(Q2,n ) = n/2. (ii) Γ t (Q2,n ) = IRt (Q2,n ) = n/2 except for n = 1, 2, 5, 6, when Γ t (Q2,n ) = IRt
(Q2,n ) = n/2 + 1.
Domination in Chessboards
Fig. 11 A set of 11 independent queens dominating Q17
(iii) Γ (Q3,n ) ≥(n + 1)/2. (iv) Γ (Q4,n ) ≥ n and Γ t (Q4,n ) ≥ n. The tables below illustrate the bounds for (Q3,n ) and t (Q4,n ) (cf. Table 2), where a private neighbor of queen Qk is the square
numbered k. Notice that in Table 2 queen Q3 is its own private neighbor. Notice also that the set of queens in Table 2 is not a total dominating set of queens because of queen Q3 . In 1994 [19] and
[20] Burger, in his master’s thesis and PhD dissertation, gives a complete listing of all minimum queens dominating sets for Qn , for 5 ≤ n ≤ 8. In 1994 Burger, Mynhardt, and Cockayne provide the
following four exact values of γ (Qn ) by exhibiting symmetric solutions. Theorem 18 (Burger, Mynhardt, Cockayne [26]) For k = 9, 12, 13, 15, γ (Q4k+1 ) = 2k + 1.
J. T. Hedetniemi and S. T. Hedetniemi
Table 2 (Q3,n ) ≥(n + 1)/2 and (Q4,n ) ≥ n Q1
1 Q1
3 Q3
5 Q5
Q2 2
Q4 4
Table 3 Best known results as of 2020 Chess pieces Queens Qn Kings Kn Rooks Rn Bishops Bn Knights Nn Grid Gn
n n
((n + 2)/3)2 n n
((n + 2)/3)2 n n
α n ((n + 1)/2)2 n 2n − 2 [39] [39]
≥ 2n − 5
n 2n − 2 [39] [39]
2n − 4 4n − 14 [39] [39]
Table 4 (Q6 ) = 7
2 Q1
7 7 Q7
As we will see below, more results for γ (Q4k+1 ) were to follow. In 1995 Weakley proves the following two results, which he presented at a conference in 1992. Theorem 19 (Weakley [92]) For all k, γ
(Q4k+1 ) ≥ 2k + 1 Theorem 20 (Weakley [92]) For k ≤ 6, and k = 8, γ (Q4k+1 ) = i(Q4k+1 ) = 2k + 1. He also proves that γ (Q7 ) = i(Q7 ) = 4, which was stated, but not proved, by W. W Rouse Ball in
1892 [81]. In 1995 [52] Fricke et al. and in 1998 [63] Hedetniemi et al. publish two comprehensive surveys of the following 36 chessboard domination-related problems. (cf. Table 3). Space limitations
do not permit us to discuss the state of knowledge of all 36 problems. Thus, we only highlight a few. The result that α(Qn ) = n is frequently attributed to Ahrens in 1910 [1], but was first shown by
Pauls in 1874 [76]. The inequality (Qn ) ≥ 2n − 5 is due to Weakley (private communication dated July 26, 1991). An illustration of Weakley’s construction of a maximum cardinality, minimal dominating
set of seven queens on Q6 is given in Table 4, with seven numbered queens and squares with an integer k indicating a private neighbor of queen Qk . Notice in Table 3 that for the rooks graph, all
formulas are known, since these graphs have a simple clique structure. The results that γ (Rn ) = i(Rn ) = α(Rn ) = n are
Domination in Chessboards
Fig. 12 IR(Rn ) = 2n − 4
Fig. 13 α(Bn ) = (Bn ) = 2n − 2 and IR(Bn ) = 4n − 14
due to Yaglom and Yaglom [96], as are the results that γ (Bn ) = i(Bn ) = n, and the result that α(Bn ) = 2n − 2. The results, that ir(Rn ) = n, (Rn ) = n and IR(Rn ) = 2n − 4, are attributed to
Hedetniemi, Hedetniemi and Wallis, but are stated as unpublished in [52] (cf. Figure 12, where the rooks in column 2 all have a private neighbor in column 1, and the rooks in row 2 all have a private
neighbor in row 1). The result that ir(Bn ) = n is attributed to Wallis, but is stated as unpublished in [52], and the results that (Bn ) = 2n − 2, and IR(Bn ) = 4n − 14 are attributed to Fricke, but
are also stated as unpublished in [52]; cf. Figure 13 and for illustrations of Fricke’s results. Because of the following theorem, proved in 1981 by Cockayne, Favaron, Payan, and Thomason, it becomes
easy to establish the values of α, , and IR for knights graphs and grid graphs, since both of these are bipartite families of graphs. Theorem 21 (Cockayne, Favaron, Payan, Thomason [39]) If G is a
bipartite graph, then α(G) = Γ (G) = IR(G). Corollary 22 (Cockayne, Favaron, Payan, Thomason [39]) For all n ≥ 1,
J. T. Hedetniemi and S. T. Hedetniemi
α(Nn ) = Γ (Nn ) = IR(Nn ) = {n2 /2 for even n; (n2 + 1)/2, for odd n}. α(Gn ) = Γ (Gn ) = IR(Gn ) = {n2 /2 for even n; (n2 + 1)/2, for odd n}. Finally, the domination numbers of all grid graphs,
that is, Cartesian products of the form Pm 2Pn , have been completely determined in a 2011 paper by Goncalves, Pinlou, Rao, and Thomasse [59], who present 16 formulas for the domination numbers of
m-by-n grid graphs; 15 different formulas for γ (Gm,n ) for 1 ≤ m ≤ 15, and one final formula for γ (Gm,n ), for all m ≥ 16. In 1995 [71] Messick, in his MS research paper, develops a genetic
algorithm for finding near-optimal solutions for IR(Qn ). One of his maximal irredundant sets of 11 queens on Q8 is shown in Table 5, in which queen Qk has as a private neighbor the square numbered
k. Messick’s genetic algorithm also established the following lower bounds for IR(Qn ), for 6 ≤ n ≤ 18 (cf. Table 6). In 1997 [27] Burger, Cockayne, and Mynhardt introduce the study of the upper
domination number (Qn ) and the upper irredundance number IR(Qn ) of the queens graph Qn and present the following three results. Theorem 23 (Burger, Cockayne, Mynhardt) For all n ≥ 1, (i) γ (Qn ) ≤
31n/54 + O(1); (ii) Γ (Qn ) ≥ 5n/2 − O(1); ( √ (iii) I R(Qn ) ≤ 6n + 6 − 8 n + n + 1. They also mention that they have determined all 638 non-isomorphic independent dominating sets of size 5 of Q8 .
In 1997 [58] Gibbons and Webb, using simulated annealing and exhaustive search techniques, extend the known values of γ (Qn ) and i(Qn ) as shown in Tables 7 and 8: Table 5 IR(Q8 ) ≥ 11
Q1 Q5 1
Q2 Q6
Q4 Q7
Table 6 Lower bounds for IR(Qn ) n IR(Qn ) ≥
Table 7 New values of γ (Qn )
n γ (Qn ) ≥
Domination in Chessboards
Table 8 New values of i(Qn )
n i(Qn ) ≥
Fig. 14 γ (Q13 ) = 7
As a by-product, the number of non-equivalent ways of covering Qn with k independent queens, for 1 ≤ n ≤ 15 and 1 ≤ k ≤ 8, as well as the case n = 16 and k = 8, are determined. As an illustration,
they present the following minimum dominating set of seven queens on Q13 (cf. Figure 14); note that three queens lie symmetrically on the main diagonal, the four queens on border are symmetrically
placed, and every other row and column contains exactly one queen. This is another case when γ (Q4k+1 ) = 2k + 1. In 2000 [23] Burger and Mynhardt provide the following two queens domination numbers:
γ (Q19 ) = 10, γ (Q31 ) = 16. In 2000 [22] Burger and Mynhardt provide the following four queens domination numbers: γ (Q30 ) = 15, γ (Q69 ) = 35, γ (Q77 ) = 39, i(Q45 ) = 23. They also provide the
following tabulation of known values of γ (Q4k+1 ) (cf. Table 9). In 2000 [22] Burger and Mynhardt add the following two values for the lower irredundance number of the queens graph: ir(Q5 ) = γ (Q5
) = 3, ir(Q6 ) = γ (Q6 ) = 3 (cf. Figure 1).
360 Table 9 γ (Q4k+1 )
Table 10 Toroidal queens chessboard
J. T. Hedetniemi and S. T. Hedetniemi k n γ k n γ
7 29 15 16 65 ≤35
• a
Q • • • • • • •
• 1
f g
In effect, they prove that there does not exist a maximal irredundant set of two queens on either Q5 or Q6 . Thus, the minimum dominating sets of cardinality three for Q5 and Q6 are also minimum
cardinality maximal irredundant sets on these two chessboards. They also state, but without proof, that ir(Q7 ) = γ (Q7 ) = 4 (cf. Figure 1). At the close of their paper, the authors offer the
following interesting comment: “The existence of a maximal irredundant set X of queens on Qn with |X| < γ (Qn ) for some n seems unlikely, as the (average) number of pns [private neighbors] per queen
seems to increase rapidly as n increases, as does the cardinality of R, the set of open [undominated] squares, and hence the cardinality of N[R]. For every square in N[R] to annihilate a queen in X
(see Theorem 1) is a tall order. “(Note: Harborth [private communication, January 2000] recently reported that ir(Qn ) = γ (Qn ) for n ≤ 10.)” In 2001 [28] Burger, Cockayne, and Mynhardt introduce
the study of domination in queens graphs on the torus, denoted Qtn , where the torus is the Cartesian product Cn 2Cn . In Qtn a diagonal is no longer a path with a beginning and an end; instead, it
is a cycle, cf. Table 10, where the queen Q on the top row dominates toroidally every square in its top row, every square in its column, and all of the labeled squares, which form to diagonal cycles
of length 8, where the black square can be labeled both 4 and d. Thus, in the toroidal queens graph, square b is adjacent to square c, square 5 is adjacent to square 6, and squares 7 and g are both
adjacent to the square labeled Q, and to each other, since they are in the same row. An example of four toroidal queens which dominate Qt8 is shown in Figure 15; note that it takes five queens to
dominate Q8 ; this is the smallest value of n, for
Domination in Chessboards
Fig. 15 Four minimum toroidal queens for Q8
which γ (Qtn ) < γ (Qn ). It is interesting to note that the authors have determined that γ t (Q15 ) = 5 < γ (Q15 ) = 9. The situation for i(Qn ) when compared with i(Qtn ) is also interesting, since
they are not comparable. For example, the authors have shown that i(Q6 ) = 4 = i(Qt6 ), i(Q7 ) = 4 < i(Qt7 ) = 5, yet i(Q8 ) = 5 < i(Qt7 ) = 4. The authors consider the independence, domination, and
independent domination numbers of graphs obtained from the moves of queens on chessboards drawn on the torus, and determine exact values for each of these parameters in infinitely many cases. In 2001
Cockayne and Mynhardt study the lower irredundance number of the queens graph, ir(Qn ). The determination of ir(Qn ) is no easy task. After some 15 pages of preliminary results and careful analysis,
the authors prove the following theorem. Theorem 24 (Cockayne, Mynhardt [36]) For any n ≥ 8, the queens graph Qn does not have a maximal irredundant set of size three. Theorem 25 (Cockayne, Mynhardt
[36]) The queens graph Q7 does not have a maximal irredundant set of size three. Since it is known that γ (Q7 ) = 4 and ir(Qn ) ≤ γ (Qn ), we can conclude the following. Corollary 26 (Cockayne,
Mynhardt [36]) For the queens graph Q7 , ir(Q7 ) = 4. In 2001 [64] Kearse and Gibbons, using probabilistic and exhaustive search techniques, such as backtracking with refinements and enhancements,
reduction methods, and local search techniques, establish the following queens domination numbers:
(i) (ii) (iii) (iv) (v) (vi)
J. T. Hedetniemi and S. T. Hedetniemi
γ (Q15 ) = γ (Q16 ) = 9, γ (Q19 ) = 10, γ (Q4k+1 ) = 2k + 1, for k = 16, 18, 20 and 21, i(Q18 ) = 10, 10 ≤ i(Q19 ) ≤ 11, and i(Q22 ) ≤ 12.
Parameters closely related to γ and i are the irredundance numbers, ir and IR, and the upper domination number . Kearse and Gibbons also show that: (vii) ir(Qn ) = γ (Qn ), for n ≤ 13, (viii) IR(Q9 )
= (Q9 ) = 13, (ix) IR(Q10 ) = (Q10 ) = 15. For the kings graphs Kn , to be discussed further below, the authors establish the following results: (x) IR(K8 ) = 17, IR(K9 ) = 25, IR(K10 ) = 27, and IR
(K11 ) = 36. Kearse and Gibbons also calculate the numbers of non-isomorphic, minimum dominating sets and independent dominating sets in Qn , for n ≤ 15 and n ≤ 18, respectively. In 2001 [74]
Östergard and Weakley publish what is arguably the most definitive paper on queens domination to date. Using a combination of known theoretical bounds for both γ (Qn ) and i(Qn ), along with advanced
computer search algorithms, the authors determine quite a few new values and bounds for these two queens parameters, which we list here. 1. γ (Qn ) = n/2, for 17 values of n. 2. i(Qn ) = n/2, for 11
values of n. 3. One or both of γ (Qn ) and i(Qn ) is equal to one of {n/2, n/2 + 1}, for 85 additional values of n. 4. γ (Q4k+1 ) = 2k + 1, for k ≤ 32. 5. For n ≤ 120, each of γ (Qn ) and i(Qn ) is
either known, or known to have one of only two consecutive values. 6. γ (Qn ) ≤ 69n/133 + (1). 7. i(Qn ) ≤ 61n/111 + O(1). 8. For all n, (n − 1)/2 ≤ γ (Qn ) ≤ i(Qn ). 9. Conjecture 27 For all n, i(Qn
) ≤n/2 + 1. 10. If n < 143 and n = 3, 11, then γ (Qn ) ≥ n/2. The authors raise the question of whether γ (Qn ) = (n − 1)/2 holds for any value of n other than n = 3 and n = 11. This question was
finally answered in 2007 by Finozhenok and Weakley. Theorem 28 (Finozhenok, Weakley [50]) The only integers n for which γ (Qn ) = (n − 1)/2 are n = 3, 11.
Domination in Chessboards
In conclusion, Östergard and Weakley [74], together with the theorem of Finozhenok and Weakley, provide the following summary results: γ (Qn ) = i(Qn ) = (n − 1)/2, only for n = 3, 11. γ (Qn ) = n/2,
for n = 1, 2, 4–7, 9, 10, 12, 13, 17 − 19, 21, 23, 25, 27, 29–31, 33, 37, 39, 41, 45, 49, 53, 57, 61, 65, 69, 71, 73, 77, 81, 85, 89, 91, 93, 97, 101, 105, 109, 113, 115, 117, 121, 125, 129–131. γ
(Qn ) = n/2 + 1, for n = 8, 14, 15, 16. γ (Qn ) ∈{n/2, n/2 + 1}, for n = 20, 22, 24–26, 28, 32, 34, 35, 36, 38, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68,
70, 72, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 87, 88, 90, 92, 94–96, 98–100, 102–104, 106–108, 110–112, 114, 116, 118–120, 122, 126, 132. i(Qn ) = n/2, for n = 1, 2, 5, 7, 9, 10, 13, 17, 21, 25,
33, 45, 57, 61, 69, 73, 77, 81, 85, 89, 93, 97, 105, 109. i(Qn ) = n/2 + 1, for n = 4, 6, 8, 12, 14 − 16, 18. i(Qn ) ∈{n/2, n/2 + 1}, for n = 19, 20, 22, 23, 24, 26–30, 31, 32, 34–44, 46– 56, 58–60,
62–68, 70–72, 74–76, 78–80, 82–84, 86–88, 90–92, 94–96, 98–104, 106–108, 110–120. In 2002 Burger and Mynhardt prove the following. Theorem 29 (Burger, Mynhardt [24]) For any n ≥ 1, γ (Qn ) ≤
8 15 n + O(1).
In 2002 Kearse and Gibbons provide the best known lower bounds for IR(Qn ). Theorem 30 (Kearse, Gibbons [65]) For Qn , 6n − O(n2/3 ) ≤ IR(Qn ). Theorem 31 (Kearse, Gibbons [65]) For even k ≥ 6, 6k 3
− 29k 2 − O(k) ≤ I R(Qk 3 ). The authors conclude their paper with the following comment: “Finally, it seems likely, although not proven, that 6n − O(n2/3 ) is also an upper bound for IR(Qn ).” In
2002 Weakley establishes improved upper bounds for the queens domination number and queens independent domination number. Theorem 32 (Weakley [93], [94]) For all n ≥ 1, γ (Qn ) ≤ 34n/63 + O(1) <
0.54n + O(1). Theorem 33 (Weakley [93], [94]) For all n ≥ 1, i(Qn ) ≤ 19n/33 + O(1) < 0.57n + O(1). In 2003 Burger and Mynhardt provide improved upper bounds, in special cases, for both γ (Qn ) and γ
(Qtn ), where Qtn is the n × n queens graph on a torus. They present a 10-page proof of the following theorem. Theorem 34 (Burger, Mynhardt [25]) For all n large enough, γ (Qn ) ≤ O(1).
101 195 n
For queens on a torus, the authors provide the following summary of known results:
J. T. Hedetniemi and S. T. Hedetniemi
⎧ ⎪ ⎪ ⎨k t γ (Q3k ) = k + 1 ⎪ ⎪ ⎩k + 2
if k ≡ 1, 5, 7, 11 (mod 12) if k ≡ 2, 10 (mod 12) if k ≡ 0, 3, 4, 6, 8, 9 (mod 12)
They also show that if n ≡ 2, 4 (mod 6), then n/3 ≤ γ (Qtn ) ≤ n2 , and if n ≡ 1, 5 (mod 6), then n/3 ≤ γ (Qtn ) ≤ γ (Qn ). The authors show that if for some fixed k there is a dominating set of
Q4k+1 of a certain type with cardinality 2k + 1, then for any n large enough, γ (Qn ) ≤ [(3k + 5)/(6k + 3)]n + O(1). The same construction shows that for any m ≥ 1 and n = 2(6m − 1)(2k + 1) − 1, γ
(Qtn ) ≤ [(2k + 3)/(4k + 2)]n + O(1). In 2003 Burger, Mynhardt, and Weakley prove the following for the queens domination number on a torus. Theorem 35 (Burger, Mynhardt, Weakley [29]) For all n ≥ 1,
γ (Qt3n ) = 2n − α(Qtn ). In 2003 [72] Mynhardt establishes improved upper bounds for γ (Qtn ) and i(Qtn ), for queens on a torus. In 2005 [2] Amirabadi, in his MS research paper, develops search
algorithms for approximating the total domination and connected domination numbers of queens graphs. His search algorithm produces results that are within three of proven lower bounds. It is known
that for the first 130 values of n, γ (Qn ) is either known or known to be one of two consecutive numbers. As a result of the author’s computations, for the first 30 values of n, γ t (Qn ) and γ c
(Qn ) are either known or known to be one of three consecutive numbers (cf. Table 11). In 2006 [11] and later in 2016 [15] Burchett initiates the study of the paired domination number γ pr (Qn ), the
total domination number γ t (Qn ), and the connected domination number γ c (Qn ) of queens graphs. Exact values for γ pr (Qn ), γ t (Qn ), and γ c (Qn ) are provided for the following values of n: γ
pr (Qn ): 2 ≤ n ≤ 10, n = 12, 13, and 15 ≤ n ≤ 20. γ t (Qn ): 2 ≤ n ≤ 10, n = 12, 15, 17, 18, 19. Table 11 Values found for γ t (Qn ) and γ c (Qn ) n γt ≥ Found γc ≥ Found n γt ≥ Found γc ≥ Found
1 x x 1 1 16 9 10 10 11
Domination in Chessboards
γ c (Qn ): 2 ≤ n ≤ 11, n = 13, 14, 16, 17, 19, 20, 22, 23. The following bounds are also provided: 4 7 (n − 1) ≤ γt (Qn ) ≤ γpr (Qn ). γt (Qn ) ≤ γpr (Qn ) ≤ 2n/3 + O(1).
2n/3 − 1 ≤ γc (Qn ) ≤ 2n/3 + O(1).
In 2008 [84] Sinko and Slater initiate the study of the border queens domination problem, that is, determining how few queens are needed to cover all of the squares of an n × n chessboard when the
queens are restricted to squares on the border. We denote this number by γ bor (Qn ). In this paper the authors give the values of γ bor (Qn ) for 1 ≤ n ≤ 13 shown in Table 12. What is particularly
interesting about this is the observation that γ bor (Q12 ) = 10 > γ bor (Q13 ) = 9. Thus, the border queens domination number is not monotonically non-decreasing. It has long been conjectured that
the queens domination number is monotonically non-decreasing, that is, for all n ≥ 1, γ (Qn ) ≤ γ (Qn+1 ). The authors present solutions to the border queens domination number for 4 ≤ n ≤ 10 as
follows: ⎧ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪4 ⎨ γbor (Qn ) = 5 ⎪ ⎪ ⎪ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎪ 6 ⎪ ⎪ ⎪ ⎩ 6
if n = 4:{(a, 2), (d, 2)} if n = 5:{(b, 5), (c, 1), (d, 5)} if n = 6:{(b, 6), (c, 1), (d, 1), (e, 6)} if n = 7:{(b, 7), (c, 7), (d, 1), (e, 7), (f, 7)} if n = 8:{(b, 8), (c, 8), (d, 1), (e, 1), (f,
8), (g, 8)} if n = 9:{(a, 7), (c, 1), (e, 1), (g, 9), (i, 3), (i, 5)} if n = 10:{(a, 1), (a, 7), (d, 1), (g, 10), (j, 1), (j, 4)}
They also establish by a computer search that 11 ≤ γ bor (Q14 ) ≤ 12 and 9 ≤ γ bor (Q15 ) ≤ 13. We illustrate a solution for γ bor (Q10 ) in Figure 16. Note that the solutions given for 4 ≤ n ≤ 8
above, all have a symmetry about the center column, while the solutions given for n = 9 and n = 10, although asymmetric, have a type of rotational symmetry. For the general case, they establish the
following bounds. Theorem 36 (Sinko, Slater [84]) For all n ≥ 4, Table 12 Values of γ bor (Qn ) k γ (Qn ) γ bor (Qn )
J. T. Hedetniemi and S. T. Hedetniemi
Fig. 16 γ bor (Q10 ) = 6
n(2 − 9/2n −
8n2 − 49n + 49/2n) ≤ γbor (Qn ) ≤ n − 2.
√ √ The authors note that limn→∞ (2 − 9/2n − 8n2 − 49n + 49/2n) = 2 − 2. For n = 3t + 1 they improve this upper bound to 2t + 1 if 3t + 1 is odd and 2t if 3t + 1 is even. In 2011 [12, 13] Burchett
studies k-tuple domination in the rooks graph, in which it is required that every square not in S be attacked at least k times with a minimum number of rooks. He also continues the study of the
border queens domination problem. In 2014 [10] Brown considers a variation on the queens domination problem posed by Bell and Stevens [3] in their survey of the n queens problem. Bell and Stevens
asked: given an n × n chessboard on which one queen has been arbitrarily placed, when is it possible to place n − 1 remaining queens to create an arrangement of n non-attacking queens? In [10] Brown
considers the possibility that a solution to this Initial Placement Problem is always possible for n > 6, and proceeds to provide solutions for n ≡ 0 (mod 6) and n ≡ 2 (mod 6). He then conjectures
that the Initial Placement Problem is solvable for all initial placements of two non-attacking queens when n > 9. In 2016 Burchett provides new upper bounds for paired, total, and connected
domination for the queens graph. Theorem 37 (Burchett [14]) For all n ≥ 1, 2n (i) 2n 3 − 1 ≤ γc (Qn ) ≤ 3 . (ii) For n ≥ 21 and n ≡ 3, 4, 5 (mod 6), γt (Qn ) ≤ 2n 3 − 1. − 1. (iii) For n ≥ 22 and n ≡
4 (mod 6), γpr (Qn ) ≤ 2n 3
Domination in Chessboards
Fig. 17 Minimum queens connected dominating set for Q21
For an illustration of this theorem, cf. Figure 17. In 2017 [6] William Bird, in his PhD thesis, adds considerably to what is known about a variety of queens domination problems. Noteworthy are three
new values of γ (Qn ): γ (Q20 ) = 11, γ (Q22 ) = 12, and γ (Q24 ) = 13. In each of these three cases the value was known to be one of two values, this value shown or one less. Thus, in each of these
three cases, the value is γ (Qn ) = n/2 + 1. In addition Bird’s sophisticated computer program was able to establish the following five new values: i(Q19 ) = 11, i(Q20 ) = 11, i(Q22 ) = 12, i(Q23 ) =
13, and i(Q24 ) = 13; again, in all of these five cases, i(Qn ) = n/2 + 1. Bird also adds many new values of the border queens domination number for 14 ≤ n ≤ 24, given in boldface in the following
Table 13.
J. T. Hedetniemi and S. T. Hedetniemi
Table 13 Values of γ bor (Qn ) n γ bor (Qn )
Table 14 Number of solutions to γ (Qn ) and i(Qn ) n γ (Qn ) Solutions i(Qn ) Solutions
15 9 25,872 9 1314
It is interesting to note from Bird’s data how frequently γ bor (Qn ) > γ bor (Qn+1 ), that is, that this function fails to be monotonic non-decreasing. But these numbers suggest the following
conjecture. Conjecture 38 For all n ≥ 1, γ bor (Qn ) ≤ γ bor (Qn+2 ). Of equal interest, Bird determines the number of minimum dominating sets and minimum independent dominating sets for Qn , for 3 ≤
n ≤ 18, as shown in Table 14. We close this section on queens domination by referring to the 2010 paper [49] by Fernau, in which he discusses three computational approaches to solving the queens
domination problem: (i) backtracking, (ii) dynamic programming on subsets, and (iii) dynamic programming using treewidth, or path decompositions. He points out that the determination of the γ (Qn )
sequence of integers is listed as Problem C18 in Richard Guy’s book entitled “Unsolved Problems in Number Theory.” [61]. At the end of this paper, Fernau discusses the perplexing problem that so
little is known about the complexity of the queens domination problem. QUEENS DOMINATING SET Instance: Positive integer n, positive integer k. Question: Does Qn have a dominating set of cardinality
at most k? It is unknown if this problem is NP-hard. This would seem unlikely, since for the first 130 values of n, γ (Qn ) is either known exactly or is known to be one of two consecutive integers.
And yet no polynomial-time algorithm is known for computing the value of γ (Qn ). Fernau also points out that since all known upper bounds for γ (Qn ) are algorithmic in nature, except for additive O
(1) constants, γ (Qn ) can be approximated up to a factor of 138 133 . For a related complexity question involving n-queens, see Gent, Jefferson and Nightingale [57].
Domination in Chessboards
5 Bishops In this section we review results about domination in bishops graphs. The bishops graph Bn is the graph whose vertices are the n2 squares of the n × n chessboard, and two vertices are
adjacent if and only if their corresponding squares lie on a common diagonal, which corresponds to a move of a bishop. In 1986 Cockayne, Gamble, and Shepherd prove the following two basic theorems
which determine the domination and total domination numbers of all bishops graphs. The fact that γ (Bn ) = n had previously been proved by Yaglom and Yaglom [96]. Theorem 39 (Cockayne, Gamble,
Shepherd [41]) For any n, γ (Bn ) = i(Bn ) = n. Proof Sketch. The set of squares of a nearest column to the center is an independent dominating set of the bishops graph; hence, γ (Bn ) ≤ i(Bn ) ≤ n.
It remains to show that γ (Bn ) ≥ n. Assume that γ (Bn ) < n. Then there must be a diagonal not having any bishop on it. Assume that the northwest to southeast running diagonals are labeled
sequentially 1, 2, . . . , 2n − 1 starting in the southwest corner and proceeding to the northeast corner. Notice that for 1 ≤ d ≤ n diagonal d has d squares, and for n + 1 ≤ d ≤ 2n − 1, diagonal d
has 2n − d squares. Let r (and b) be the labels of the red (black) diagonal closest to the main diagonal which has no bishop. Without loss of generality, we may assume that {r, b}⊂{1, 2, . . . , n}.
Diagonal r has r squares and these must be dominated. By the definition of r, there are bishops on each diagonal strictly between r and 2n − r, else there is a row closer to the main diagonal which
has no bishop. Hence, the number of red bishops in any dominating set satisfies nr ≥ max{r, n − r − 1}. Similarly, nb ≥ max{b, n − b − 1}. From these two inequalities we can deduce that γ (Bn ) ≥ n.
Theorem 40 (Cockayne, Gamble, Shepherd [41]) For any n ≥ 3, 2 γt (Bn ) = 2 (n − 1). 3 Proof Sketch. The bishops graph Bn is the disjoint union of the red bishops graph Rn and the black bishops graph,
Bn . We summarize only the proof that γt (Bn ) = 2 23 (n − 1) for n even.
J. T. Hedetniemi and S. T. Hedetniemi
Fig. 18 Diamond-shaped chessboard
Notice that a bishops total dominating set of Bn is precisely a rook total dominating set of the diamond-shaped chessboard Sn , which has n rows and n − 1 columns (cf. Figure 18). For ease of
presentation, we use rooks, rows, and columns, rather than bishops and diagonals. Lemma 41 ([41]) For any n, Sn has a minimum rooks total dominating set with rooks on consecutive rows and columns. It
follows from this lemma that some minimum rooks total dominating set of Sn may be used to construct a rooks total dominating set of an m × p rectangular board with property REL, i.e., a rook on every
line (row or column). It is shown that such a board satisfies m + p ≥ n − 1, and hence, if s(m, p) equals the minimum number of rooks in an REL total dominating set of an m × p board, then γ t (Bn )
= minm+p≥n−1 {s(m, p)}. Lemma 42 ([41]) For p ≤ m ≤ 2p + 2, s(m, p) = 23 (m + p), and for m > 2p + 2, s(m, p) = m. Proof. By establishing and solving a recurrence for s(m, p). 2 One may deduce from
this that γt (Bn ) ≥ 23 (n − 1). The final part of the proof 2 exhibits a rooks total dominating set of Sn with 23 (n − 1) rooks. Figure 19 illustrates a minimum independent dominating set of bishops
on B8 and a minimum total dominating set of bishops on B8 . In 1994 [66] Koehler, in his MS research paper, initiates the study of chessboard domination problems in three-dimensional chessboards. For
three-dimensional bishops graphs Bn3 , Koehler obtains the following results, cf. Table 15.
Domination in Chessboards
Fig. 19 Minimum bishops independent and total dominating set for B8 Table 15 Values of γ (Bn3 )
n γ (Bn3 )
5 ≤13
6 ≤18
7 ≤27
Table 16 Minimal dominating set of 8 bishops on B43
Table 16 illustrates a minimum dominating set of 8 bishops on the threedimensional bishops graph B43 . Each level is a 4 × 4 bishops graph, and there are four levels from left to right. A bishop
dominates every square in its two diagonals on its level, and all squares above and below it on ascending or descending diagonals. In 2002 [51] Fisher and Thalos consider bishops graphs on
rectangular k × n boards, which we denote by Bk,n . They extend the result by Yaglom and Yaglom [96], and independently by Cockayne, Gamble, and Shepherd [41], that γ (Bn,n ) = n, as follows. Theorem
43 (Fisher, Thalos [51]) For Bk,n , (i) If k < n, then γ (Bk,n ) = 2n/2. (ii) For 2 < 2k < n, γ (Bk,n ) ≤ 2(k + n)/3. The authors then make the following conjecture. Conjecture 44 (Fisher, Thalos)
For 2 < 2k < n, γ (Bk,n ) = 2(k + n)/3. They show that this conjecture is true when k ≤ 3 or n ≤ 2k + 5.
J. T. Hedetniemi and S. T. Hedetniemi
In 2016 [14] Burchett introduces the study of the k-tuple domination number, denoted γ ×k (Bn ), which equals the minimum number of bishops in a set S so that every square not in S is attacked by at
least k bishops, and every bishop is attacked by at least k − 1 bishops. In this paper, for odd n, and k ≤ n2 , the k-tuple domination number of Bn is shown to equal one of two possible values, and
for even n, the ktuple domination number is shown to be bounded between nk − k and nk for k ≤ n2 . In 2016 [16] Burchett and Buckley introduce the concept of the kth border bishop’s domination
number. When k = 1, the kth border is the set of outermost squares on the board. For k = 2, the kth border consists of all squares adjacent to first border squares and, in general, the set of kth
border square equals the set of square adjacent to the set of k − 1 border squares. Let γ bor,k (Bn ) denote the minimum number of bishops which can be placed only on kth border square in order to
dominate all squares not containing a bishop. The authors point out that as k grows large with respect to n, kth border dominating sets might not exist. In fact, they show that for k > n/4 + 1, no
kth border dominating sets exist. In this paper, they prove the following result. Theorem 45 (Burchett, Buckley) If a kth border dominating sets exist for Bn , then (i) γ bor,k (Bn ) = 2n − 4k + 2,
for n ≡ 2, 3 (mod 4), and (ii) γ bor,k (Bn ) = 2n − 4k + 2, for n ≡ 0, 1 (mod 4), unless k = n/4 + 1, in which case γ bor,k (Bn ) = 2n − 4k + 4, and γ bor,k (Bn ) = 2n − 4k + 3, respectively. In 2017
[70] Low and Kapbasov introduce the study of the vertex independence number of bishops α(Bm,n ) and kings α(Km,n ) on m × n rectangular and cylindrical chessboards, where on a cylindrical chessboard
the left and right edges of the board are identified. The authors only consider narrow boards, 1 × n, 2 × n, and 3 × n, and for each of these they determine the number of non-attacking bishops or
kings positions.
6 Knights The study of knights domination dates at least back to 1896, in L’Intermédiaire des Mathématiciens, Gauthier-Villars, Paris, Tome III (1896), p. 58, Tome IV (1897), p. 15, and Tome V
(1898), p. 87 (cf. Ball [80]). In 1910 Ahrens [1] presents known results for this problem, giving a covering for the 11 × 11 chessboard using 22 knights, which was known since 1896. The value γ (N11
) ≤ 21 was provided by Lemaire in 1973 [69] (cf. Figure 20). In 1967, in his Scientific American column Mathematical Games, Martin Gardner [53] discusses the knights covering problem for the n × n
chessboard, and gives the best known solutions for various values of n (cf. Table 17).
Domination in Chessboards
Fig. 20 Lemaire’s dominating set of 21 knights on N11
Table 17 Values of γ (Nn ) and number of solutions
n γ (Nn ) # solutions
9 14 1?
10 16 1?
Gardner gives solutions for γ (Nn ), for 3 ≤ n ≤ 8, suggesting that the readers find solutions for n = 9, 10, stating that both of these solutions were thought to be unique. Answers appeared in the
following issue [54]. In the January, 1968 issue [55], Gardner presents a second solution for γ (N10 ) that had been found by his readers. Also in that issue, Gardner gives the best known solutions
for γ (Nn ) for n = 11, 12, 13, 14, 15, which use 22, 24, 28, 34, and 37 knights, respectively. Proofs of the optimality of the values of γ (Nn ) given in Table 17 by Gardner, along with figures
showing optimal solutions for 3 ≤ n ≤ 10, due to Frank Rubin, can be found at the website: http://www.contestcen.com/knight.htm. Figure 21 illustrates two solutions. In 1987 [62] Hare and Hedetniemi
present a dynamic programming algorithm for computing the knights domination number γ (Nn ) on rectangular m × n chessboards, which is linear in n but exponential in m. Figure 22 illustrates a
minimum dominating set of knights on N8,10 that is the only minimum dominating set for this knights graph. The authors present the following values of γ (Nm,n ) in Table 18. The authors make the
following conjectures. Conjecture 46 (Hare, Hedetniemi [62]) For k = 3 and n > 8, γ (Nk,n ) is given by the following:
J. T. Hedetniemi and S. T. Hedetniemi
Fig. 21 Independent and total dominating knights for N6 Fig. 22 Unique knights dominating set for N8,10
Table 18 Values of γ (Nm,n )
γ (Nk,n ) =
γ (Nm,n ) 3 4 5 6 7 8 9 10
⎧ ⎪ ⎪(2n + 4)/3 ⎨
(2n + 5)/3 ⎪ ⎪ ⎩4n/6
for n ≡ 1 (mod 6) for n ≡ 2 (mod 6) otherwise
Domination in Chessboards
Conjecture 47 (Hare, Hedetniemi [62]) For k = 4 and n > 7, γ (Nk,n ) is given by the following: γ (Nk,n ) =
. (2n + 4)/3 4n/6
for n ≡ 1 (mod 6) otherwise
Conjecture 48 (Hare, Hedetniemi [62]) For k = 6 and n > 5, γ (Nk,n ) is given by the following:
γ (Nk,n ) =
⎧ ⎪ ⎪ ⎨n + 1
(2n + 5)/3 ⎪ ⎪ ⎩4n/4
for n ≡ 1 (mod 4) for n ≡ 2 (mod 6) otherwise
In 1994 Wallis, in his PhD thesis, introduces the study of domination in kdimensional chessboards and gives the following theorem. k
Theorem 49 (Wallis [89]) For any n, α(Nnk ) = (Nnk ) = I R(Nnk ) = n2 . Proof Sketch. It is well known that the knights graph Nn is bipartite. Wallis shows that for any k, the k-dimensional knights
graph is still bipartite. The theorem then follows from this well-known result. Theorem 50 (Cockayne, Favaron, Payan, Thomason [39]) For any bipartite graph G, α(G) = Γ (G) = IR(G). In 1995 [56]
Garnick and Nieuwejaar initiate the study of total domination on rectangular chessboards by considering knights graphs and kings graphs. They observe that Rouse Ball had the idea of total domination
in 1892. It is easy to see that for all n, γ t (Rn ) = n for rooks graphs. And it is immediate that queens diagonal domination provides an upper bound, that is, γ t (Qn ) ≤ γ diag (Qn ). For knights
total domination they provide the following results. Theorem 51 (Garnick, Nieuwejaar [56]) For all m, n > 4, (i) mn/8 < γ t (Nm,n ), (ii) γ t (Nm,n ) ≤ (mn + 5m + 6n + 56)/8, for m ≡ n (mod 2), (iii)
γ t (Nm,n ) ≤ (mn + 5m + 5n + 43)/8, for m odd and n even. Using a backtracking search algorithm, the authors were able to determine the following values of γ t (Nn ) for square chessboards:
J. T. Hedetniemi and S. T. Hedetniemi
γt (Nn ) =
⎧ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎪ 7 ⎪ ⎪ ⎪ ⎨8 ⎪ 10 ⎪ ⎪ ⎪ ⎪ ⎪ 14 ⎪ ⎪ ⎪ ⎩ 18
if n = 4 if n = 5 if n = 6 if n = 7 if n = 8 if n = 9
They also provide improved upper bounds for γ t (Nn ) as shown in Table 19.
7 Kings As was mentioned in Section 3 Early Chessboard Domination, the following formulas are known for the kings graph. 2 Theorem 52 (Yaglom and Yaglom [96]) For kings graphs, γ (Kn ) = n+2 3 .
Theorem 53 (Yaglom and Yaglom [96]) For rectangular kings graphs, γ (Km,n ) =
m+2 n+2 . 3 3
Theorem 54 (Yaglom and Yaglom [96]) For kings graphs, i(Kn ) = α(Kn ) = 2 n+1 2 . Notice that while for any graph G, we have γ (G) ≤ γ t (G) ≤ 2γ (G), for the kings graph Kn , both of these bounds
can be achieved, since γ (K4 ) = γ t (K4 ) = 4 and γ (K7 ) = γ t (K7 ) = 9, but γ t (K6 ) = 2γ (K6 ) = 8 (cf. Figure 23). In 1995 Garnick and Nieuwejaar initiate the study of total domination on
rectangular chessboards for kings graphs. For narrow boards 1 ≤ m ≤ 4, it is easy to determine the kings total domination number. Theorem 55 (Garnick, Nieuwejaar [56]) ⎧ ⎪ ⎪ ⎨n/2 γt (Km,n ) = n/2 + 1
⎪ ⎪ ⎩2n/3
For n > 1 and m ≤ 3, for n ≡ 0 (mod 4) for n ≡ 1, 2, 3 (mod 4) for m = 4
Table 19 Upper bounds for γ (Nn ) n γ t (Nn ) ≤
Domination in Chessboards
Fig. 23 Four dominating kings (left) and eight total dominating kings (right) on a 6 × 6 chessboard Table 20 Values of γ t (Kn ) n γ t (Kn )
Table 21 Upper bounds for γ t (Kn ) n γ (Kn ) ≤
They provide the following general lower and upper bounds for γ t (Kn ). Theorem 56 (Garnick, Nieuwejaar [56]) For all m, n ≥ 5, mn/7 ≤ γ t (Km,n ) ≤ (mn + 2n + 89)/7. They provide both exact values
and improved upper bounds for γ t (Kn ), as shown in Tables 20 and 21. In 2002 [91] Watkins and Ricci initiate the study of kings domination on a torus. In 2003 Favaron, Fricke, Pritikin, and Puech
establish the following results involving irredundant sets of kings. Theorem 57 (Favaron, Fricke, Pritikin, Puech [48]) For n ≥ 6, (n − 1)2 /3 ≤ IR(Kn ) ≤ n2 /3. Theorem 58 (Favaron, Fricke,
Pritikin, Puech [48]) For n ≥ 6, (n − 2)2 /3 + 3 ≤ Γ (Kn ) ≤ n2 /3. Theorem 59 (Favaron, Fricke, Pritikin, Puech [48]) For n ≥ 1, n2 /9≤ ir(Kn ) ≤(n + 2)/32 , and ir(Kn ) = n2 /9 when n ≡ 0 (mod 3).
In Table 22 the authors prove the first few values of ir(Kn ), (Kn ), and IR(Kn ), cf. Figure 24 for a minimaximal irredundant set of kings on K7 .
378 Table 22 Values of ir(Kn ), γ (Kn ), IR(Kn )
J. T. Hedetniemi and S. T. Hedetniemi n ir(Kn ) (Kn ) IR(Kn )
Fig. 24 ir(K7 ) = 8
The authors also offer an intermediate value, or interpolation, theorem for the cardinalities of maximal independent sets in kings graphs Kn . Theorem 60 (Favaron, Fricke, Pritikin, Puech [48]) For
any n ≥ 1 and positive integer t such that i(Kn ) ≤ t ≤ α(Kn ), there exists a maximal independent set of t kings on Kn .
8 Rooks The structure of rooks graphs Rn is the simplest of all chessboard graphs. Therefore, the values of all seven domination parameters are fairly easy to establish. Theorem 61 (Yaglom and Yaglom
Yaglom and Yaglom [96]) For n ≥ 1, γ (Rn ) = i(Rn ) = α(Rn ) = n. Corollary 62 For n ≥ 2, γ (Rn ) = γ t (Rn ). The following three results are given, but stated as unpublished, in [52]. Theorem 63
(Hedetniemi, Hedetniemi, Wallis) For n ≥ 1, ir(Rn ) = n. Theorem 64 (Hedetniemi, Hedetniemi, Wallis) For n ≥ 1, Γ (Rn ) = n. Theorem 65 (Hedetniemi, Jacobson, Wallis) For n ≥ 4, IR(Rn ) = 2n − 4, IR
(R1 ) = 1, IR(R2 ) = 2, and IR(R3 ) = 3.
Domination in Chessboards
Table 23 Lower bound of 31 ≤ (R53 ) R R R
R R R
R R R R
R R
R R R R
R R R R
R R R R R
In 1994 Koehler, in his MS research paper, initiates the study of chessboard domination problems in three-dimensional rooks graphs. Theorem 66 (Koehler [66]) For n ≥ 1, γ (Rn3 ) = i(Rn3 ) =
n2 2 .
Theorem 67 (Koehler [66]) For n ≥ 1, α(Rn3 ) = n2 . Theorem 68 (Koehler [66]) For n ≥ 3, (Rn3 ) ≥ 3(n − 2)2 + 4. Table 23 illustrates a minimal dominating set of 31 rooks on the threedimensional
rooks graph R53 . Each level is a 5 × 5 rooks graph, and there are five levels. A rook dominates every vertex in its row and column on its level, and all rooks above and below it. In 2008 [33] Chen
and Ho initiate the study of rooks domination on what are called sawtoothed chessboards, or STC for short. These are chessboards whose boundary forms two staircases from left down to right without
any holes inside. A rook at square (i, j) still dominates all squares in row i and column j. In this paper, the authors represent an STC by two particular graphs: a rooks graph and a board graph.
They show that for an STC, the rooks graph is the line graph of the board graph, and the board graph is a bipartite permutation graph. Thus, the rooks domination problem on STCs can be solved by any
algorithm for solving the edge domination problem on bipartite permutation graphs.
9 Other Varieties of Chessboard Domination Problems In this concluding section we briefly mention a variety of other types of chessboard domination problems that have been considered in the
literature; they are mixed and quite varied. 1. In 2009 [32] Chatham, Doyle, Fricke, Reitmann, Skaggs, and Wolff consider the general problem of placing a fixed number k of pawns on a chessboard in
such a way as to influence either the maximum number of independent pieces
5. 6. 7.
J. T. Hedetniemi and S. T. Hedetniemi
of one kind that can be placed on the board, or the minimum number of pieces necessary to cover all squares. The main result in this paper is that for each positive integer k and each n > max{87 + k,
25k}, it is possible to place k pawns and n + k independent queens on Qn . The authors consider the same problem for bishops and rooks. In 1998 [87] Theron and Geldenhuys consider queens domination
in beehive or hexagonal chessboards, in which each square is a hexagon. For example, in square n × n hexagonal chessboards they show that the diagonal queens domination number equals n − 1. In 1999
[21] Burger and Mynhardt study queens domination on hexagonal boards and show that on hexagonal boards with n ≥ 1 rows and diagonals, for n ≡ 3 (mod 4), there are only two types of minimum dominating
sets of queens. The authors also study the queens irredundance numbers on 5 × 7 hexagonal boards. In 2000 [7] Bode and Harborth study the independence numbers of chess-like pieces on boards whose
cells are either triangles or hexagons, and for many of these pieces they determine the independence numbers. In 2000 [86] Theron and Burger study queens domination on hexagonal boards. In 2003 [9]
Bode, Harborth, and Harborth study the kings independence numbers on triangle-cell chessboards. In 2003 [8] Bode and Harborth study three types of knights on triangle-cell and hexagonal boards, and
determine the independence numbers for two of these types of knights, and for one residue class mod 4 for the third type. In 2005 [82] Sinko and Slater introduce the study of several
domination-related parameters in chessboards, called influence parameters. The influence of a vertex v in a graph G equals I(v) = |N[v]|, the number of vertices it dominates. The influence of a set S
equals the sum of the influences of its vertices, that is, I(S) = v ∈ S I(v) = v ∈ S |N[v]|. A vertex set S is called an efficient dominating set, or a perfect code if for every vertex v ∈ V , |N[v]
∩ S| = 1. Since not every graph G has an efficient dominating set, one can instead consider the maximum number of vertices that can be dominated by a set S subject to the restriction that no vertex
is dominated more than once; this is called the efficient domination number, denoted F(G). This restriction means that the set S must be a packing, that is, for every u, v ∈ S, d(u, v) ≥ 3. Thus, F
(G) = max{I(S) : S is a packing }. Similarly one might seek to minimize the total amount of domination, given that every vertex must be dominated at least once. This gives rise to a parameter called
the total redundance R(G) = min{I(S) : S dominates V (G)}. In this paper, the authors consider the values of these and several other related parameters on rectangular rooks, kings, and knights
graphs. In 2006 [83] Sinko and Slater study the efficient domination number F(Nm,n ) on rectangular knights graphs Nm,n . They provide the following initial values in Table 24. In Table 25 we
illustrate a set of three knights, labeled N, dominating a set of 19 squares at most once, labeled X.
Domination in Chessboards Table 24 Small values of F(Nm,n )
381 Nm,n F(Nm,n )
N1,n n
N2,n 2n
N3,8t 20t
Table 25 F(N5,5 ) = 19
Table 26 Values of knights γ (Nm,n ) and number of solutions
N3,3 7
N3,4 12 X X X X N
m/n 3 4 5 6 7 8
3 4/8
4 4/15 4/9
5 4/6 4/3 5/47
6 4/2 4/1 6/46 8/127
N4,4 12
X X X X
N5,5 19 X
N N X
7 6/10 6/1 7/47 8/4 10/10
X X
X X X X
8 8/1192 8/579 7/1 8/1 11/2 12/2
9. In 2009 [18] Burchett, Lane, and Lachniet consider the problem of the minimum number of rooks in set S such that every unoccupied square is covered by at least k rooks in S (k-domination), or in
such a way that every square, including squares occupied by a rook, are covered at least k times (k-tuple domination). 10. In 2010 [85] Steinbach and Posthoff develop a computation methodology based
on Boolean models to compute the domination, independent domination, and vertex independence numbers of rectangular m × n bishops graphs. 11. In 2011 [12] Burchett continues the study of k-tuple
domination in the rooks graph, as well as the border queens domination problem. 12. In 2012 [4] Berghammer initiated the study of domination, independent domination, and total domination in
rectangular m × n chessboards, by describing a simple computing technique, based on relational modeling, which is applicable to a variety of other chessboard problems. He presents tables for the
domination and independence numbers, and the number of solutions, for rooks, kings, knights, and bishops, for 3 ≤ m ≤ 8 and 3 ≤ n ≤ 8, a sample of which is given in Table 26. 13. In 2013 [45] DeMaio
and Tran study the domination number and vertex independence number of triangular-shaped hexagonal boards, having n hexagons on each of three exterior sides. We denote such boards by TRn for rooks,
TRn for bishops, TNn for knights, and TKn for rooks. They show the following: (i) n/5 γ (TRn ) = n, (ii) α(TRn ) = n, (iii) α(TBn ) = n, and (iv) γ (T Nn ) ≤ i=1 (4i − 1). 14. In 2014 [42] Cooper,
Pikhurko, Schmitt, and Warrington solve the following problem posed by Martin Gardner: What is the smallest number of queens you can put on Qn so that no additional queen can be added without
creating three
15. 16.
J. T. Hedetniemi and S. T. Hedetniemi
in a row, column, or diagonal? The authors prove that this number is at least n, unless n ≡ 3 (mod 4), in which case n − 1 may suffice. In 2014 [44] DeMaio and Lightcap study kings total domination
numbers on square n × n hexagonal boards. In 2018 [17] Burchett and Chatham study more chessboard separation problems, such as the maximum number of independent rooks and bishops that can be placed
on an n × n board containing k pawns, and all the values of k for which there is a placement of k pawns that allows the placement of n + k independent rooks on an n × n board. They also study the
same problem for bishops. For queens, they find lower bounds on the queens domination-, total domination-, paired domination-, and connected domination-separation numbers. In 2018 [31] Chatham
considers the domination number, the independence number, and the independent domination number of dragon king boards and dragon horse boards. A dragon king moves like a rook and a king, while a
dragon horse moves like a bishop and a king. These are pieces from the chesslike game called shogi. In 2018 [75] Pahlavsay, Palezzato, and Torielli consider 3-tuple total domination in rectangular
rooks graphs. The authors give a formula for the 3-tuple total domination number of an m × n rooks graph.
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Domination in Digraphs Teresa W. Haynes, Stephen T. Hedetniemi, and Michael A. Henning
AMS Subject Classification: 05C69
1 Introduction Domination in digraphs is relatively unexplored if compared to its counterpart in graphs. In this chapter, we present selected results on domination in digraphs and give some
background on the related topics of bases and kernels. The first two Ph.D. dissertations devoted to the study of domination in digraphs were written by Changwoo Lee [62] in 1994 and by Lisa Hansen
[46] in 1997. A survey of results prior to 1998 on domination in directed graphs by Ghoshal, Laskar, and Pillone [43] is given in Chapter 15 of [54]. For completeness, many of these results are
T. W. Haynes () Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN, USA Department of Mathematics and Applied Mathematics, University of Johannesburg,
Johannesburg, South Africa e-mail: [email protected] S. T. Hedetniemi School of Computing, Clemson University, Clemson, SC, USA e-mail: [email protected] M. A. Henning Department of Mathematics and
Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. W. Haynes et
al. (eds.), Structures of Domination in Graphs, Developments in Mathematics 66, https://doi.org/10.1007/978-3-030-58892-2_13
T. W. Haynes et al.
here. We first present some terminology. For terminology and notation not found here, we refer the reader to the glossary in chapter “Glossary of Common Terms” of this volume.
1.1 Basic Terminology and Notation Throughout this chapter, we let D = (V, A) be a finite directed graph, or digraph, with a finite vertex set V = V (D) and an arc set A = A(D) ⊆ V × V , which is a
subset of the Cartesian product V × V , consisting of all ordered pairs of vertices in V , where neither loops (u, u) nor multiple arcs (u, v) and (u, v) are allowed, although pairs of opposite arcs,
such as (u, v) and (v, u), are allowed. Also, G = (V, E) stands for a simple, finite, undirected graph with vertex set V (G) and edge set E(G), which consists of a subset of the set of all unordered
pairs uv = vu of distinct vertices in V . For two vertices u, v ∈ V and an arc (u, v) ∈ A, we say that: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)
(u, v) is an arc from u to v, u is adjacent to v, v is adjacent from u, v is an out-neighbor of u, u is an in-neighbor of v, v is a successor of u or the terminal vertex of the arc, u is a
predecessor of v or the initial vertex of the arc, u and v are incident to arc (u, v), and arc (v, u) is the reverse of arc (u, v).
We also denote an arc (u, v) by u → v. If both arcs (u, v) and (v, u) are in A, we denote this by u ↔ v; and this is called a bidirected or symmetric arc. A digraph D = (V, A) is called oriented or
anti-symmetric if for every (u, v) ∈ A, we have (v, u)∈A, that is, D has no symmetric arcs. Equivalently, an oriented digraph can be obtained from a graph G by assigning a direction, either u → v or
v → u, to each edge uv of G. + The outset or out-neighborhood of a vertex u ∈ V is the set of vertices ND (u) = − {v | u → v ∈ A}, while the inset or in-neighborhood of vertex u is the set ND (u) = +
{v | u ← v ∈ A}. The outdegree of vertex u, denoted odD (u) or dD (u) in the + − (u)|, while the indegree of u, denoted idD (u) or dD (u) in the literature, equals |ND − literature, equals |ND (u)|.
The maximum indegree of a digraph D, denoted − (D), is the maximum indegree among the vertices in D. The maximum outdegree of D is defined as expected and is denoted + (D). Similarly, the minimum
indegree and minimum outdegree of D are denoted δ − (D) and δ + (D), respectively. The degree of a vertex v in D is dD (v) = odD (v) + idD (v). We note that v∈V (D)
odD (v) =
v∈V (D)
idD (v).
Domination in Digraphs
A digraph is r-regular if odD (v) = idD (v) = r for every vertex v of D. We also + + define the closed out-neighborhood of a vertex v to equal ND [v] = ND (v) ∪ {v} − − and similarly the closed
in-neighborhood to equal ND [v] = ND (v) ∪ {v}. The + + (S) = ∪v∈S ND (v), and the closed out-neighborhood of a set S of vertices is ND + + out-neighborhood of S is ND [S] = ∪v∈S ND [v]. And finally,
the in-neighborhood − − − of S is ND (S) = ∪v∈S ND (v), and the closed in-neighborhood of S is ND [S] = − ∪v∈S ND [v]. Let S ⊆ V and u ∈ S. A vertex v ∈ V \ S is called a private out-neighbor of u
with − respect to S if ND (v)∩S = {u}, that is, v is an out-neighbor of u, u → v, but is not an out-neighbor of any other vertex in S. The set of all private out-neighbors of u with respect to S is
denoted by pn+ D (u, S). Similarly, a vertex v ∈ V \ S is called a private + in-neighbor of u with respect to S if ND (v) ∩ S = {u}, that is, v is an in-neighbor of u, u ← v, but is not an
in-neighbor of any other vertex in S. The set of all private in-neighbors of u with respect to S is denoted by pn− D (u, S). If the digraph D is clear from context, we omit the subscript D from the
above notational definitions. For example, we simply write id(u), od(u), N− (u), N+ (u), − + pn+ (u, S), and pn− (u, S), rather than idD (u), odD (u), ND (u), ND (u), pn+ D (u, S), − and pnD (u, S),
respectively. A vertex u is called: (i) an isolated vertex if od(u) = id(u) = 0, (ii) a source or transmitter if id(u) = 0 and od(u) > 0, and (iii) a sink or receiver if od(u) = 0 and id(u) > 0.
Given two sets R, S ⊆ V , we let (R, S) denote the set of all arcs in A from R to S, that is, (R, S) = {(u, v) ∈ A | u ∈ R, v ∈ S}. For any integer k ≥ 1, we use the standard notation [k] = {1, . . .
, k} and [k]0 = [k] ∪{0} = {0, 1, . . . , k}. A directed walk in a digraph D = (V, A) from a vertex u to a vertex w, called a (u, w)-walk, is a sequence of vertices of the form u = v0 , v1 , . . . ,
vk = w such that for every i ∈ [k], we have (vi−1 , vi ) ∈ A. Such a (u, w)-walk has length k. A directed walk having no repeated edges is called a directed trail. A directed walk having no repeated
vertices is called a directed path. A directed walk in which v0 = vk is called a closed directed walk, and a closed walk in which all vertices, except v0 and vk , are distinct is called a directed
cycle or a circuit. Let C'n denote the directed cycle on n vertices. The distance dD (u, v) from a vertex u to a vertex v in a digraph D is the minimum length of a directed (u, v)-path. If the
digraph D is clear from the context, we write d(u, v) rather than dD (u, v). Given a digraph D = (V, A), the underlying graph of D is the undirected graph G(D) = (V, E), where uv ∈ E if and only if u
→ v ∈ A, u ← v ∈ A, or u ↔ v ∈ A. A digraph D is connected or weakly connected if its underlying graph G(D) is connected. A digraph D is said to be strongly connected if for every u, w ∈ V , there
exist a directed (u, w)-path and a directed (w, u)-path. We note that one could consider the class of digraphs having the property that for every u, w ∈ V either there is a directed walk from u to w
or there is a directed walk from w to u. A digraph D = (V, A) is said to be transitive if (u, v), (v, w) ∈ A implies that the arc (u, w) ∈ A. In other applications, a digraph D of order n is said to
have a transitive orientation if there is an ordering of the vertices v1 , v2 , . . . , vn such that for every
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i ∈ [n − 1], we have (vi , vi+1 ) ∈ A. A digraph is complete if for every u, v ∈ V , either (u, v), (v, u), or both arcs are in A. A tournament is an oriented complete graph. We denote the degree of
a vertex v in an undirected graph G by dG (v), or simply by d(v) if the graph G is clear from context. The average degree in G is denoted by dav (G). The minimum degree among the vertices of G is
denoted by δ(G) and the maximum degree by (G).
1.2 Domination and Independence In this section we define independence and the types of domination in digraphs that will be discussed in this chapter. Let D = (V, A) be a digraph with vertex set V
and arc set A. Definition 1 A set S of vertices in a digraph D is independent if no two vertices u, v ∈ S are joined by an arc, that is, (u, v)∈A and (v, u)∈A. The maximum cardinality of an
independent set in a digraph D is called the vertex independence number of D and is denoted α(D), while the minimum cardinality of a maximal independent set of vertices in a digraph is the lower
vertex independence number, denoted αmin (D). Definition 2 A set S of vertices in a digraph D is an out-dominating set, or just a dominating set, if for every vertex v ∈ V \ S, there exists a vertex
u ∈ S such that u → v ∈ A, that is, every vertex in V \ S is adjacent from a vertex in S. In other words, S is a dominating set of D if V \ S ⊆ N+ [S]. The minimum cardinality of dominating set in D
is called the out-domination number, or simply the domination number, of D and is denoted γ + (D), or just γ (D). In general, we adopt the simplified terminology for out-dominating sets by omitting
“out” and simply referring to dominating sets, domination number, and γ (D). Definition 3 A set S of vertices in a digraph D is an in-dominating set (also called a converse dominating set in the
literature) if for every vertex v ∈ V \ S, there exists a vertex u ∈ S such that v → u ∈ A, that is, every vertex in V \ S is adjacent to a vertex in S. In other words, S is an in-dominating set of D
if N+ (v) ∩ S=∅. The minimum cardinality of an in-dominating set in a directed graph D is called the in-domination number of D and is denoted γ − (D). Definition 4 A set S of vertices in a digraph D
is a twin dominating set of D if it is both an in-dominating set and out-dominating set of D. The minimum cardinality of a twin dominating set is the twin domination number γ ± (D) of D (also denoted
γ ∗ (D) in the literature). To illustrate the above definitions, consider the digraph D shown in Figure 1. The darkened vertices in Figure 1(a) and 1(b) form a dominating set and an indominating set,
respectively, of D, while the darkened vertices in Figure 1(c) form a twin dominating set of D.
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(a) γ(D) = 2
(b) γ− (D) = 2
(c) γ± (D) = 3
Fig. 1 A digraph D with γ (D) = γ − (D) = 2 and γ ± (D) = 3
2 Background and History In this section, we recognize and honor Dénes König for his pioneering work on domination in digraphs. His work on the basis of a digraph, which we shall see is an
independent dominating set, comes some 30 years before any other mention of domination in the literature. Since König was the originator of domination in digraphs, we give several of his theorems
along with their proofs. In the second part of this section, we present a brief overview of kernels in digraphs, which we shall see are independent in-dominating sets. We include some of Berge’s
early results on kernels with a sampling of proofs. We also give some results on the existence of kernels in digraphs. A survey of the expansive literature on kernels is beyond the scope of this
chapter, so our brief overview is not meant to be complete. For more information we refer the reader to surveys by Boros and Gurvich [12] and Frankel [37], respectively.
2.1 Basis of the Second Kind The concept of domination in digraphs was introduced as early as 1936 by König [61]. We present his original ideas in what follows, as they form a foundation on which
many ideas for domination in digraphs can be built. For any vertex a ∈ V in a digraph D = (V, A), let Va equal the set consisting of a together with all vertices x for which there exists a directed
path from a to x. If there is no vertex b ∈ V such that Va ⊂ Vb , then Va is called a basic set with source a. Theorem 1 ([61]) Every vertex a ∈ V of a finite directed graph D = (V, A) is a member of
some basic set of D. Proof Let a ∈ V . If Va is a basic set, then clearly a is a member of a basic set. By definition, if Va is not a basic set, then there exists a vertex b ∈ V such that Va ⊂ Vb ,
which implies that there must exist a directed path from b to a. Thus, if Vb is a basic set, then a is a member of the basic set Vb . Again, if Vb is not a basic set, then by definition, there exists
a vertex c ∈ V such that Va ⊂ Vb ⊂ Vc . If Vc is a basic set, then a is a member of the basic set Vc . Since V is a finite set, this process must end with a vertex x ∈ V , such that Vx is a basic set
containing a.
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König pointed out that this theorem does not hold for infinite directed graphs, using the example of an infinite directed path v1 , v2 , v3 , . . . , in which every arc has the form (vi+1 , vi ). It
is easy to see that this infinite directed path has no basic set. Theorem 2 ([61]) No proper subset of a basic set is a basic set. Proof Suppose, to the contrary, that a basic set Vb contains a basic
set Va as a proper subset. Since there is a directed path from b to a, and since Va is a basic set, Va cannot be properly contained in another basic set. Thus, it follows that there must be a
directed path from a to b. From this it follows that Vb must be a subset of Va and thus that Va = Vb . But this means that Va is not a proper subset of Vb , a contradiction. We can now define a basis
of a directed graph. Definition 5 A basis of a directed graph D = (V, A) is a set B ⊂ V having the following two properties: (i) for every vertex v ∈ V \ B, there exist a vertex u ∈ B and a directed
path from u to v. (ii) for every pair of vertices u, v ∈ B, there is no directed path from u to v. Theorem 3 ([61]) Every finite directed graph D = (V, A) has a basis. Proof Let V = {Va , Vb , . . .
, Vk } be the set of all basic sets of a finite directed graph D = (V, A), and let B = {a, b, . . . , k} be sources for each of these basic sets. We claim that the set B is a basis of D. Note that
Theorem 1 says that every vertex v ∈ V is a member of some basic set, say v ∈ V ∈ V. Assume that v ∈ V \ B. But V = Vw for some Vw ∈ V and w ∈ B, since V contains all basic sets. Thus, by definition
there must be a directed path from w to v, and property (i) in Definition 5 is satisfied. In order to show that B satisfies property (ii) in Definition 5, suppose, to the contrary, that for two
sources a and b in B, where Va =Vb , there is a directed path from a to b. But in this case, it follows that Vb ⊆ Va . However, if Vb ⊂ Va , then Vb cannot be a basic set, a contradiction. On the
other hand, if Vb = Va , then we contradict the supposition that Va =Vb . Theorem 4 ([61]) If a vertex a ∈ V is contained in a basis B in a directed graph D = (V, A), then Va is a basic set. Proof
Assume that a vertex a ∈ V is contained in a basis. Suppose, to the contrary, that Va is not a basic set. Then there must exist a vertex b ∈ V not contained in Va such that Va is a proper subset of
Vb . Therefore, there must be a directed path from b to a. But if this is the case, then b does not belong to the basis B, since by property (ii) there can be no directed path between two vertices in
a basis. Therefore, there must be a directed path from a vertex c of B to b, where c=a, for otherwise b would belong to Va . The directed paths from c to b and from b to a imply, by Theorem 1, that
there exists a directed path from c to a, contradicting property (ii) in the definition of a basis.
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Theorem 5 ([61]) Every basis B in a digraph D = (V, A) consists of one source from each basic set. Proof By Theorem 4, every vertex of a basis B is a source of a basic set. In addition, two distinct
vertices in B are never sources of the same basic set, since by property (ii) there can be no directed path between two vertices in B. It only remains to show that every basic set has a source in B.
Suppose there exists a basic set Va with source a such that a ∈ B. By the definition of basis, there is a vertex b ∈ B such that there is a directed path from b to a. But b is the source of a basic
set Vb , and so the basic set Va is a proper subset of the basic set Vb , contradicting Theorem 2. Corollary 6 ([61]) Every basis of a digraph D has the same cardinality, which equals the number of
source vertices in D. Proof By Theorem 5, since every basis has one source from each basic set, every basis has a cardinality equal to the number of basic sets in D. In his book, König pointed out
that if every edge of a digraph D is symmetric, and the digraph D is basically an undirected graph, then the number of basic sets equals the number of components. König then defined a basis of the
second kind as follows. Definition 6 A basis of the second kind in a directed graph D = (V, A) is a set B ⊂ V satisfying the following two conditions: (i) if v is a vertex in V \ B, then there is an
arc (u, v) from a vertex u ∈ B to v, and (ii) there is no arc between two vertices in B. Notice that by property (i) a basis of the second kind is a dominating set of D and by (ii) a basis of the
second kind is an independent set of D. König noted that Corollary 6 is no longer true for bases of the second kind, i.e., for independent dominating sets. In the case where a digraph D is symmetric,
König’s basis of the second kind appears to be the first time in the literature where an independent dominating set is defined in an undirected graph. It also, of course, defines an independent
dominating set in a digraph for the first time. To illustrate a minimum independent dominating set in an undirected graph, König used as an example the classical problem of covering an 8 × 8
chessboard with the minimum number of queens. The Queen’s graph consists of 64 vertices (one for each square on the chessboard), where two vertices/squares are adjacent if and only if a queen placed
on one square can occupy the second square in 1 move. Thus, two vertices are adjacent if and only if they are in the same row, column, or diagonal. The minimum number of queens needed to cover the
chessboard (the domination number of the Queen’s graph) is 5. König’s example of five queens, placed at the locations shown in Figure 2, covers the board with the added constraint that no two queens
can attack each other, that is, this placement of these five queens represents a minimum independent dominating set of the Queen’s graph.
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Fig. 2 Minimum independent dominating set of queens
An independent dominating set of a digraph is also called a solution in the literature. In the context of games, a solution is defined by Von Neumann and Morgenstern in their now classic book [92].
We formally state the definition of a solution in terms of digraphs and give notation for a minimum independent dominating set. Definition 7 A solution in a digraph D is an independent dominating set
of D. The solution number of D, denoted i+ (D), equals the minimum cardinality of a solution in D, that is, i + (D) = αmin (D). Richardson [79] showed that every digraph with no odd cycles has at
least one solution.
2.2 Kernels in Digraphs In 1958, Berge [6] defined an in-dominating set, which he called an absorbant set. Although he called the in-domination number the absorption number and denoted it by β(D), we
shall continue with the terminology in-domination and denote the in-domination number as γ − (D), as defined in Section 1.2. Definition 8 A kernel in a digraph D is an independent, in-dominating
(absorbant) set of D. The kernel number of D equals the minimum cardinality of a kernel in D and is denoted i− (D). The topic of kernels in digraphs has its roots in game theory and was introduced by
Von Neumann and Morgenstern in 1944 [92]. Kernel applications have grown from n-person games and Nim-type games to more recent applications in artificial intelligence, combinatorics, and coding
Domination in Digraphs
Fig. 3 A graph with a solution but no kernel
We note that not every digraph has a kernel; for example, a directed cycle C'5 does not. Neither does C'5 have a solution. The graph in Figure 3 has a solution, consisting of the three vertices of
indegree zero, but it has no kernel. For digraphs with kernels, Berge [6] proved the following. Theorem 7 ([6]) If S is a kernel, then S is both a maximal independent set and a minimal in-dominating
set. Proof Let S ⊆ V be a kernel in a digraph D = (V, A). Since S is an in-dominating set, for each vertex u ∈ V \ S, there is an arc (u, v) ∈ A where v ∈ S. Hence, S ∪{u} is not an independent set,
and so, S is a maximal independent set. Similarly, if u ∈ S, then S \{u} is not an in-dominating set since S is an independent set, and therefore there is no arc (u, v) for any v ∈ S \{u}. Thus, S is
a minimal in-dominating set. Since not all digraphs have kernels, a natural question to ask is: What structural properties of digraphs imply the existence of a kernel? The existence of a kernel in a
given digraph has been studied in many papers, including [5, 25, 26, 41, 79]. Berge [7] gave a necessary and sufficient condition for a vertex set to be a kernel in terms of its characteristic
function. Recall that the characteristic function φ S : V →{0, 1} of a set S is defined as: φ S (x) = 1 if x ∈ S and φ S (x) = 0 if x∈S. We will assume that if a vertex x has no out-neighbors, then
max{φS (y) | y ∈ N + (x)} = 0. Theorem 8 ([7]) A set S ⊆ V is a kernel of a digraph D = (V, A) if and only if for every x ∈ V , φS (x) = 1 − max{φS (y) | y ∈ N + (x)}. Proof Let S be a kernel in a
digraph D, and assume that φ S is the characteristic function defined on it. If x ∈ S, then φ S (x) = 1. Since S is an independent set, no out-neighbor of x is in S. Thus, max{φS (y) | y ∈ N + (x)} =
0, and therefore, φS (x) = 1 = 1 − max{φS (y) | y ∈ N + (x)}. If x∈S, then φ S (x) = 0. Since S is an in-dominating set, it follows that there must be a vertex v ∈ S and an arc (x, v) ∈ A. Thus, max
{φS (y) | y ∈ N + (x)} = 1, and therefore, φS (x) = 0 = 1 − max{φS (y) | y ∈ N + (x)}. Conversely, let S be a set for which, for every x ∈ V , φS (x) = 1 − max{φS (y) | y ∈ N + (x)}. If x ∈ S, then φ
S (x) = 1. Thus, since φS (x) = 1 − max{φS (y) | y ∈ N + (x)}, it must follow that max{φS (y) | y ∈ N + (x)} = 0, but this means that no out-neighbor of x is in S. If an in-neighbor of x, say y, is
in S, then x is an outneighbor of y, and therefore, φ S (y) = 1. But max{φS (x) | x ∈ N + (y)} = 1, and so, 1 − max{φS (x) | x ∈ N + (y)} = 0, a contradiction. Therefore, S is an independent set.
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Similarly, if x∈S, then φ S (x) = 0. But since, for every x ∈ V , φS (x) = 1 − max{φS (y) | y ∈ N + (x)}, this must mean that max{φS (y) | y ∈ N + (x)} = 1. Hence, at least one neighbor of x, say y,
is in S. Therefore, S is an in-dominating set. As early as 1936, König [61] proved the following result. A digraph D = (V, A) is called transitive if whenever (u, v) ∈ A and (v, w) ∈ A, then (u, w) ∈
A. Theorem 9 ([61]) If D = (V, A) is a transitive digraph, then every minimal indominating set has the same cardinality. Furthermore, a set S ⊆ V is a kernel if and only if S is a minimal
in-dominating set. Corollary 10 Every transitive digraph has a kernel, and all of its kernels have the same cardinality. In 1990 De la Vega [29] showed that although not all digraphs have kernels,
probabilistically speaking, almost all digraphs do. Let D(n, p) = (V, A) denote a random digraph of order n where for every u, v ∈ V , the arc (u, v) is chosen with probability p. Theorem 11 ([29])
For any probability p, where 0 ≤ p ≤ 1, the probability that the random digraph D(n, p) has a kernel goes to 1 as n →∞. Algorithms for determining all the kernels of a digraph D have been presented
by Rudeanu [81] in 1966 and Roy [80] in 1970. Many of the existence results for kernels are proved under an even stronger condition that the digraph is kernel-perfect. A digraph D is said to be
kernelperfect if D has a kernel and every induced subdigraph of D has a kernel. Meyniel conjectured that if every circuit of a digraph D has at least two chords, then D is kernel-perfect. Although
Galeana-Sánchez [39] proved this conjecture to be false, the searching for a proof motivated results on sufficient conditions for the existence of a kernel in a digraph. The proof we present of the
following result of Von Neumann and Morgenstern [92] is due to Berge [7]. Theorem 12 ([92]) Every digraph D without directed cycles is kernel-perfect and has a unique kernel. Proof Given a digraph D
having no directed cycles, define the set S0 as the collection of sinks of D, and for each k ≥ 1, define Sk as the set of all vertices u such that a longest (directed) path from u to a vertex in S0
has length k. Thus, S0 = {v ∈ V | N + (v) = ∅}. S1 = {v ∈ V | N + (v) ⊆ S0 }. S2 = {v ∈ V | N + (v) ⊆ (S0 ∪ S1 )}. And in general, Sk = {v ∈ V | N+ (v) ⊆ (S0 ∪ S1 ∪ . . . ∪ Sk−1 )}. Since D contains
no directed cycles, the sets Sk form a partition of V (D). One can then define a characteristic function φS (x) = 1 − max{φS (y) | y ∈ N + (x)}
Domination in Digraphs
iteratively, starting with the vertices u ∈ S0 for each of which φ S (u) = 1, and then each vertex in S1 receives the value 0. After this, a vertex x can be assigned a value φ S (x) only after all of
the vertices in N+ (x) have been assigned a value, at which point the value of max{φS (y) | y ∈ N + (x)} can be determined. By Theorem 8, S = {v ∈ V | φ S (v) = 1} is a kernel of D. Since every
subdigraph of D is acyclic, it follows that D is kernel-perfect. Moreover, the set S0 of sinks is nonempty and unique, and so by definition, Sk is unique for each k ≥ 1. The uniqueness of S follows
from the fact that any kernel of D must contain S0 , and hence the vertices of Sk ∩ S. We next mention classical results due to Richardson [79] and Duchet [25]. Theorem 13 ([79]) Every digraph D
without odd directed cycles is kernel-perfect. Theorem 14 ([25]) If every circuit in a digraph D has at least one symmetric arc, then D is kernel-perfect. Recall that a kernel in a digraph D is an
independent set S such that every vertex not in S dominates some vertex in S, where as usual by “dominates” we mean “outdominates,” that is, a vertex u dominates a vertex v if there is an arc (u, v)
from u to v. We next define a semi-kernel in a digraph. Recall that the distance dD (u, v) from a vertex u to a vertex v in a digraph D is the shortest directed path from u to v. We note that dD (u,
v) may be very different from dD (v, u). Definition 9 A set S of vertices in a digraph D is a semi-kernel if S is an independent set and every vertex not in S either dominates some vertex in S or
dominates a vertex which in turn dominates some vertex in S. Thus, S is a semi-kernel in D if S is an independent set and for every vertex v ∈ V (D) \ S, there is a vertex u ∈ S such that dD (v, u) ≤
2. As observed earlier, not all digraphs have kernels. However, every digraph has a semi-kernel. This result is attributed to Chvátal and Lovász [24]. However, in this paper they proved Theorem 16,
which we state shortly. It is not clear if Theorem 16 immediately implies Theorem 15. The proof of the following result is due to Bondy [11]. Theorem 15 ([11]) Every digraph has a semi-kernel. Proof
Let D be a digraph and let H be a maximal induced acyclic subdigraph of D. By Theorem 12, the acyclic digraph H has a (unique) kernel. Let S be the kernel of H. We claim that S is a semi-kernel of D.
Since S is a kernel of H, every vertex of H − S dominates some vertex of S. Let v be an arbitrary vertex outside H, and so v ∈ V (D) \ V (H). By our choice of H, there is a directed cycle C in the
subdigraph of D induced by V (H) ∪{v}. The vertex v therefore dominates its successor v+ on C. Since v+ ∈ V (H), either v+ ∈ S, in which case v dominates a vertex of S, or v+ ∈S, in which v+
dominates a vertex of S and therefore v dominates a vertex which in turn dominates some vertex of S. Thus, S is a semi-kernel of D.
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Definition 10 For any integer k ≥ 2, a set S of vertices in a digraph D is a kdominating set if S is an independent set and every vertex not in S can be reached from a vertex of S by a directed path
of length at most k, that is, for every vertex v ∈ V (D) \ S, there is a vertex u ∈ S such that d(u, v) ≤ k. We note that a 1-dominating set of a digraph D is an independent, (out-) dominating set of
D. For k ≥ 1, every k-dominating set is a (k + 1)-dominating set. In particular, every 1-dominating set is a 2-dominating set. Not every digraph has a 1-dominating set; for example, C' 5 does not. In
1974 Chvátal and Lovász [24] proved that every digraph has a 2-dominating set. Theorem 16 ([24]) Every digraph has a 2-dominating set. Proof We proceed by induction on the order n of a digraph D. For
n = 1 or n = 2, the result is immediate. Let n ≥ 3 and assume that every digraph of order less than n has + a 2-dominating set. Let w be an arbitrary vertex of D. If V (D) = ND (w), then the set {w}
is a 1-dominating set and therefore also a 2-dominating set. Hence, we may + assume that V (D) = ND (w). Let D be the subdigraph of D induced by the set of vertices at distance at least 2 from w in
D. Thus, V (D ) = {v ∈ V (D) | dD (w, v) ≥ 2}. Further, (x, y) ∈ A(D ) if and only if x, y ∈ V (D ) and (x, y) ∈ A(D). Applying the inductive hypothesis, the digraph D contains a 2-dominating set S .
Suppose firstly that there is an arc from u to w for some vertex u ∈ S . Therefore, dD (u, w) = 1, and + every vertex in ND (w) is reachable from u by a directed path of length at most 2, + [w]. In
this case, let S = S . Suppose that is, dD (u, x) ≤ 2 for every vertex x ∈ ND secondly that there is no arc from a vertex in S to the vertex w, and so dD (u, w) ≥ 2 for all vertices u ∈ S . In this
case, we let S = S ∪{w}. In both cases, the set S is a 2-dominating set of D. As observed earlier, not every digraph has a 1-dominating set. In 1996 Jacob and Meyniel [59] proved that a digraph with
no 1-dominating set contains at least three 2-dominating sets. Theorem 17 ([59]) Every digraph with no 1-dominating set contains at least three 2-dominating sets. Kernels have relations to Grundy
functions in digraphs. We conclude this subsection with some results relating the two. Definition 11 A non-negative function g: V → [n]0 from the vertex set V of a digraph D to the integers [n]0 is
called a Grundy function if for every vertex u ∈ V, g(u) is the smallest non-negative integer not belonging to {g(v) | v ∈ N+ (u)}. It follows, therefore, that if g is a Grundy function, then the
following hold. (1) g(u) = k implies that for each 0 ≤ j < k, there is a vertex v ∈ N+ (u) with g(v) = j. (2) g(u) = k implies that for every v ∈ N+ (u), g(v)=g(u). Proposition 18 ([7]) If a digraph
D has a Grundy function, then D has a kernel.
Domination in Digraphs
Proof Let g: V → [n]0 be a Grundy function on a digraph D = (V, A), and let S = {u ∈ V | g(u) = 0}. From condition (2) in Definition 11, we know that g(u) = 0 implies that for every v ∈ N+ (u), g(v)=
g(u) = 0, and therefore, S is an independent set. If a vertex v∈S, then g(v) = k > 0. From condition (1) in Definition 11, we know that g(u) = k > 0 implies that for each j < k, there is a vertex u ∈
N+ (u) with g(u) = j, and in particular there is a vertex w ∈ N+ (u) with g(w) = 0. Thus, S is an indominating set. Therefore, S is a kernel. While it can be verified that if a graph has a kernel, it
need not have a Grundy function, the following interesting connection to kernel-perfect digraphs was shown by Berge [7]. Theorem 19 ([7]) Every kernel-perfect digraph has a Grundy function. Proof Let
D = D0 be a kernel-perfect digraph, and let S0 be a kernel of D0 . It follows from the definition of a kernel-perfect digraph that the digraph D1 = D0 − S0 is a kernel-perfect digraph. Therefore, let
S1 be a kernel of D1 . Let D2 = D1 − S1 and let S2 be a kernel in D2 . In general for k ≥ 1, let Sk be a kernel of the subdigraph Dk . The resulting sets S0 , S1 , . . . , Sk form a partition of V
(D). Define a function g: V → [k]0 by g(u) = j if and only if u ∈ Sj . It follows that g is a Grundy function of D. If g(u) = j, then vertex u is a vertex in every digraph D0 , D1 , . . . , Dj−1 .
And S0 , S1 , . . . , Sj−1 are in-dominating sets of these digraphs, respectively. Therefore, for each i < j, there is a vertex w ∈ Si where w ∈ N+ (u). Thus, condition (1) of a Grundy function (see
Definition 11) is satisfied. If g(u) = j, then u ∈ Sj , which is an in-dominating set of the digraph Dj . This means that the set Sj is an independent set. Therefore, if g(u) = j, then each v ∈ N+
(u) satisfies g(v)=j. Therefore, every kernelperfect digraph D has a Grundy function g. Fraenkel [36] has determined that deciding whether a finite digraph D has a kernel or a Grundy function is
NP-complete, even when restricted to cyclic planar digraphs with od(x) ≤ 2, id(x) ≤ 2, and od(x) + id(x) ≤ 3, and these bounds are best possible, since decreasing any of them results in a decision
problem that can be solved in polynomial time. The proof of this theorem uses a simple transformation from 3-Satisfiability.
3 Bounds on In, Out, and Twin Domination Numbers In this section, we present bounds on the domination, in-domination, and twin domination numbers of digraphs.
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3.1 (Out)-Domination We begin with some well-known results of Ore [75] on dominating sets of graphs. Theorem 20 ([75]) If G is a graph having no isolated vertices, then the complement V \ S of any
minimal dominating set S is a dominating set of G. Corollary 21 The vertices of any graph G having no isolated vertices can be partitioned into two dominating sets. Corollary 22 For any graph G of
order n having no isolated vertices, γ (G) ≤ 12 n. Fu was interested in possible analogs of these results of Ore for digraphs. For example, can the vertices of a digraph D without isolated vertices
be partitioned into two (directed) dominating sets? Fu [38] obtained the following results on dominating sets of directed graphs. Theorem 23 ([38]) A dominating set S in a digraph D is a minimal
dominating set if for each u ∈ S, there is no arc (u, v) for any vertex v ∈ S. Proof Assume that S is a dominating set of a digraph D having the property that for no two vertices u, v ∈ S, (u, v) ∈
A, that is, S is an independent set. Then it follows that for every u ∈ S, S \{u} is a not a dominating set since there is no vertex in S \{u} that dominates vertex u. Thus, S is a minimal dominating
set of D. As observed by Fu [38], in order that a digraph D has a dominating set S such that its complement V \ S is also a dominating set, it is necessary and sufficient that each vertex u ∈ S is
dominated by a vertex in V \ S and each vertex in V \ S is dominated by a vertex in S. Moreover, in order that a digraph D has a dominating set S whose complement V \ S is an in-dominating set, it is
necessary and sufficient that each vertex in S dominates at least one vertex in V \ S. Fu defined a digraph D to be cyclic or strongly connected if every pair of vertices are contained in a directed
cycle. Theorem 24 ([38]) A strongly connected digraph D has a dominating set S whose complement S = V \ S is also a dominating set if and only if D contains a directed cycle of even length. Proof For
the necessity part, assume that D has a dominating set S whose complement S = V \ S is also a dominating set. Assume that no vertices are colored. Select an arbitrary vertex u ∈ S. Color it blue.
Since the complement S is a dominating set, there must be a vertex v ∈ S and an arc (v, u). Color vertex v red. Since S is a dominating set, there are a vertex w ∈ S and an arc (w, v). If w = u, then
we have found a directed cycle of length 2. If w=u, color vertex w blue. There must be a vertex z ∈ S which dominates w. If z has been previously colored, we have found a directed cycle beginning and
ending in S and therefore having even length. If z has not been colored, color it red. Continuing in this way, all vertices encountered will either be in S and colored blue or in S and colored red.
Domination in Digraphs
Sooner or later we will have to encounter a previously colored vertex and hence have constructed a directed cycle of even length. To prove the sufficiency, we assume that there is a directed cycle of
even length, and we need only show that there is a way to assign the vertices of D either to S or S, in such a way that both sets are dominating sets. We begin with any directed cycle C0 of even
length and alternately assign its vertices to S and S. Thus, all of the vertices on C0 are assigned to a dominating set of C0 . If this includes all vertices of D, then the theorem is proved. Thus,
we may assume that there is an unassigned vertex, say w. Since D is strongly connected, w and u are on a directed cycle for any vertex u on C0 . We may then find a directed path from u to w and
continue until a vertex is encountered which has already been assigned. The vertices on this directed path can be alternately assigned to either S or S. This directed path may end with two
consecutive vertices assigned to the same set, but each vertex thus encountered is always dominated by the vertex which precedes it on the directed path. Since w is an arbitrary unassigned vertex,
every vertex of D can be assigned to one of S and S. Corollary 25 ([38]) A strongly connected digraph D has a dominating set S whose complement S is also a dominating set, and furthermore both S and
S are indominating sets if and only if every vertex of V is in some directed cycle of even length. Corollary 26 ([38]) In order that a strongly connected digraph D has a dominating set S whose
complement S is an in-dominating set, it is sufficient that D contains a directed cycle of even length. Corollary 27 ([38]) If D is a strongly connected digraph of order n having a cycle of even
length, then γ (D) ≤ 12 n. We observe that if D is a Hamiltonian digraph of order n, then γ (D) ≤ n2 . In 1998 Lee [63] improved the result of Corollary 27 as follows. Theorem 28 ([63]) If D is a
strongly connected digraph of order n, then 1 ≤ γ (D) ≤ n2 . In order to prove Theorem 28, Lee [63] proved that if D is a directed tree of order n that contains a vertex u such that every vertex in D
is reachable from u, that is, for every v in D different from u, there is a directed path from u to v, then 1 ≤ γ (D) ≤ n2 . The proof of this result given in [63] is algorithmic in nature and finds
a dominating set S in such a directed tree D satisfying 1 ≤ |S| ≤ n2 . From this result, we can readily deduce Theorem 28, noting that a strongly connected digraph has as a subdigraph a directed
spanning tree with the desired property. Lee [62] proved the following upper bound on the domination number of a digraph D in terms of its order and the minimum indegree δ − (D). Theorem 29 ([62]) If
D is a digraph of order n with δ − (D) = δ − ≥ 1, then
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γ (D) ≤
δ− + 1 n. 2δ − + 1
As a consequence of Theorem 29, we have the following upper bound on the domination number of a digraph in which every vertex has indegree at least 1. Corollary 30 ([62]) If D is a digraph of order n
with δ − (D) ≥ 1, then γ (D) ≤ 23 n. Using standard probabilistic arguments, Lee [62] established the following upper bound on the domination of a digraph. Theorem 31 ([62]) If D is a digraph of
order n with δ − (D) = δ − ≥ 1, then ⎛ γ (D) ≤ ⎝1 −
1 1 + δ−
1 δ−
1 1 + δ−
1+δ−− δ
⎞ ⎠ n.
We remark that when the minimum indegree δ − (D) is small, namely, δ − (D) ∈{1, 2}, then the upper bound given by Theorem 29 is better than that given by Theorem 31. As before, let D(n, p) = (V, A)
denote a random digraph of order n where for every u, v ∈ V , the arc (u, v) is chosen with probability p. Let Q be a property of digraphs. If A is the set of digraphs of order n with property Q and
the probability Pr(A) of A has limit 1 as n →∞, then we say almost all digraphs have property Q or a random digraph has property Q almost surely. Lee [62] established the following result for random
digraphs. Theorem 32 ([62]) For a fixed p with 0 < p < 1, a random digraph D ∈ D(n, p) satisfies γ (D) = k ∗ + 1
γ (D) = k ∗ + 2
almost surely, where k ∗ = log n − 2 log log n + log log e and where log denotes the logarithm with base 1/(1 − p). Ghoshal, Laskar, and Pillone [43] determined lower and upper bounds on the
domination number of a digraph in terms of its order and maximum outdegree. Theorem 33 ([43]) If D is a digraph of order n, then n ≤ γ (D) ≤ n − + (D). 1 + + (D) Proof Let x ∈ V be any vertex having
maximum outdegree in D, that is, od(x) = + (D). Let S = V \ N+ (x). It follows that S is an out-dominating set. Thus, γ (D) ≤|S| = n − + (D). This establishes the upper bound. To prove the lower
bound, let S ⊆ V be a minimum dominating set of D, that is, γ + (D) = |S|.
Domination in Digraphs
Every vertex in S dominates at most + (D) vertices outside S, implying that n −|S| = |V \ S|≤|S|· + (D), and so γ + (D) = |S|≥ 1/(1 + + (D)). Hao and Qian [52] strengthened the lower bound of Theorem
33. The Slater number sl(D) of a digraph D is the smallest integer t such that adding t to the sum of the first t terms of the non-increasing outdegree sequence of D is at least as large as the order
of D. Theorem 34 ([52]) If D is a digraph of order n, then n ≤ sl(D) ≤ γ (D). 1 + + (D) the authors [52] showed that the difference between sl(D) and Moreover, n can be arbitrarily large. 1++ (D)
3.2 In-Domination We turn our attention to bounds on the in-domination number of a digraph and give the following classical 1973 results due to Berge [7]. Proposition 35 ([7]) If D is a digraph of
order n and size m, then γ − (D) ≥ n − m. Proof Let S ⊆ V be a minimum in-dominating set, that is, γ − (D) = |S|. Since for every vertex w ∈ V \ S, there exist a vertex v ∈ S and an arc (w, v), it
follows that n −|S| = |V \ S|≤ m, and so γ − (D) = |S|≥ n − m. Proposition 36 ([7]) For any digraph D of order n having maximum indegree Δ− (D),
n 1 + − (D)
≤ γ − (D) ≤ n − − (D).
Proof Let x ∈ V be any vertex having maximum indegree in D, that is, id(x) = − (D). Let S = V \ N− (x). It follows that S is an in-dominating set. Thus, γ − (D) ≤|S| = n − − (D). This establishes the
upper bound. To prove the lower bound, let S ⊆ V be a minimum in-dominating set of D, that is, γ − (D) = |S|. Every vertex in S is dominated by at most − (D) vertices outside S, implying that n −|S|
= |V \ S|≤|S|· − (D), and so γ − (D) = |S|≥ 1/(1 + − (D)). We note that both bounds of Proposition 36 are sharp for a digraph of order n having − (D) = n − 1.
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3.3 Domination and In-Domination In 1999 Chartrand, Harary, and Yue [19] proved the following upper bound on the sum of the domination number and the in-domination number of a digraph. Recall that
C'3 denotes the directed cycle on three vertices and an endvertex is a vertex of degree 1. Theorem 37 ([19]) If D is a digraph of order n with δ − (D) ≥ 1 and δ + (D) ≥ 1, then γ (D) + γ − (D) ≤
4 n. 3
Further, equality holds if and only if D = C'3 , or if every vertex of D is an endvertex or is adjacent to exactly one endvertex and adjacent from exactly one endvertex. In 2015 Hao and Qian [51]
improved the upper bound of Theorem 37 as follows. Theorem 38 ([51]) Let D be a digraph of order n with δ − (D) ≥ 1 and δ + (D) ≥ 1. If 2k + 1 is the length of a shortest odd circuit of D, then γ (D)
+ γ − (D) ≤
2k + 2 n. 2k − 1
As a consequence of Theorem 38, we have the following result. Corollary 39 ([51]) If D is a digraph of order n with δ − (D) ≥ 1 and δ + (D) ≥ 1 with no odd directed cycle, then γ (D) + γ − (D) ≤ n.
3.4 Twin Domination In this section, we present results on the twin domination number of a digraph. We first present the following key lemma. Recall that for r ≥ 1 an integer, a graph G is
r-degenerate if every induced subgraph of G has minimum degree at most r. When we say that digraph D is minimal with respect to some property P, we mean arcminimal, that is, removing any arc from D
destroys property P. Lemma 40 If a digraph D is minimal with respect to the property of every vertex of D having indegree and outdegree at least k, then the underlying graph is 2kdegenerate. Proof
Let D be a digraph that is minimal with respect to the property P that every vertex of D has indegree and outdegree at least k. Let G be the underlying (undirected) graph of D. We show that G is
2k-degenerate. Suppose, to the contrary, that there is a set V of vertices such that the subgraph, say G , of G induced by the set V has minimum degree at least 2k + 1. Let D be the subdigraph of D
Domination in Digraphs
Fig. 4 A digraph D with γ ± (D) = 23 n
induced by the set V , and so G is the underlying graph of the digraph D . Each vertex v ∈ V has an excess of in- or out-arcs in D , noting that dG (v) ≥ 2k + 1. Suppose there is an arc av whose
removal from D destroys the property of v having indegree and outdegree at least k. If odD (v) ≥ k + 1, then av is an arc into v and in this case idD (v) = k. If idD (v) ≥ k + 1, then av is an arc
out of v and in this case odD (v) = k. Thus, the number of arcs incident to v whose removal from D destroys property P is either zero or k. Hence, there are at most k|V | arcs in D whose removal
destroys property P. But every arc removal from D destroys property P for some vertex of D , implying that there are at most k|V | arcs in D . This in turn implies that every vertex has indegree and
outdegree exactly k in D , and therefore G is a (2k)-regular graph, contradicting the supposition that δ(G ) ≥ 2k + 1. In 2003 Chartrand, Dankelmann, Schultz, and Swart [20] established the following
upper bound on the twin domination number of a digraph. We present here a simple proof of this result, using the key Lemma 40. Our proof is based on the fact that a k-degenerate graph has chromatic
number at most k + 1, as shown by Szekeres and Wilf [88] in 1968. Recall that a vertex and an edge cover each other in a graph G if they are incident in G. A vertex cover in G is a set of vertices
that covers all the edges of G. The vertex cover number β(G) (also denoted by τ (G) or vc(G) in the literature) is the minimum cardinality of a vertex cover in G. Theorem 41 ([20]) If D is a digraph
of order n with δ − (D) ≥ 1 and δ + (D) ≥ 1, then γ ± (D) ≤ 23 n. Proof We may assume the digraph D is minimal with respect to this property of δ − (D) ≥ 1 and δ + (D) ≥ 1, since adding arcs cannot
increase the twin domination number. With this assumption, the underlying graph G of D is 2-degenerate by Lemma 40 and hence 3-colorable. Thus, the independence number of G is at least n/3, which
means that the vertex cover number of G is at most 2n/3. But a vertex cover of G is a twin dominating set in D since δ − (D) ≥ 1 and δ + (D) ≥ 1. Thus, γ ± (D) ≤ 23 n. The simplest example of a
digraph achieving equality in the upper bound of Theorem 41 is C'3 . As a further small example, the digraph D shown in Figure 4 has order n = 6 and satisfies γ ± (D) = 4 = 23 n, where the darkened
vertices form a twin dominating set of D of cardinality 4.
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In 2013 Arumugam, Ebadi, and Sathikala [4] gave the following upper bound on the twin domination number. Theorem 42 ([4]) If D is a digraph of ordern and (D) is the length of a longest directed path
in D, then γ ± (D) ≤ n − (D) . 2 The bound of Theorem 42 is attained, for example, by directed paths and also by any digraph D obtained from a directed path Pk : u1 , u2 , . . . , uk by adding a new
vertex u i and arc (u i , ui ) for each ui for i ∈ [k].
3.5 Reverse Domination The digraph obtained from a digraph D by reversing all the arcs of D is called the reverse digraph (also called the converse in the literature) of D, denoted D− . We note that
γ (D) = γ − (D− ) for every digraph D. Thus by Theorem 37, if D is a digraph of order n with δ − (D) ≥ 1 and δ + (D) ≥ 1, then γ (D) + γ (D − ) ≤ 43 n. For r ≥ 1, let Dr be the class of r-regular
strongly connected digraphs. We note that the only 1-regular strongly connected digraphs are the directed cycles, and so D1 = {C' n | n ≥ 3}. Since a directed cycle is isomorphic to its reverse, if D
∈ D1 , then γ (D− ) − γ (D) = 0. For r ≥ 2, the difference γ (D− ) − γ (D) can be arbitrarily large in the class Dr , as shown by Gyürki [45] in the case when r = 2 and by Niepel and Knor [73] for
all r ≥ 3. However, for a fixed r ≥ 2, it remains an open problem to determine the greatest ratio γ (D− )/γ (D) of an r-regular strongly connected digraph. The best known results to date are the
following. 4 γ (D − ) ≥ . 3 D∈D2 γ (D)
Theorem 43 ([45]) For digraph D ∈ D2 , sup
7 γ (D − ) ≥ . γ (D) 6 D∈Dr
Theorem 44 ([45, 73]) For r ≥ 3, we have sup
4 Domination in Digraph Products Vizing’s conjecture [90] asserts that the domination number of the Cartesian product of two graphs is at least as large as the Cartesian product of their domination
numbers. This conjecture was first stated in 1963 as a problem in [89] and later in 1968 formally posed as a conjecture in [90]. It is considered by many to be the main open problem in the area of
domination in graphs. It is natural then that the study of domination in digraphs considers results for Cartesian products of digraphs. The Cartesian product of two digraphs G = (V (G), A(G)) and H =
(V (H), A(H)), denoted by G2H , is the digraph with vertex set V (G) × V (H), and there exists an arc ((u1 , v1 ), (u2 , v2 )) ∈ A(G2H ) if and only if either (u1 , u2 ) ∈ A(G) and v1 = v2
Domination in Digraphs
or (v1 , v2 ) ∈ A(H) and u1 = u2 . Much of the work on Cartesian products in digraphs considers directed cycles. In 2009 Shaheen [84] and in 2010 Liu, Zhang, Chen, and Meng [64, 93] independently
determined the domination number of C'm 2 C'n for m ≤ 6 and arbitrary n ≥ 2. Theorem 45 ([64, 84, 93]) For n ≥ 2, the following hold. (a) γ (C' 2 2 C'n ) = n. (b) γ (C' 3 2 C'n ) = n if n ≡ 0 (mod
3); otherwise, γ (C'3 2 C'n ) = n + 1. (c) γ (C' 4 2 C'n ) = 32 n if n ≡ 0 (mod 8); otherwise, γ (C'4 2 C'n ) = n + n+1 . 2 (d) γ (C' 5 2 C'n ) = 2n. (e) γ (C' 6 2 C'n ) = 2n + 2. Zhang et al. [93]
also determined γ (C' m 2 C'n ) when both m and n are divisible by 3. Theorem 46 ([93]) If m ≡ 0 (mod 3) and n ≡ 0 (mod 3), then γ (C'm 2 C'n ) = 13 mn. In 2013, Mollard [71] determined the exact
values of γ (Cm 2Cn ) for m congruent to 2 modulo 3, with the exception of one subcase. Theorem 47 ([71]) If m, n ≥ 2, m ≡ 2 (mod 3), k = m/3, and = n/3, then ⎧ ⎪ n(k + 1) if n = 3 ⎪ ⎪ ⎪ ⎨n(k + 1) if
n = 3 + 1 and 2 ≥ k γ (C'm 2 C'n ) = ⎪ n(k + 1) if n = 3 + 2 and n ≥ m ⎪ ⎪ ⎪ ⎩ m( + 1) if n = 3 + 2 and n ≤ m. Furthermore, γ (C'm 2 C'n ) if n = 3 + 1 and 2 < k. Zhang et al. [93] conjectured that
if k ≥ 2 where k = m3 , then γ (C'm 2 C'n ) = k(n+1) for n≡0 (mod 3), but Mollard [71] disproved this conjecture by showing that it doesn’t always hold whenn ≡ 1(mod 3). For example, they noted that
γ (C'3k 2 when k≡0 (mod 8), while the conjecture claims C'4 ) = γ (C'4 2 C'3k ) = 3k + 3k+1 2 ' ' that γ (C4 2 C3k ) = 5k. These values are different for k ≥ 3. Mollard [71] also established the
following bounds. Theorem 48 ([71]) If m, n ≥ 2 and k = m3 , then
γ (C'm 2 C'n ) ≥
⎧ ⎪ ⎪ ⎨nk
if m ≡ 0 (mod 3)
nk + ⎪ ⎪ ⎩nk + n n 2
if m ≡ 1 (mod 3) if m ≡ 2 (mod 3).
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In 2013 Shao, Zhu, and Lang [86] determined upper and lower bounds on γ (C'm 2 C'n ) for the case when m is congruent to 1 modulo 3. Theorem 49 ([86]) If k ≥ 1 and n ≥ 3 are integers, then
(2k + 1)n 2
≤ γ (C' 3k+1 2 C'n ) ≤
(2k + 1)n + k. 2
Based on the bounds of Theorem 49, Shao et al. [86] determined the exact values of γ (C'm 2 C'n ) for m ∈{7, 10}. We conclude this section by noting that Liu, Zhang, and Meng [65] investigated
domination numbers of Cartesian products of directed paths in 2011, and Ma and Liu [67] studied the twin domination number of the Cartesian products of directed cycles in 2016. For domination and
twin domination in other types of digraph products, see [15, 66, 68, 69, 74].
5 Domination in Oriented Graphs Recall that an oriented graph D is a digraph that can be obtained from a graph G by assigning a direction to (i.e., orienting) each edge of G. The resulting digraph D
is called an orientation of G. Thus, if D is an oriented graph, then for every pair u and v of distinct vertices of D, at most one of (u, v) and (v, u) is an arc of D. For example, a tournament is an
oriented complete graph. Recall also that the independence number of a directed graph D is denoted by α(D). As before, unless otherwise stated, we refer to an out-dominating set in a digraph simply
as a dominating set.
5.1 Oriented Graphs In 1996 Chartrand, Vanderjagt, and Yue [18] studied domination in oriented graphs. They defined the lower orientable domination number of a graph G, which they denoted as dom(G)
(denoted by γ d (G) in [17]), to equal the minimum domination number over all orientations of G. Further, they defined the upper orientable domination number, or simply the orientable domination
number, of a graph G, which they denoted as DOM(G) (denoted by d (G) in [17]), as the maximum domination number over all orientations of G. Thus, dom(G) = min{γ (D) | over all orientations D of G}
DOM(G) = max{γ (D) | over all orientations D of G}. The orientable domination number of a complete graph was first studied by Erd˝os in 1963 [28], albeit in disguised form. In 1962, Schütte [28]
raised the question of
Domination in Digraphs
given any positive integer k > 0, does there exist a tournament Tn(k) on n(k) vertices in which for any set S of k vertices, there is a vertex u that dominates all vertices in S. Erd˝os [28] showed,
by probabilistic arguments, that such a tournament Tn(k) does exist, for every positive integer k. The proof of the following bounds on the orientable domination number of a complete graph is along
identical lines to that presented by Erd˝os [28]. This result can also be found in [78]. Here, log is to the base 2. Theorem 50 ([28]) For n ≥ 2, log n − 2 log(log n) ≤ DOM(Kn ) ≤ log(n + 1). This
notion of orientable domination in a complete graph was subsequently extended to orientable domination of all graphs by Chartrand et al. [18]. They proved the following result. Theorem 51 ([18]) For
every graph G, dom(G) = γ (G). In view of Theorem 51, it is not interesting to ask about the lower orientable domination number, dom(G), of a graph G since this is precisely its domination number,
which is very well studied. We therefore focus our attention on the (upper) orientable domination number of a graph. Chartrand et al. [18] determined DOM(G) for special classes of graphs, including
paths, cycles, complete bipartite graphs, and regular complete tripartite graphs. They also proved the following result. Theorem 52 ([18]) For every graph G and for every integer c with dom(G) ≤ c ≤
DOM(G), there exists an orientation D of G such that γ (D) = c. In 2010 Blidia and Ould-Rabah [8] continued the study of domination in oriented graphs. For an oriented graph D, let α (D) denote the
matching number of D and let s(D) denote the number of support vertices in the underlying graph of D. The authors in [8] proved the following result. In fact, they proved a slightly stronger result
involving the irredundance number of an oriented graph (which we do not define here). Theorem 53 ([8]) If D is an oriented graph of order n, then s(D) ≤ γ (D) ≤ n − α (D). Blidia and Ould-Rabah [8]
characterized the oriented trees T satisfying γ (T) − α (T) and the oriented graphs D satisfying γ (D) = s(D) and s(D) = n − α (D). In 2011 Caro and Henning [16] also studied domination in oriented
graphs. In this paper, they proved a Greedy Partition Lemma, which they used to present an upper bound on the orientable domination number of a graph in terms of its independence number. To state
their result, let α ≥ 1 be an integer and let Gα be the class of all graphs G with α ≥ α(G). Theorem 54 ([16]) For α ≥ 1 an integer, if G ∈ Gα has order n ≥ α, then n DOM(G) ≤ α 1 + 2 ln . α
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The next result follows as a consequence of Theorem 54, where χ (G) denotes the chromatic number of G and dav (G) denotes the average degree in G. Corollary 55 ([16]) If G is a graph of order n, then
the following hold. (a) DOM(G) ≤ α(G) (1 + 2 ln (χ (G))). (b) DOM(G) ≤ α(G) (1 + 2 ln (dav (G) + 1)). For any integer d ≥ 1, let Fd be the class of all graphs G whose complement is a d-degenerate
graph. The property of being d-degenerate is a hereditary property that is closed under induced subgraphs, as is the property of the complement of a graph being d-degenerate. Applying their Greedy
Partition Lemma for domination in oriented graphs, the authors in [16] proved the following result. Theorem 56 ([16]) For any integer d ≥ 1, if G ∈ Fd has order n, then DOM(G) ≤ 2d + 1 + 2 ln
n − 2d + 1 . 2
The following upper bound on the orientable domination number of a K1,m -free graph is established in [16], where a graph is F-free if it does not contain F as an induced subgraph. Theorem 57 ([16])
For m ≥ 3, if G is a K1,m -free graph of order n with δ(G) = δ, then δ+m−1 . DOM(G) < 2(m − 1)n ln δ+m−1 Let Gn denote the family of all graphs of order n. We define NGmin (n) = min{DOM(G) + DOM(G)}
NGmax (n) = max{DOM(G) + DOM(G)} where the minimum and maximum are taken over all graphs G ∈ Gn . The following Nordhaus-Gaddum-type bounds for the orientable domination of a graph were established
in [16]. Theorem 58 ([16]) The following hold. (a) c1 log n ≤ NGmin (n) ≤ c2 (log n)2 for some constants c1 and c2 . (b) n + log n − 2 log(log n) ≤ NGmax (n) ≤ n + n2 . Caro and Henning continued
their study of the orientable domination number in [17]. They defined the maximum average degree in a graph G, denoted by mad(G), as the maximum of the average degrees taken over all subgraphs H of
G, that is, mad(G) = max
H ⊂G
% 2|E(H )| . |V (H )|
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Theorem 59 ([17]) If G is a graph of order n, then the following hold. (a) (b) (c) (d)
DOM(G) ≥ α(G) ≥ γ (G). DOM(G) ≥ n/χ (G). DOM(G) ≥(diam(G) + 1)/2). DOM(G) ≥ n/(mad(G)/2 + 1).
Proof We present here only a proof of part (a). Let I be a maximum independent set in G, and let D be the digraph obtained from G by orienting all arcs from I to V \ I and orienting all arcs in G[V \
I], if any, arbitrarily. Every dominating set of D contains the set I, and so γ (D) ≥|I|. However, the set I itself is a dominating set of D, and so γ (D) ≤|I|. Consequently, DOM(G) ≥ γ (D) = |I| = α
(G) ≥ γ (G). As remarked in [17], since mad(G) ≤ (G) for every graph G, as an immediate consequence of Theorem 59(d), we have that DOM(G) ≥ n/((G)/2 + 1). The following lemma is useful when
establishing upper bounds on the orientable domination number of a graph. Lemma 60 ([17]) Let G = (V, E) be a graph and let V1 , V2 , . . . , Vk be subsets of V , not necessarily disjoint, such that
∪ki=1 Vi = V . If Gi = G[Vi ] for i ∈ [k], then DOM(G) ≤
DOM(Gi ).
Proof Consider an arbitrary orientation D of G. Let Di be the orientation of the edges of Gi induced by D, and let Si be a γ -set of Di for each i ∈ [k]. By Theorem 59(a), DOM(Gi ) ≥ γ (Di ) = |Si |
for each i ∈ [k]. Since the set S = ∪ki=1 Si is a dominating set of D, we have that γ (D) ≤ |S| ≤
k i=1
|Si | ≤
DOM(Gi ).
Since this is true for every orientation D of G, the desired upper bound of DOM(G) follows. As a consequence of Lemma 60, the authors in [17] proved the following upper bounds on the orientable
domination number of a graph. Theorem 61 ([17]) If G is a graph of order n, then the following hold. (a) (b) (c) (d) (e)
DOM(G) ≤ n − α (G). If G has a perfect matching, then DOM(G) ≤ n/2. DOM(G) ≤ n with equality if and only if G = K n . If G has minimum degree δ and n ≥ 2δ, then DOM(G) ≤ n − δ. DOM(G) = n − 1 if and
only if every component of G is a K1 -component, except for one component which is either a star or a complete graph K3 .
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Proof We present here only a proof of part (a). Let M = {u1 v1 , u2 v2 , . . . , ut vt } be a maximum matching in G, and so t = α (G). Let Vi = {ui , vi } for i ∈ [t]. If n > 2t, let (Vt+1 , . . . ,
Vn−2t ) be a partition of the remaining vertices of G into n − 2t subsets each consisting of a single vertex. By Lemma 60, DOM(G) ≤
DOM(Gi ) = t + (n − 2t) = n − t = n − α (G).
Applying results on the size of a maximum matching in a regular graph established in [57], we have the following consequence of Theorem 61(a). Theorem 62 ([17]) For r ≥ 2, if G is a connected
r-regular graph of order n, then
DOM(G) ≤
⎧ 2 % r + 2r n n+1 ⎪ ⎪ max if r is even × , ⎪ ⎪ ⎨ 2 2 r2 + r + 2 ⎪ ⎪ ⎪ (r 3 + r 2 − 6r + 2) n + 2r − 2 ⎪ ⎩ 2(r 3 − 3r)
if r is odd.
The orientable domination number of a bipartite graph is precisely its independence number. Recall that König [60] and Egerváry [27] showed that if G is a bipartite graph, then α (G) = β(G). Hence by
Gallai’s Theorem [42], if G is a bipartite graph of order n, then α(G) + α (G) = n. Theorem 63 ([17]) If G is a bipartite graph, then DOM(G) = α(G). Proof Since G is a bipartite graph, we have that n
− α (G) = α(G). Thus, by Theorem 59(a) and Theorem 61(a), we have that α(G) ≤DOM(G) ≤ n − α (G) = α(G). Consequently, we must have equality throughout this inequality chain. In particular, DOM(G) = α
(G). In 2018 Harutyunyan, Le, Newman, and Thomassé [53] observed that in general there is no upper bound on the orientable domination number of a graph solely in terms of its independence number.
Nevertheless, they showed that these two quantities can be related. Theorem 64 ([53]) If G is a graph of order n, then DOM(G) ≤ α(G) · log n. Theorem 64 implies that when the independence number of
an oriented graph is sufficiently large, it is possible to bound the orientable domination number of the graph purely in terms of its independence number. Theorem 65 ([53]) If D is a graph of order n
and α(G) ≥ log n, then DOM(D) ≤ (α(D))2 .
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Harutyunyan et al. [53] concluded their paper with the following conjecture. Conjecture 1 There exists an integer k such that for any C'3 -free oriented graph D with α(D) = α, we have γ (D) ≤ α k .
The following result establishes an upper bound on the orientable domination number of a graph in terms of its independence number and chromatic number. Theorem 66 ([17]) If G is a graph of order n,
then the following hold. (a) DOM(G) ≤ α(G) ·χ (G)/2. (b) DOM(G) ≤ n −χ (G)/2. (c) DOM(G) ≤ (n + α(G))/2. The following result establishes an upper bound on the orientable domination of a graph in
terms of the chromatic number of its complement. Theorem 67 ([17]) If G is a graph of order n, then DOM(G) ≤ χ (G) · log
n χ (G)
+1 .
As a consequence of Theorem 67, we have the following result on the orientable domination number of a graph with sufficiently large minimum degree. Theorem 68 ([17]) If G is a graph of order n with
minimum degree δ(G) ≥ (k − 1)n/k where k divides n, then DOM(G) ≤ nk log(k + 1). Let Mop(n) = max{DOM(G)}, where the maximum is taken over all maximal outerplanar graphs of order n. Theorem 69 ([17])
For maximal outerplanar graphs of order n, Mop(n) = n2 .
5.2 Tournaments Since a tournament is an oriented complete graph, many applications interpret a tournament as a competition graph. That is, a tournament on n vertices represents a competition between
n teams (each represented by a vertex) in which the teams play each other once. No ties are allowed, and there is an arc from a vertex u to a vertex v if and only if u defeats v. The score of a
vertex v is its outdegree (the number of teams it defeats). Hence, a dominating set S of a tournament represents a collection of teams such that every team not in S is defeated by at least one team
in S. Tournaments are popular, in part, because of this pairwise comparison and ranking of competitors. The following result is attributed by Moon to Erd˝os (cf. Moon [72] p. 28). As before, unless
otherwise stated, log is to the base 2. Theorem 70 (Erd˝os) If T is a tournament with n ≥ 2 vertices, then γ (T ) ≤ log n.
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Proof The sum of the outdegrees of the vertices in a tournament T = (V, A) of order n is the number of arcs in T, that is, u∈V
odT (u) =
1 n(n − 1). 2
Thus, there must be a vertex x ∈ V with odT (x) ≥ 12 (n − 1). We remove this vertex x and all out-neighbors of x, thereby removing at least half the vertices. We now repeat this process on the
remaining tournament, which has at most 12 (n − 1) vertices, by again selecting a vertex which dominates at least half of the remaining vertices and then deleting this second vertex and all of its
out-neighbors. Repeating this process will produce a dominating set with no more than log n vertices. A random tournament is obtained by orienting the edges of a complete graph randomly,
independently, with equal probabilities. Let Tn be the probability space consisting of the random tournaments on n vertices. In 1997 Bollobás and Szabó [9] showed that the domination number of a
random tournament is one of two values, where log is to the base 2. We remark that this result was obtained by Lee [62] in 1994. Theorem 71 ([9, 62]) A random tournament T ∈ Tn has domination number
k+1 or k + 2, where k = log(n) − 2 log(log(n)) + log(log(e)). By Theorem 71, there are tournaments having arbitrarily large domination numbers. This leads to the question: Which tournaments have
bounded domination number (not dependent on the order n of the tournament)? To partially answer this question, we first define a k-majority tournament. Definition 12 As usual, by a linear order in a
tournament, we mean with respect to the transitive orientation of the tournament. A tournament T is a k-majority tournament if there are 2k − 1 linear orders of V (T) such that for all distinct
vertices u and v in T, if u is adjacent to v, then u is before v in at least k of the 2k − 1 orders. Let F(k) be the supremum of the size of a minimum dominating set in a k-majority tournament, where
the supremum is taken over all k-majority tournaments, with no restriction on their size. Trivially, F(1) = 1. In 2006 Alon, Brightwell, Kierstead, Kostochka, and Winkler [2] proved that F(2) = 3. To
do this, they first showed that every 2-majority tournament has a dominating set of size at most 3, that is, F(2) ≤ 3. We omit their proof. To show that F(2) ≥ 3, Alon et al. [2] provided the
following example. Recall that if there is an integer x with 0 < x < p such that x2 ≡ q (mod p), then q is a quadratic residue modulo p. In practice, it suffices to restrict the range of x to 0 < x
≤p/2 because of the symmetry (p − x)2 ≡ x2 (mod p). For example, the quadratic residues modulo 7 are given by 1, 2, 4 since 11 ≡ 1 (mod 7), 22 ≡ 4 (mod 7), and 32 ≡ 2 (mod 7). Let T be the quadratic
residue tournament whose vertices are the elements of the finite field GF(7) in which i → j if and only if i − j is a quadratic residue modulo 7,
Domination in Digraphs
Fig. 5 A 2-majority tournament T with γ (T ) = 3
i.e., (i − j) mod 7 ∈{1, 2, 4}. Since the edges of T are preserved under translation, it suffices for us to consider the subtournament T of T with vertex set {0, 1, . . . , 6} as illustrated in
Figure 5. No two vertices dominate T , while the set {0, 1, 2}, for example, is a dominating set of T , and so γ (T ) = 3. Further, T is a 2-majority tournament realized by the orders P1 , P2 , and
P3 , where P1 : 0 < 1 < 2 < 3 < 4 < 5 < 6, P2 : 4 < 6 < 1 < 3 < 5 < 0 < 2, P3 : 5 < 2 < 6 < 3 < 0 < 4 < 1. Thus, T is a 2-majority tournament satisfying γ (T ) = 3. As observed earlier, the edges of
T are preserved under translation, implying that T is a 2-majority tournament satisfying γ (T) = 3. This example shows that F(2) ≥ 3. As observed earlier, F(2) ≤ 3. Consequently, F(2) = 3. We state
this result formally as follows. Theorem 72 ([2]) For 2-majority tournaments, F(2) = 3. The value of F(k) has yet to be determined for any value of k ≥ 3. The following nontrivial result shows that F
(3) ≥ 4. Theorem 73 ([2]) There exists a 3-majority tournament T with γ (T) = 4, that is, F(3) ≥ 4. As observed earlier, there are tournaments having arbitrarily large domination numbers. Kierstead
and Trotter (see [2] for a discussion) conjectured that this is not the case for k-majority tournaments for some fixed k. Alon et al. [2] proved this conjecture and showed that F(k) is finite for
each fixed k. Theorem 74 ([2]) For an arbitrary fixed integer k ≥ 1, if T is a k-majority tournament, then γ (T ) ≤ 20(2 + o(1))k log(k(2 log 2)) ≤ (80 + o(1))k log(k).
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We remark that their paper was the first to introduce the idea of using the VC dimension to study domination in tournaments, where the VC dimension (VapnikChervonenkis dimension) of a hypergraph H is
the largest cardinality of a vertex subset X shattered by H, that is, for any Y ⊆ X, the hypergraph H has an edge A such that A ∩ X = Y . The upper bound in the following theorem follows as a
consequence of Theorem 74. Theorem 75 ([2]) For an arbitrary fixed integer k ≥ 1,
1 k + o(1) ≤ F (k) ≤ (80 + o(1))k log(k). 5 log k
A tournament is k-transitive if its edge set can be partitioned into k sets each of which is transitively oriented. András Gyárfás made the conjecture that k-transitive tournaments have bounded
domination number, and this was explored in 2014 by Pálvölgyi and Gyárfás [76]. Conjecture 2 (Gyárfás) For each positive integer k, there exists a (least) p(k) such that every k-transitive tournament
has a dominating set of at most p(k) vertices. We proceed further with the following definitions. Definition 13 A class C of tournaments has bounded domination if there exists a constant c such that
every tournament in C has domination number at most c. If S and T are tournaments, then T is called S-free if no subtournament of T is isomorphic to S. A tournament S is a rebel if the class of all
S-free tournaments has bounded domination. In 2018 Chudnovsky, Ringi, Chun-Hung, Seymour, and Thomassé [23] investigated the following conjecture posed by HeHui Wu. Conjecture 3 (HeHui Wu) Every
tournament is a rebel. Chudnovsky et al. [23] disproved Conjecture 3. For this purpose, they defined the notion of a poset tournament. Definition 14 A tournament T is a poset tournament if its vertex
set can be ordered {v1 , . . . , vn } such that for all 1 ≤ i < j < k ≤ n, if vj is adjacent from vi and adjacent to vk , then vi is adjacent to vk ; that is, the “forward” edges under this linear
order form the comparability graph of a partial order. Chudnovsky et al. [23] observed that not every tournament is a poset tournament. Thereafter, they proved the following result, hence disproving
Conjecture 3. Theorem 76 ([23]) Every rebel is a poset tournament. However, it remains an open problem to determine if every poset tournament is a rebel. Since Wu’s Conjecture, that every tournament
is a rebel, is false, it naturally raises the question: Which tournaments are rebels? Theorem 76 provides a partial
Domination in Digraphs
Fig. 6 The non-2-colorable tournament T∗
answer to this question. To further answer this question, we need the definition of a coloring of a tournament. Definition 15 A k-coloring of a tournament T is a partition of V (T) into k transitive
sets, or, equivalently, into k acyclic sets. A tournament T with a k-coloring is called k-colorable. Chudnovsky et al. [23] proved that Conjecture 3 is true for 2-colorable tournaments. Their proof
followed from a direct application of VC dimension. Theorem 77 ([23]) All 2-colorable tournaments are rebels. A breakthrough in their paper [23] is that Chudnovsky et al. overcame the unboundedness
of the VC dimension by showing that large shattered sets in a hypergraph are sparse, which turns out to be enough to carry over the proof of Theorem 76. This enabled them to give a non-2-colorable
tournament T∗ on seven vertices that satisfies Conjecture 3. Such a tournament T∗ is constructed from a cyclic triangle by substituting a copy of a cyclic triangle for two of the three vertices of an
original cyclic triangle. A sketch of the tournament T∗ is given in Figure 6, where the arrow from v to the cyclic triangle T1 indicates that all three arcs from v to T1 are arcs out of v while the
arrow from the cyclic triangle T2 to v indicates that all three arcs from T2 to v are arcs into v. Further, the arc from T1 to T2 indicates that every vertex in T1 is adjacent to every vertex in T2 .
Theorem 78 ([23]) The non-2-colorable tournament T∗ is a rebel. Thus, Theorem 78 gives a counterexample to the converse of Theorem 77, that all rebels are 2-colorable. As a consequence of Theorem 78,
the following result is proven, where the odd girth of a tournament T is the smallest k for which there exists a subtournament of T with k vertices that is not 2-colorable (and is undefined if T is
2-colorable). Theorem 79 ([23]) For k ≥ 8, the class of tournaments with odd girth at least k has bounded domination. We close this section on domination in tournaments, with a brief discussion on
what we define next as a domination graph of a digraph. Definition 16 Two vertices x and y dominate an oriented graph D = (V, A) if the set {x, y} is a dominating set of D, that is, every vertex z
different from x and y is adjacent from at least one of x and y, and so (x, z) ∈ A or (y, z) ∈ A. The domination graph of an oriented graph D is the graph G with V (G) = V (D) and with an edge
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between two vertices x and y if x and y dominate T, that is, if every other vertex loses to at least one of x and y. Domination graphs were introduced and studied by Fisher et al. [30–35] and [21,
22], who largely considered the domination graphs of tournaments. In particular, Fisher et al. showed that the domination graph of a tournament is either an odd cycle with or without isolated and/or
pendant vertices or a forest of caterpillars. They also showed that any graph consisting of an odd cycle with or without isolated and/or pendant vertices is the domination graph of some tournament.
6 Total Domination in Digraphs There are several possibilities for defining the counterpart of a total dominating set in a digraph D. We consider four such versions in the following subsections.
6.1 Total Domination: Version 1 In this version of total domination, we define a set S in a digraph D to be a total in-dominating set if S is an in-dominating set in D with the added property that
the subdigraph induced by S has no isolated vertices. Here we define the total indomination number γti− (D) of a digraph D to equal the minimum cardinality of such a set S according to Version 1. We
note that if the underlying graph of D has no isolated vertices, then V (D) is vacuously a total in-domination set of D, and so γti− (D) is well-defined and γti− (D) ≤ |V (D)|.
6.2 Total Domination: Version 2 In this version of total domination, a set S in a digraph D is a total dominating set if S is a dominating set in D with the added property that the subdigraph induced
by S has no isolated vertices. This is a version defined by Arumugam, Jacob, and Volkmann [3] in 2007 and Hao [49] in 2017. We define the total domination number γ t (D) of a digraph D with no
isolated vertices to equal the minimum cardinality of such a set S according to Version 2. As with version 1 above, we note that γ t (D) is well-defined and γ t (D) ≤|V (D)|. Arumugam et al. [3]
established the following lower bound on the total domination number of a digraph. Theorem 80 ([3]) If D is a digraph of order n, with maximum outdegree Δ+ and without isolated vertices, then
Domination in Digraphs
γt (D) ≥
2n . 2+ + 1
Hao and Chen [50] improved the lower bound in Theorem 80. For this purpose, they define the out-Slater number of a digraph D of order n as sl+ (D) = min{k : k/2 + (d1+ + d2+ + · · · + dk+ ) ≥ n},
where d1+ , d2+ , . . . , dk+ are the first k largest outdegrees of D. Theorem 81 ([50]) If D is a digraph of order n, with maximum outdegree Δ+ and without isolated vertices, then
2n + . γt (D) ≥ sl (D) ≥ 2+ + 1 Further, the gap between the rightmost two numbers can be arbitrarily large. The authors in [50] also determined the following lower bound on the total domination
number of an oriented tree in terms of its order and number of vertices of outdegree 0. Theorem 82 ([50]) If T is an oriented tree of order n ≥ 2, with n0 vertices of outdegree 0 and with
non-increasing outdegree sequence d1+ , d2+ , . . . , dn+ , then γt (T ) ≥ sl+ (D) ≥
2 (n − n0 + 1), 3
+ with equality if and only if n − n0 ≡ 2 (mod 3) and dk+1 ≤ 1, where k = 23 (n − n0 + 1).
6.3 Total Domination: Version 3 In this version of total domination, a set S in a digraph D = (V, E) is a total indominating set if every vertex in V is adjacent to a vertex in S, that is, N− (S) = V
. This is equivalent to saying that S is an in-dominating set and the subdigraph induced by S has no isolated vertices and no sources. The minimum cardinality of such a set could be called the total
absorption number, denoted γt− (D). We note that every digraph D with δ − (D) ≥ 1 has a total dominating set according to this definition since V (D) is such a set. For example, the digraph D shown
in Figure 7 satisfies γt− (D) = 3, where the darkened vertices form a total dominating set of D of cardinality 3. For a digraph : V → R, the weight of D = (V, E) and for a real-valued function f f is
w(f ) = v ∈ V f (v). Further, for S ⊆ V , we define f (S) = v ∈ S f (v); in particular,
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Fig. 7 A digraph D with γt− (D) = 3
this means that w(f ) = f (V ). Let f : V →{0, 1} be a function which assigns to each vertex of a graph an element of the set {0, 1}. We say f is a total dominating function if for every v ∈ V , the
sum of the function values under f in every out-neighborhood of a vertex is at least 1, that is, for every vertex v ∈ V , we have
f (u) ≥ 1.
u∈N + (v)
The total absorption number of D can be defined as γt− (D) = min{w(f ) | f is a total dominating function on D}. In order to present a lower bound on the total absorption number of a digraph,
St-Louis, Gendron, and Hertz [87] in 2012 considered the fractional version of a total dominating set where vertices have fractional weights in the range [0, 1]. A real-valued function f : V → [0, 1]
is called a fractional total dominating function of a digraph D if u∈N + (v) f (u) ≥ 1 for each v ∈ V . The minimum weight of a fractional total dominating function of D is the fractional total
domination number, which we denote here by γtf− (D). Thus, γtf− (D) = min {w(f ) | f is a fractional total dominating function for D}. We remark that the fractional total domination number is readily
viewed as a linear program. Thus we can talk of minimum, rather than infimum in the above definition. By definition, γt− (D) ≥ γtf− (D), and so the fractional version provides a lower bound on the
total absorption number of D. The girth g(D) of a digraph D is the number of vertices of the smallest directed cycle in D. St-Louis et al. [87] posed two conjectures, one of which is the following.
Conjecture 4 ([87]) If D is a digraph with δ + (D) ≥ 1, then γtf− (D) > g(D) − 1. St-Louis et al. [87] proved that Conjecture 4 is equivalent to the 1978 CaccettaHäggkvist Conjecture which we state
below. Conjecture 5 ([14]) If D is a digraph of order n with δ + (D) ≥ r ≥ 1, then g(D) ≤ nr .
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Fig. 8 A digraph D with γto+ (D) = 3
6.4 Total Domination: Version 4 In this version of total domination, a set S in a digraph D = (V, E) is a total dominating set if every vertex in V is adjacent from a vertex in S, that is, N+ (S) = V
. This is equivalent to saying that S is a dominating set and the subdigraph induced by S has no isolated vertices and no sinks. This is a version defined by Hansen, Lai, and Yue [47] in 1999 and by
Schaudt [83] in 2012. We shall call this type of + total dominating set a total open dominating set and let γto (D) equal the minimum cardinality of a total open dominating set in a digraph D. For
example, the digraph + D shown in Figure 8 satisfies γto (D) = 3, where the darkened vertices form a total open dominating set of D of cardinality 3. In 1999 Hansen et al. [47] defined the lower
orientable open domination number dom1 (G) of a graph G as the minimum total open domination number among all orientations of G. The upper orientable total open domination number DOM1 (G) equals the
maximum such total open domination number. Theorem 83 ([47]) For a connected graph G, dom1 (G) and DOM1 (G) exist if and only if G is not a tree. Hansen et al. [47] also investigated the function
DOM1 (Kn ). They showed this to be a non-decreasing function and unbounded and determined specific values. Analogous to Theorem 52, they proved the following result. Theorem 84 ([47]) For every
integer c with dom1 (Kn ) ≤ c ≤DOM1 (Kn ), there + (D) = c. exists an orientation D of Kn such that γto In 2012 Schaudt [83] studied efficient total domination in digraphs, where an efficient total
dominating set of a digraph D is a total open dominating set S with the property that for each vertex v of D, there is a unique vertex u ∈ S that is adjacent to v. Graphs that permit an orientation
having such a set were studied in [83]. Further, complexity results and characterizations were given.
6.5 Fractional Domination in Digraphs In Section 6.3, we considered the fractional version of total domination in digraphs. In this section, we present results on the fractional version of domination
in digraphs. Adopting our earlier notation, a real-valued function f : V → R in a
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digraph D is a dominating function if for every v ∈ V , the sum of the function values under f in every closed out-neighborhood of a vertex is at least 1, that is, for every vertex v ∈ V , we have
f (u) ≥ 1.
u∈N + [v]
The domination number of D can be defined as γ (D) = min{w(f ) | f is a dominating function on D}. A real-valued function f : V → [0, 1] is called a fractional dominating function of a digraph D if
u∈N + [v] f (u) ≥ 1 for each v ∈ V . The minimum weight of a fractional dominating function of D is the fractional domination number, which we denote here by γ f (D). Thus, γf (D) = min {w(f ) | f is
a fractional dominating function for D}. In 1982 Sands, Sauer, and Woodrow [82] (also due to Erd˝os) posed the following conjecture. Conjecture 6 ([82]) For each n, there is a (least) positive
integer f (n) so that every finite tournament whose edges are colored with n colors contains a set S of f (n) vertices with the property that for every vertex u not in S, there is a monochromatic
directed path from u to a vertex of S. A complete multidigraph is a directed graph in which multiple arcs and circuits of length 2 are allowed and such that there always exists an arc between two
distinct vertices. A tournament, for example, is a complete multidigraph in the special case when the directed graph is simple (and contains no multiple arcs or circuits of length 2). As remarked in
[13], the transitive closure of each color class is a quasi-order (i.e., a transitive digraph); hence, the Erd˝os-Sands-Sauer-Woodrow conjecture can be restated as follows. Conjecture 7 ([82]) For
every k, there exists an integer f (k) such that if T is a complete multidigraph whose arcs are the union of k quasi-orders, then γ (T) ≤ f (k). In 2019 Bousquet, Lochet, and Thomassé [13] succeeded
in proving this longstanding 1982 Erd˝os-Sands-Sauer-Woodrow conjecture. The main ingredient in their proof is that the fractional domination number of complete multidigraphs (and therefore of
tournaments) is bounded. Theorem 85 ([13]) For every k, if T is a complete multidigraph whose arcs are the union of k quasi-orders, then γ (T ) = O(ln(2k) · k k+2 ).
Domination in Digraphs
Harutyunyan, Le, Newman, and Thomassé [53] continued the study of fractional domination in digraphs. Recall that in general there is no upper bound on the domination number of an oriented graph
solely in terms of its independence number. However, by Theorem 64, if G is a graph of order n, then DOM(G) ≤ α(G) · log n. In contrast to this result, Harutyunyan et al. [53] showed that for any
digraph, its fractional domination number is at most twice its independence number. Theorem 86 ([53]) For every digraph D, we have γ f (D) ≤ 2α(D), and this bound is sharp. The authors in [53]
presented two proofs of Theorem 86. The first proof uses the duality of linear programming, while the second proof is by induction. To show sharpness of the bound, given an arbitrary small real
number > 0, for any integer k ≥ 1, they constructed a digraph D such that α(G) = k and γ f (D) > 2k − . Further, they showed that almost surely a random tournament has fractional domination number
close to the upper bound of 2.
7 The Oriented Version of the Domination Game In 2002 Alon, Balogh, Bollobás, and Szabó [1] introduced and first studied the oriented domination game, which belongs to the growing family of
competitive optimization graph games. The oriented domination game describes a process in which two players with conflicting goals alternately orient an edge of a graph G until all of the edges are
oriented. One player’s goal is to minimize the domination number of the resulting oriented graph, while the other player wants to maximize it. Formally, the oriented domination game on a graph G
consists of two players, Minimizer and Maximizer (called Dominator and Avoider in [1]), who take turns orienting an unoriented edge of a graph G, until all edges are oriented. The goal of Minimizer
is to minimize the domination number of the resulting digraph, while the goal of Maximizer is to maximize the domination number. The Minimizer-start oriented domination game is the oriented
domination game when Minimizer plays first. The oriented game domination number γ og (G) of G is the minimum possible domination number of the resulting digraph when both players play according to
the rule that on each move a player may only orient an unoriented edge. To illustrate the game, Alon et al. [1] determined the oriented game domination number of a complete graph. Proposition 87
([1]) For a complete graph Kn of order n ≥ 4, we have γ og (Kn ) = 2. Proof Minimizer’s strategy is to pick two arbitrary vertices, say u and v. On each of his turns, Minimizer orients an edge from u
or v to a vertex w different from u and v. His strategy is to orient these edges in such a way that at least one of u and v is oriented towards w. He can always achieve his goal as follows. Whenever
Maximizer orients the edge uw from w to u, then Minimizer immediately replies by orienting the edge vw from v to w, if it is not already oriented. Analogously,
T. W. Haynes et al.
whenever Maximizer orients the edge vw from w to v, then Minimizer immediately replies by orienting the edge uw from u to w, if it is not already oriented. In this way, he ensures that the set {u, v}
is a dominating set in the resulting oriented graph. Thus, γ og (Kn ) ≤ 2. To show that γ og (Kn ) ≥ 2, Maximizer adopts the following strategy. Maximizer can clearly prevent a source in the oriented
graph resulting when n = 4. In the case when n ≥ 5, there exists a collection of n edge-disjoint paths of length 2, one centered at each of the n vertices of Kn (see [10]). Maximizer’s strategy is
whenever Minimizer orients one of these edges from a central vertex on one of these paths, Maximizer responds by orienting the other edge of the corresponding path towards the central vertex. In this
way, Maximizer guarantees that the indegree of each vertex in the resulting oriented graph becomes at least 1, implying that γ og (Kn ) ≥ 2. Consequently, γ og (Kn ) ≥ 2. In [1], the authors obtained
a sharp lower bound for the oriented game domination number of trees. Theorem 88 ([1]) If G is a tree of order n, then 12 n ≤ γog (G) ≤ 23 n. The proof of Theorem 88 implies that the upper bound
holds for any connected graph G, as Minimizer can concentrate his attention on a spanning tree T of G and play according to his strategy in the tree T. Whenever Maximizer orients an edge not in T,
Minimizer continues to orient edges according to his strategy in the tree. As shown in [1], both bounds in Theorem 88 are sharp. For graphs with minimum degree at least 2, the following improved
upper bound was given in [1]. Theorem 89 ([1]) If G is a graph of order n with δ(G) ≥ 2, then γog (G) ≤ 12 n. If G is a graph of order n with maximum degree , then a trivial lower bound on the
domination number is γ (G) ≥ n/ . In the oriented domination game, Maximizer orients half of the edges. As observed by Alon et al. [1], Maximizer might succeed in decreasing the outdegree of each
vertex to about /2, in which case the resulting domination number is at least 2n/ . This prompted them to pose the following conjecture. Conjecture 8 ([1]) If G is a graph of order n with maximum
degree Δ, then γog (G) ≥
2 n. (1 + o(1))
Conjecture 8 has yet to be settled. The best general lower bound to date on the oriented game domination number in terms of the maximum degree and order of the graph is the following result in [1].
Theorem 90 ([1]) If G is a graph of order n with maximum degree Δ, then γog (G) ≥
4 n. 3 + 7
Domination in Digraphs
Nordhaus-Gaddum-type inequalities for the oriented domination game are given in [1]. Here, G denotes the complement of a graph G. Theorem 91 ([1]) If G is a graph of order n, then γog (G) + γog (G) ≤
n + 2, and this bound is sharp. We note that if G is the complete graph Kn where n ≥ 4, then γog (G) = n and, by Proposition 87, γ og (G) = 2. Thus, if G = Kn , then γog (G)+γog (G) = n+2, showing
sharpness of the bound in Theorem 91. We close this section with the following conjecture posed in [1], that the inequality in Theorem 91 can be strengthened for connected graphs. Conjecture 9 ([1])
If both G and its complement G are connected graphs of order n, then γog (G) + γog (G) ≤
2 n + 3. 3
8 Concluding Comments In this chapter, we have surveyed selected results on domination in digraphs. Many results have been omitted to prevent the chapter from growing too large. For example, topics
such as signed domination in digraphs, efficient domination in digraphs, packing in digraphs, reinforcement numbers of digraphs, rainbow domination in digraphs, and Roman domination in digraphs, to
name a few, are omitted. Additional references on domination in digraphs can be found in [40, 44, 48, 55, 56, 58, 70, 77, 85, 91]. Due to space limitations, we have also omitted proofs of many
important results on domination in digraphs presented in this chapter, including the proofs of results due to Alon, Brightwell, Kierstead, Kostochka, and Winkler [2]; Chudnovsky, Ringi, Chun-Hung,
Seymour, and Thomassé [23]; Harutyunyan, Le, Newman, and Thomassé [53]; and Bousquet, Lochet, and Thomassé [13] which have significantly impacted the latest developments in the field of domination in
digraphs and tournaments. We apologize for these omissions.
References 1. N. Alon, J. Balogh, B. Bollobás, T. Szabó, Game domination number. Discrete Math. 256, 23–33 (2002) 2. N. Alon, G. Brightwell, H.A. Kierstead, A.V. Kostochka, P. Winkler, Dominating
sets in kmajority tournaments. J. Combin. Theory Ser. B 96(3), 374–387 (2006) 3. S. Arumugam, K. Jacob, L. Volkmann, Total and connected domination in digraphs. Australas. J. Combin. 39, 283–292
(2007) 4. S. Arumugam, K. Ebadi, L. Sathikala, Twin domination and twin irredundance in digraphs. Appl. Anal. Discrete Math. 7, 275–284 (2013)
T. W. Haynes et al.
5. C. Balbuena, H. Galeana-Sánchez, M. Guevara, A sufficient condition for kernel-perfectness of a digraph in terms of semikernel modulo F. Acta Math. Appl. Sin. 28, 340–356 (2012). English Series 6.
C. Berge, Théorie des graphes et ses applications (French), in Collection Universitaire de Mathematiques, vol. II (Dunod, Paris, 1958), viii+277 pp. 7. C. Berge, Graphs and Hypergraphs (North
Holland, New York, 1973) 8. M. Blidia, L. Ould-Rabah, Bounds on the domination number in oriented graphs. Australas. J. Combin. 48, 231–241 (2010) 9. B. Bollobás, T. Szabó, Domination in oriented
graphs. Proceedings of the Twenty-eighth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1997). Congr. Numer. 123, 55–64 (1997) 10. B. Bollobás, T.
Szabó, The oriented cycle game. Discrete Math. 186, 55–67 (1998) 11. J.A. Bondy, Short proofs of classical theorems. J. Graph Theory 44, 159–165 (2003) 12. E. Boros, V. Gurvich, Perfect graphs,
kernels, and cores of cooperative games. Discrete Math. 306, 2336–2354 (2006) 13. N. Bousquet, W. Lochet, S. Thomassé, A proof of the Erd˝os-Sands-Sauer-Woodrow conjecture. J. Combin. Theory Ser. B
137, 316–319 (2019) 14. L. Caccetta, R. Häggkvist, On minimal digraphs with given girth. Congr. Numer. 21, 181–187 (1978) 15. H. Cai, J. Liu, L. Qian, The domination number of strong product of
directed cycles. Discrete Math. Algorithms Appl. 6(2), 1450021, 10 pp. (2014) 16. Y. Caro, M.A. Henning, A greedy partition lemma for directed domination. Discrete Optim. 8, 452–458 (2011) 17. Y.
Caro, M.A. Henning, Directed domination in oriented graphs. Discrete Appl. Math. 160, 1053–1063 (2012) 18. G. Chartrand, D. Vanderjagt, B.Q. Yue, Orientable domination in graphs. Congr. Numer. 119,
51–63 (1996) 19. G. Chartrand, F. Harary, B.Q. Yue, On the out-domination and in-domination numbers of a digraph. Discrete Math. 197/198, 179–183 (1999) 20. G. Chartrand, P. Dankelmann, M. Schultz,
H.C. Swart, Twin domination in digraphs. Ars Combin. 67, 105–114 (2003) 21. H.H. Cho, F. Doherty, S.R. Kim, J.R. Lundgren, Domination graphs of regular tournaments II. Congr. Numer. 130, 95–111
(1998) 22. H.H. Cho, S.R. Kim, J.R. Lundgren, Domination graphs of regular tournaments. Discrete Math. 252, 57–71 (2002) 23. M. Chudnovsky, R. Kim, C.-H. Liu, P. Seymour, S. Thomassé, Domination in
tournaments. J. Combin. Theory Ser. B 130, 98–113 (2018) 24. V. Chvátal, L. Lovász, Every directed graph has a semi-kernel, in Hypergraph Seminar, ed. by C. Berge, D.K. Ray-Chaudhuri (Springer, New
York, 1974), p. 175 25. P. Duchet, Graphes Noyau-parfaits. Ann. Discrete Math. 9, 93–101 (1980) 26. P. Duchet, A sufficient condition for a digraph to be kernel-perfect. J. Graph Theory 11, 81–85
(1987) 27. E. Egerváry, On combinatorial properties of matrices. Mat. Lapok 38, 16–28 (1931) 28. P. Erd˝os, On Schütte problem. Math. Gaz. 47, 220–222 (1963) 29. W.F. De la Vega, Kernels in random
graphs. Discrete Math. 82, 213–217 (1990) 30. D.C. Fisher, J.R. Lundgren, S.K. Merz, K.B. Reid, Domination graphs of tournaments and digraphs. Congr. Numer. 108, 97–107 (1995) 31. D.C. Fisher, J.R.
Lundgren, S.K. Merz, K.B. Reid, The domination and competition graphs of a tournament. J. Graph Theory 29, 103–110 (1998) 32. D.C. Fisher, J.R. Lundgren, S.K. Merz, K.B. Reid, Connected domination
graphs of tournaments. J. Combin. Math. Combin. Comput. 31, 169–176 (1999) 33. D.C. Fisher, D. Guichard, J.R. Lundgren, S.K. Merz, Domination graphs with nontrivial components. Graphs Combin. 17(2),
227–228 (2001)
Domination in Digraphs
34. D.C. Fisher, D. Guichard, J.R. Lundgren, S.K. Merz, K.B. Reid, Domination graphs of tournaments with isolated vertices. Ars Combin. 66, 299–311 (2003) 35. D.C. Fisher, D. Guichard, J.R. Lundgren,
S.K. Merz, K.B. Reid, Domination graphs with 2 or 3 nontrivial components. Bull. ICA 40, 67–76 (2004) 36. A. Fraenkel, Planar kernel and Grundy with d ≤ 3, dout ≤ 2, din ≤ 2 are NP-complete. Discrete
Appl. Math. 3, 257–262 (1981) 37. A.S. Fraenkel, Combinatorial games: selected bibliography with a Succinct Gourmet introduction. Electron. J. Combin. 42, #DS2 (2007) 38. Y. Fu, Dominating set and
converse dominating set of a directed graph. Amer. Math. Mon. 75, 861–863 (1968) 39. H. Galeana-Sánchez, A counterexample t
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Basic Calculator - calculator
How to use the Basic Calculator
Basic Calculator: To use the calculator, enter numbers by clicking the buttons. Select an operator (addition, subtraction, multiplication, division) and then enter the second number. Finally, press '
=' to see the result. Use 'C' to clear the input. The calculator performs basic arithmetic operations and displays the result in the text box. It is designed for simplicity and ease of use.
What operations can I perform?
You can perform addition, subtraction, multiplication, and division. Just use the corresponding buttons to select the operation after entering your numbers.
Can I use decimals?
Yes, you can use decimal numbers. Just type the decimal point while entering your numbers. The calculator will handle decimal calculations accurately.
What happens if I divide by zero?
The calculator will show an error message if you attempt to divide by zero, as this operation is undefined in mathematics.
Is there a memory function?
This basic calculator does not have a memory function. It performs calculations in real-time without storing previous results.
Can I clear the result?
Yes, you can clear the input by clicking the 'C' button. This will reset the calculator, allowing you to start a new calculation.
Is this calculator responsive?
Yes, the calculator is designed to work on various screen sizes, ensuring usability on both desktop and mobile devices.
Can I use keyboard shortcuts?
Currently, the calculator only responds to mouse clicks on the buttons. Keyboard shortcuts are not implemented in this version.
Is the calculator secure?
The calculator operates entirely in the browser and does not transmit any data, ensuring that your calculations are secure and private.
What technologies were used?
This calculator was built using HTML, CSS, and JavaScript. These technologies provide a simple yet effective way to create interactive web applications.
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Algebraic Structure of Generalized Splines
Acik erisim
Show full item record
Given a fnite graph G=(V,E), a commutative ring R with unity and an edge labeling function α that assigns the ideals of R to the edges of G, the pair (G,α) is called an edge labeled graph. A vertex
labeling F ⊂ R |V | is said to be a generalized spline on (G,α) if the difference of the labels on adjacent vertices is an element of the ideal on the corresponding edge. The collection of all
generalized splines on (G,α) over the base ring R is denoted by R(G,α) . There exists a ring and an R-module structure on R(G,α) . The module structure is studied with number of methods such as
fow-up bases, the Chinese Remainder Theorem and linear algebra techniques in this thesis. We focus on the problems of freeness and finding bases for generalized spline modules. We give a
combinatorial method to find the smallest leading entries of flow-up classes on any graph over principal ideal domains. We introduce a basis criteria for generalized spline modules on cycle graphs,
diamond graphs and trees over greatest common divisor domains by using some determinantal techniques. We define the homogenization of an edge labeled graph to get more information about the
generalized spline modules. This thesis includes six chapters. We give a survey of the literature on classical and generalized spline theory in Chapter 1. We give a detailed movitation of generalized
spline theory. We summarize the results of the thesis. In Chapter 2, we give the necessary background knowledge such as the properties of CGD and LCM, the Chinese Remainder Theorem and the
fundamentals of module theory and graph theory. In Chapter 3, we introduce basic definitions and properties of generalized spline theory. We study algebraic properties of the set R(G,α) and
investigate the effect of changing the ordering of the vertices of (G,α) on the module structure of R(G,α) . Also, we define the matrix M(G,α) which is used for finding R-module generators of R(G,α)
. In Chapter 4, we focus on a specific type of generalized splines, which is called flow-up classes. We formulate the smallest leading entries of flow-up classes on any graphs over any principal
ideal domains by using some combinatorial techniques. We also investigate the existence of flow-up bases for R (G,α) . Moreover, we give an algorithm to compute flow-up classes that have smallest
leading entries on arbitrary ordered cycles. In Chapter 5, we give a basis criteria for generalized spline modules on cycles, dia- mond graphs and trees over greatest common divisor domains by using
determinantal techniques and flow-up classes. We generalize some previous works which are done for cycles and diamond graphs over integers and we introduce a basis criteria for generalized spline
modules on trees. In order to do this, we use the smallest leading entries of flow-up classes that we formulate in Chapter 4. In Chapter 6, we defne the homogenization of an edge labeled graph in
order to give a graded module structure to the set of generalized splines. We study the freeness relation between R(G,α) and the module of its homogenization. We also give some applications of the
basis criteria that we introduce in Chapter 5.
The following license files are associated with this item:
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Mathematical Analysis 2 is a 9 CFU course, part of the Engineering Sciences bachelor course. These pages contain the practical details related to the course.
Teams code: ojz9mnk (use this to join on Microsoft Teams)
Classroom: Aula 8
Lecture notes
A pdf of lecture notes is available for download. This text will be updated as the course progresses. As such, text corresponding to lectures which have not yet been delivered should be considered as
a draft and subject to change.
If you wish to have a reference book, we recommend Mathematical Analysis II by Canuto and Tabacco.
General advice
• Develop your intuition, it's a powerful skill – But don’t trust it completely
• Don’t aim to memorize but rather seek to understand – It is easy to remember anything when you understand it.
• Question always, be skeptical of all statements presented to you. Don’t accept them until you are sure they are believable.
• Observe, question how everything fits together, notice all the details.
• Part of the process of mathematical reasoning is creative - to be creative we must drop our inhibitions and be ready to be wrong, repeatedly.
See the lesson diary for full details.
What is MA2?
Much of what we do in this course builds on ideas established in Mathematical Analysis 1. In particular many of the ideas are extended to the higher dimensional setting.
Mathematical Analysis 1 Mathematical Analysis 2
(Functions) (Scalar fields)
(Vector fields)
(Derivative) (Partial derivatives)
(Directional derivative)
(Derivative of path)
(Jacobian matrix)
(Extrema) (Extrema)
Lagrange multiplier method
Integral Multiple integral
Line integral
Surface integral
Additional info
Course material from previous years and other instructors is available.
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Effective interest rate computation bsp
The effective interest method of amortization causes the bond's book value to increase from $95,000 January 1, 2017, to $100,000 prior to the bond's maturity. The issuer must make interest Effective
Period Rate = Nominal Annual Rate / n. Effective annual interest rate calculation. The effective interest rate is equal to 1 plus the nominal interest rate in percent divided by the number of
compounding persiods per year n, to the power of n, minus 1. Effective Rate = (1 + Nominal Rate / n) n - 1 . Effective interest rate calculation The effective interest rate is the interest rate on a
loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears. It is used to compare the annual interest between loans with
different compounding terms (daily, monthly, quarterly, semi-annually, annually, or other).
Hence 5.063 is the effective interest rate for semi annual, 5.094 for quarterly, 5.116 for monthly, and 5.127 for daily compounding. Just … Effective Period Rate = 5% / 12months = 0.05 / 12 =
0.4167%. Effective annual interest rate calculation. The effective annual interest rate is equal to 1 plus the nominal interest rate in percent divided by the number of compounding persiods per year
n, to the power of n, minus 1. Effective Rate = (1 + Nominal Rate / n) n - 1. Example. What is the effective annual interest rate for nominal annual interest rate of 5% compounded monthly? Solution:
Effective Rate = (1 + 5% / 12) 12 - 1 If you have an investment earning a nominal interest rate of 7% per year and you will be getting interest compounded monthly and you want to know effective rate
for one year, enter 7% and 12 and 1. If you are getting interest compounded quarterly on your investment, enter 7% and 4 and 1. The effective interest rate is the usage rate that a borrower actually
pays on a loan. It can also be considered the market rate of interest or the yield to maturity . This rate may vary from the rate stated on the loan document, based on an analysis of several factors;
a higher effective rate might lead a borrower to go to a different lender .
to the correct computation of the interest rates imposed by Lending Companies (“LCs”) and Financing Companies (“FCs”). Adopting BSP Memorandum No. M-2011-040 in the case of LCs and FCs, an effective
interest rate (“EIR”) calculation model for a loan, founded on established principles of discounted cash flow
1 Jan 2012 BSP Memo No. M-2011-040:Effective Interest Rate Calculation Models for All Types of Loans. The percentage that the finance charge bears to the total amount to be financed expressed as a
simple annual rate or an effective annual interest rate (EIR) as 6 Jun 2019 Adopting BSP Memorandum No. M-2011-040 in the case of LCs and FCs, an effective interest rate. (“EIR”) calculation model
for a loan, Uniform interest rate calculation for every loan. • Actual price of the Require MFIs to disclosure effective rates. • Establish an official Loan amount, interest and other finance
charges must be disclosed Philippines: BSP Memorandum.
Effective Period Rate = 5% / 12months = 0.05 / 12 = 0.4167%. Effective annual interest rate calculation. The effective annual interest rate is equal to 1 plus the nominal interest rate in percent
divided by the number of compounding persiods per year n, to the power of n, minus 1. Effective Rate = (1 + Nominal Rate / n) n - 1. Example. What is the effective annual interest rate for nominal
annual interest rate of 5% compounded monthly? Solution: Effective Rate = (1 + 5% / 12) 12 - 1
Changing the loan amount in the calculator back to $200,000, and trying out a few interest rates, shows that an interest rate of 4.11% would produce that same $968 monthly payment. Therefore this
loan's effective interest rate, or APR, is 4.11%.
21 Oct 2019 “BSP Governor Benjamin E. Diokno gave hints of an end to reverse ““The recent headline regarding the RRR cut of bond issuances from 6% to 3% effective Nov. The reserve ratios of thrift
banks will also be cut to five percent from the 26 policy meetings — to bring the interest rate on its overnight
Tenor, Monthly Rate, Factor Rate, Contractual Interest Rate, Effective Interest Rate* *Based on Loan Amount of Php100,000 and Processing Fee of Php1, 900. Per BSP Circular No.730, EIR is the rate
that exactly discounts estimated future 6 Aug 2019 Average inflation to date is 3.3 percent while inflation rate in July 2018 on a stronger REER (real effective exchange rate), higher real interest
22 May 2019 The effective interest rate (EIR) is “the rate that exactly discounts estimated The Bangko Sentral ng Pilipinas (BSP) uses this EIR definition as well. SSS deducts a 1% service fee from
the SSS salary loan amount, while 27 Sep 2018 They also welcomed the BSP's decision to delay the bank reserve Real effective exchange rate (2005=100). 109.9. 109.5 corporations' interest payment
deductions to a specified percentage of earnings before interest, 9 May 2019 The Monetary Board of the Bangko Sentral ng Pilipinas (BSP) on Thursday to 5.0%, and the overnight deposit rate to 4.00%,
effective Friday, May 10. However, the BSP in June 2016 implemented the interest rate corridor the target range of 3.0% ± 1.0 percentage point for both 2019 and 2020, while Subject: Effective
Interest Rate Calculation Models for all Types of Loans Relative to the implementation of Circular No. 730 dated 20 July 2011 on updated rules in implementing the Truth in Lending Act to enhance loan
transaction transparency, Effective Interest Rate (EIR) calculation models illustrative of common loan features are presented
Subject: Effective Interest Rate Calculation Models for all Types of Loans Relative to the implementation of Circular No. 730 dated 20 July 2011 on updated rules in implementing the Truth in Lending
Act to enhance loan transaction transparency, Effective Interest Rate (EIR) calculation models illustrative of common loan features are presented
For loans with contractual interest rates stated on monthly basis, the effective interest rate may be expressed as a monthly rate. In accordance with the Philippine Accounting Standards (PAS)
definition, effective interest rate is the rate that exactly discounts estimated future cash flows through the life of the loan to the net amount of loan The effective interest method of amortization
causes the bond's book value to increase from $95,000 January 1, 2017, to $100,000 prior to the bond's maturity. The issuer must make interest Effective Period Rate = Nominal Annual Rate / n.
Effective annual interest rate calculation. The effective interest rate is equal to 1 plus the nominal interest rate in percent divided by the number of compounding persiods per year n, to the power
of n, minus 1. Effective Rate = (1 + Nominal Rate / n) n - 1 . Effective interest rate calculation The effective interest rate is the interest rate on a loan or financial product restated from the
nominal interest rate as an interest rate with annual compound interest payable in arrears. It is used to compare the annual interest between loans with different compounding terms (daily, monthly,
quarterly, semi-annually, annually, or other). Based on a payment of $900 to buy the bond, three interest payments of $50 each, and a principal payment of $1,000 upon maturity, Muscle derives an
effective interest rate of 8.95%. Using this rate, Muscle's controller creates the following amortization table for the bond discount: Monthly effective rate will be equal to 1.6968%. The nominal
percent is 1.6968% * 12 = is 20.3616%. The effective annual rate is: The monthly fees increased till 22, 37%. But in the loan contract will continue to be the figure of 18%. However, the new law
requires banks to specify in the loan agreement to the effective annual interest rate. Finally, multiply the result by 100 to find the effective interest rate for the discounted bond. Effective
Interest Rate Example For example, say there is a 10-year bond with a face value of $2,000 that pays 5 percent interest every year and returns the principal when the bond matures.
Hence 5.063 is the effective interest rate for semi annual, 5.094 for quarterly, 5.116 for monthly, and 5.127 for daily compounding. Just … Effective Period Rate = 5% / 12months = 0.05 / 12 =
0.4167%. Effective annual interest rate calculation. The effective annual interest rate is equal to 1 plus the nominal interest rate in percent divided by the number of compounding persiods per year
n, to the power of n, minus 1. Effective Rate = (1 + Nominal Rate / n) n - 1. Example. What is the effective annual interest rate for nominal annual interest rate of 5% compounded monthly? Solution:
Effective Rate = (1 + 5% / 12) 12 - 1 If you have an investment earning a nominal interest rate of 7% per year and you will be getting interest compounded monthly and you want to know effective rate
for one year, enter 7% and 12 and 1. If you are getting interest compounded quarterly on your investment, enter 7% and 4 and 1.
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I'm working on probability HW and working on word problems. Do
you have any tips as...
I'm working on probability HW and working on word problems. Do you have any tips as...
I'm working on probability HW and working on word problems. Do you have any tips as to how to differentiate between Binomials, Geometric and Negative Binomials when doing word problems .
Binomial is simplest one in this we have number of trials given 'n' and probability of success 'p' and they ask for probability of getting success 'r' times.
For example: When we toss a coin 3 times what's probability of getting 2 heads. Here n = 3, p = 0.5 (because success is getting head, and that is 0.5)
Geometric: Geometric is extended form of Binomial distribution almost same just in this it ask for probability of first success in 'r' th trial.
For example: When we toss a coin 3 times what's the probability of getting 1st head on 2nd toss.
Negative binomial: This is one more step ahead it's same as Geometric it ask for 'k' th success in 'r' th trial.
For example: When we toss a coin 3 times what's the probability of getting 2nd head in 3rd toss.
I took same experiment to explain how it change from one distribution to another. I hope this will help.
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Finite Element Analysis By Hand - The 1D Bar Problem - Fidelis Engineering Associates
The behavior of any physical system can be defined mathematically. Solving these mathematical equations allows for the analysis of the system’s behavior. Depending on the complexity of the equation,
finding solutions directly may not be straightforward. Finite element analysis is an approximation technique to transform the mathematical representation of a physical system into a set of computer
solvable algebraic equations and calculate the numerical solution at a set of discrete locations.
There are several types of mechanical problems, such as static, dynamic, vibration, shock, thermal and corrosion. Static problems deal with structures attaining equilibrium under the applied loads.
These problems involve analyzing forces and moments acting on a structure to ensure that it remains in state of balance without any motion. Principles such as equilibrium equations, Newton’s laws of
motion and summation of forces and moments are used to solve these static problems. In this article let’s understand the steps needed to solve static problems using FEA by hand.
FEA Procedure Overview
The FEA procedure can be categorized into 3 parts as shown in the flow chart below.
In this article, let’s discuss the general steps for defining a finite element problem and calculating a numerical solution using simple 1D elastic bar problem.
Problem Formulation and discretization
Defining The Physical Problem
It is very important to determine the physicality of the problem at hand first. For this, we need to answer certain questions, such as whether the problem is 1D, 2D or 3D. Is it a solid mechanics,
continuum mechanics, heat transfer, computational fluid dynamics problem? Is it a static or dynamic problem? Is it a single or multi- body problem? Does it require plastic and damage properties for
In this article, we are considering a simple solid bar with fixed cross-sectional area (A). The bar is elastic with Youngs modulus (E). The left-hand side of the bar is fixed in all degrees of
freedom, and it is subjected to compressive pressure (t) on the right-hand side. The stresses, displacement and strains in the bar are unknown variables. This can be reduced to a 1D problem as the
deformation and stresses are almost uniform across the cross section of the rod and they only vary along its axis. This is a quasi-static problem as we are calculating its deformation when the bar
attains equilibrium under the influence of applied loads.
Mathematical Model
Mathematical equations relating input and output variables need to be established in this step. This can be obtained using fundamental laws such as equilibrium equations, elasticity law, heat
transfer law and diffusion law.
For the current problem at hand, we need to obtain a relationship between the applied force and elastic deformation of the rod. For this, we use the equation of equilibrium and Hooke’s material law
to arrive at the mathematical model. In this step, we also need to identify the boundary conditions and include them in the mathematical model. For the 1D problem, the boundary conditions are the
body force b and the pressure t applied at x = 6 in. The mathematical model for the 1D bar problem can be obtained as,
Deriving Weak Form
In this step, optimization requirements can be applied to the strong form equation established in the previous step to obtain weak form. There are different variational principles as shown in the
figure below to derive the weak form.
For the 1D bar problem, the weak form is derived to be,
“Weak” refers to the fact that the continuity requirement of the displacement variable u is weakened in this newly derived weak form equation. For the strong form differential equation to hold
everywhere on the bar, the first derivation of u (du/dx) has to be continuous, and the second derivative of u(d^2u/dx^2) needs to exist. Whereas for the weak form integral equation to hold everywhere
on the bar, u and its variation (δu) should be continuous, and the first derivatives can have discontinuity at finite number of locations and thus the continuity requirement is weakened in this step.
In this step, we break the computational domain of the problem to a set of smaller segments. Each segment is termed as an element. This process of discretizing the computational domain is called
meshing. The meshing process is demonstrated here by diving the 1D bar into 6 elements (1 in length each). The adjacent elements are connected through a common node. The elements and nodes are
numbered for tracking purposes. The meshing process for 1D problems is pretty straight forward. This becomes more involved for 2D and 3D domains with complex geometries. This can be achieved
numerically by using algorithms.
Calculating The Solution
Defining Unknown Variables For The Elements
In this step, the unknown variables such as displacement are approximated over the elements in the computational domain. Let’s assume the unknown displacement in the 1D bar problem at nodes 1, 2…6 is
u1, u2 …u6 respectively. These displacements are termed as nodal displacements. The displacements over each of the elements can be assumed either to be linear, quadratic, or other higher polynomials
as shown in the figure below.
A general polynomial function in x for u can be written as,
Let’s assume linear approximation for the displacement over each element in our model. Calculating the linear polynomial coefficients gives linear shape functions for each element as shown below.
The shape functions and their first derivative values must be calculated for all the elements in this step. For more information on shape functions and their properties please refer to our previous
blog here.
Weak Form Discretization
In this step, the global integral in the weak form can be broken down into element integrals:
The unknown displacement function and its derivates must be replaced by finite element polynomial approximations. For the first element, these polynomial approximations are derived to be,
Substituting the finite element approximations into the weak form for all elements gives the elemental stiffness matrix and force vectors.
Global Equation Systems
The elemental stiffness matrices [k] and force vectors {f} should be assembled in the global matrix and vector as shown in the figure below. Each element and its nodal DOF are given a unique global
index, which forms the basis for building global matrices. Note that, based on mesh connectivity, some of the elemental stiffness terms will get added in the global matrix.
Loads and Boundary Conditions
There are two types of boundary conditions (BCs): natural and essential. Essential boundary conditions are specified directly to the solution matrices (u, f) while the natural boundary conditions
describe phenomenon that affect solution matrices. Examples of essential BCs are fixed displacement or nodal forces and examples of natural BCs are body force, fluxes, and rate changes. The natural
BCs are already included in the global vectors during formulation of element matrices, but the essential BC should be applied in this step. There are different approaches like direct substitution
method, penalty method and Lagrange multipliers to apply essential BCs. In the 1D bar problem, the essential boundary condition is u|x=0 =0. This has to be applied in this step.
Solving the global equation system
The overall global system of equations (KU =F) with all the boundary conditions applied, has to be solved in this step for solution vector {u}. With increase in size and complexity of the problem
domain, solving this equation system becomes tricky and computationally expensive. There are a number of direct and iterative methods available to solve this equation system as shown in the table
Iterative methods Direct methods
SOR (successive over relaxation)Conjugate gradient methodMinimal residual methodGauss-Siedel Gaussian eliminationGauss-Jordan eliminationLU decomposition
The nodal DOF are only obtained by solving the global system of equations. But we often need to calculate other physical quantities such as strain and stress in the element and visualize them in the
contour plots or graphs. These calculations are performed in this step.
In the 1D bar problem, we can calculate strain and stress in each element by,
These quantities can be plotted on the graph, or they can be visualized using contour plots or animations.
Final Thoughts
Hopefully this article has given an overview on steps involved in formulating and solving a simple 1D FEA problem. The complexity of meshing process, global matrices and vectors increases with
increase in complexity and dimensionality of the computational domains. Thankfully we have commercial FEA software like Abaqus to ease this problem formulation process. Using this software, the
repetitive tasks like element formulations, assembling global matrices and solving equations are automated, and solutions can be obtained much faster. Hence the commercial FEA softwares play a vital
role in modern engineering design and analysis, facilitating the development of innovative and reliable products.
If you are interested in solving mechanical problems using FEA simulations, feel free to reach us. Our Engineers have many years of experience in commercial FEA, and we would love to talk to you!
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Abstract Algebra
Hw7 Problem 11
Return to Homework 7, Glossary, Theorems
Problem 11
Let $G$ be a group and let $a\in G$ be a fixed element of $G$. Show that the map $\lambda_{a}: G\rightarrow G$, given by $\lambda_{a}(g)=ag$ for all $g\in G$, is a permutation of the set $G$.
A permutation on a set $G$ is a function $\phi : G \rightarrow G$ that is 1-1 and onto.
Let $g,h \in G$. Suppose $\lambda_{a}(g)=\lambda_{a}(h)$. Now, $ag=ah$ and by cancellation, $g=h$. Hence, $\phi$ is 1-1.
Let $h \in G$. Note $\lambda_{a}(a^{-1}h)=h$. Now,
We know $G$ is a group, therefore $a^{-1} \in G$ and the function $\phi$ is onto.
Therefore, the map $\lambda_{a}:G \rightarrow G$ is a permutation on $G$.
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Sum fields for histogram
I have a dataset that is structured like this:
I need to tally the number of each category that has a value of 1 for each row like this:
i want to then plot the “Totals” field as a histogram to see the frequency distribution of total categories across all “Companies”.
My calculated field is structured exactly how you would calculate this in excel and i can verify that the column data types are all integers:
sum({Category1} + {Category2} + {Category3} + {Category4} + {Category5} + {Category6} + {Category7})
However, when i try to plot this calculated field as a histogram, i get the following error:
I am not using SQL and this is a static, csv dataset so this error is especially confusing.
Does anyone know what the issue may be?
Hi @jtroxel,
Welcome to the QuickSight community!
Currently, it is not possible to create a Histogram visual type for this.
Thanks and regards,
Biswajit Dash
1 Like
@Biswajit_1993 my mind is blown once again! thank you!
@Biswajit_1993 just closing the loop on this, i was able to generate the desired histogram by removing the sum() function from my calculated field and simply writing it out as a plain expression:
{Category1} + {Category2} + {Category3} + {Category4} + {Category5} + {Category6} + {Category7}
1 Like
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Math Aptitude Test 34 - Online Aptitude Test
Math Aptitude Test 34
This math aptitude Test #34 will measure your aptitude in a variety of math subjects. The questions could be addition, subtraction, multiplication, division, algebra, geometry, square roots or
measurements. The questions could be expressed in numbers or written as word problems.
There are 10 questions on the test. You have TWO MINUTES to answer all 10 questions. You may not skip any of the questions on this math aptitude test. You must answer all 10 questions in two minutes
in order to receive your score. Your score will be shown immediately after you complete the test.
You’re welcome to take the tests as many times as you’d like. The tests should contain different questions but they will all be of the same difficulty. We have hundreds of general aptitude practice
questions in our database. You’re welcome to retake the test as many times as you’d like.
When you’ve completed the test there should be a button to View Answers. Wrong answers are highlighted in red. The correct answer is shown in a box highlighted in green.
GOOD LUCK!!!
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Tight Hamilton cycles in 3-uniform quasirandom graphs
We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry-quasirandom 3-graphs.Our first result asserts
that for any fixed real α > 0, cherry-quasirandom 3-graphs of sufficiently large order n having minimum 2-degree at least α(n − 2) have a tight Hamilton cycle.
Our second result concerns the minimum 1-degree sufficient for such 3-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every d, α > 0 satisfying d + α > 1, any sufficiently
large n-vertex such 3-graph H of density d and minimum 1-degree at least α n−12 has a tight Hamilton cycle.
Original language American English
Title of host publication Large Networks and Random Graphs, Frankfurt, Germany
State Published - 2018
Dive into the research topics of 'Tight Hamilton cycles in 3-uniform quasirandom graphs'. Together they form a unique fingerprint.
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It is proved that the variety of relevant disjunction lattices has the finite embeddability property. It follows that Avron's relevance logic RMI
has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron's result that RMI
is decidable. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The turbulent flow in a compound meandering channel with a rectangular cross section is one of the most complicated turbulent flows, because the flow behaviour is influenced by several kinds of
forces, including centrifugal forces, pressure‐driven forces and shear stresses generated by momentum transfer between the main channel and the flood plain. Numerical analysis has been performed for
the fully developed turbulent flow in a compound meandering open‐channel flow using an algebraic Reynolds stress model. The boundary‐fitted coordinate system is introduced as a method for coordinate
transformation in order to set the boundary conditions along the complicated shape of the meandering open channel. The turbulence model consists of transport equations for turbulent energy and
dissipation, in conjunction with an algebraic stress model based on the Reynolds stress transport equations. With reference to the pressure–strain term, we have made use of a modified pressure–strain
term. The boundary condition of the fluctuating vertical velocity is set to zero not only for the free surface, but also for computational grid points next to the free surface, because experimental
results have shown that the fluctuating vertical velocity approaches zero near the free surface. In order to examine the validity of the present numerical method and the turbulent model, the
calculated results are compared with experimental data measured by laser Doppler anemometer. In addition, the compound meandering open channel is clarified somewhat based on the calculated results.
As a result of the analysis, the present algebraic Reynolds stress model is shown to be able to reasonably predict the turbulent flow in a compound meandering open channel. Copyright © 2005 John
Wiley & Sons, Ltd.
A time discrete scheme is used to approximate the solution toa phase field system of Penrose Fife type with a non-conservedorder parameter. An
a posteriori
error estimate is presentedthat allows the estimation of the difference between continuousand semidiscrete solutions by quantities that can be calculatedfrom the approximation and given data.
Trabecular bone fracture is closely related to the trabecular architecture, microdamage accumulation, and bone tissue properties. Primary constituents of trabecular tissue are hydroxyapatite (HA)
mineralized type-I collagen fibers. In this research, dynamic fracture in two dimensional (2-D) micrographs of ovine (sheep) trabecular bone is modeled using the mesoscale cohesive finite element
method (CFEM). The bone tissue fracture properties are obtained based on the atomistic strength analyses of a type-I collagen + HA interfacial arrangement using molecular dynamics (MD). Analyses show
that the presented framework is capable of analyzing the architecture dependent fracture in 2-D micrographs of trabecular bone.
The paper addresses the problem of calculation of the local stress field and effective elastic properties of a unidirectional fiber reinforced composite with anisotropic constituents. For this aim,
the representative unit cell approach has been utilized. The micro geometry of the composite is modeled by a periodic structure with a unit cell containing multiple circular fibers. The number of
fibers is sufficient to account for the micro structure statistics of composite. A new method based on the multipole expansion technique is developed to obtain the exact series solution for the micro
stress field. The method combines the principle of superposition, technique of complex potentials and some new results in the theory of special functions. A proper choice of potentials and new
results for their series expansions allow one to reduce the boundary-value problem for the multiple-connected domain to an ordinary, well-posed set of linear algebraic equations. This reduction
provides high numerical efficiency of the developed method. Exact expressions for the components of the effective stiffness tensor have been obtained by analytical averaging of the strain and stress
The viscoelastic properties of binary blends of nitrile rubber (NBR) and isotactic polypropylene (PP) of different compositions have been calculated with mean‐field theories developed by Kerner. The
phase morphology and geometry have been assumed, and experimental data for the component polymers over a wide temperature range have been used. Hashin's elastic–viscoelastic analogy principle is used
in applying Kerner's theory of elastic systems for viscoelastic materials, namely, polymer blends. The two theoretical models used are the discrete particle model (which assumes one component as
dispersed inclusions in the matrix of the other) and the polyaggregate model (in which no matrix phase but a cocontinuous structure of the two is postulated). A solution method for the coupled
equations of the polyaggregate model, considering Poisson's ratio as a complex parameter, is deduced. The viscoelastic properties are determined in terms of the small‐strain dynamic storage modulus
and loss tangent with a Rheovibron DDV viscoelastometer for the blends and the component polymers. Theoretical calculations are compared with the experimental small‐strain dynamic mechanical
properties of the blends and their morphological characterizations. Predictions are also compared with the experimental mechanical properties of compatibilized and dynamically cured 70/30 PP/NBR
blends. The results computed with the discrete particle model with PP as the matrix compare well with the experimental results for 30/70, 70/30, and 50/50 PP/NBR blends. For 70/30 and 50/50 blends,
these predictions are supported by scanning electron microscopy (SEM) investigations. However, for 30/70 blends, the predictions are not in agreement with SEM results, which reveal a cocontinuous
blend of the two. Predictions of the discrete particle model are poor with NBR as the matrix for all three volume fractions. A closer agreement of the predicted results for a 70/30 PP/NBR blend and
the properties of a 1% maleic anhydride modified PP or 3% phenolic‐modified PP compatibilized 70/30 PP/NBR blend in the lower temperature zone has been observed. This may be explained by improved
interfacial adhesion and stable phase morphology. A mixed‐cure dynamically vulcanized system gave a better agreement with the predictions with PP as the matrix than the peroxide, sulfur, and
unvulcanized systems. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1417–1432, 2004
在喷气Z pinch内爆等离子体研究中，雪铲模型是一种常用的、比较简单的物理模型。根据实验中提供的电流波形，负载线质量和初始半径，可以通过雪铲模型来估算内爆到心的时刻。根据一维运动方程和不同构形下的解析解
以及部分实验结果相结合，讨论了雪铲模型的适用范围。数值计算的内爆时间和实验（Gamble II, Double EAGLE, BLACKJACK 5）测量值符合得较好。结果表明，雪铲模型在喷气Z pinch实验的负载优化设计研究中是很有参考价
For three‐dimensional flows with one inhomogeneous spatial coordinate and two periodic directions, the Karhunen–Loeve procedure is typically formulated as a spatial eigenvalue problem. This is
normally referred to as the direct method (DM). Here we derive an equivalent formulation in which the eigenvalue problem is formulated in the temporal coordinate. It is shown that this so‐called
method of snapshots (MOS) has some numerical advantages when compared to the DM. In particular, the MOS can be formulated purely as a matrix composed of scalars, thus avoiding the need to construct a
matrix of matrices as in the DM. In addition, the MOS avoids the need for so‐called weight functions, which emerge in the DM as a result of the non‐uniform grid typically employed in the
inhomogeneous direction. The avoidance of such weight functions, which may exhibit singular behaviour, guarantees satisfaction of the boundary conditions. The MOS is applied to data sets recently
obtained from the direct simulation of turbulence in a channel in which viscoelasticity is imparted to the fluid using a Giesekus model. The analysis reveals a steep drop in the dimensionality of the
turbulence as viscoelasticity is increased. This is consistent with the results that have been obtained with other viscoelastic models, thus revealing an essential generic feature of polymer‐induced
drag reduced turbulent flows. Published in 2006 by John Wiley & Sons, Ltd.
Mass distributions of fragments in the low-energy fission of nuclei from
Ir to
At have been analysed. This analysis has shown that shell effects in symmetric-mode fragment mass yields from the fission of pre-actinide nuclei could be described if one assumes the existence of two
strongly deformed neutron shells in the arising fragments with neutron numbers
≈ 52 and
≈ 68. A new method has been proposed for quantitatively describing the mass distributions of the symmetric fission mode for pre-actinides with
≈ 180–220.
In [5] Phillips proved that one can obtain the additive group of any nonstandard model *? of the ring ? of integers by using a linear mod 1 function
h : F
?, where
is the α-dimensional vector space over ? when α is the cardinality of *?. In this connection it arises the question whether there are linear mod 1 functions which are neither addition nor
quasi-linear. We prove that this is the case.
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What is the least common denominator of 3 4 and 5? | - The Game ArchivesWhat is the least common denominator of 3 4 and 5? | - The Game ArchivesWhat is the least common denominator of 3 4 and 5? |
The least common denominator of 3, 4 and 5 is 2.
The “what are the common multiples of 3, 4 and 5” is a question that will be answered in this blog.
2 2 3 5 = 60 2 2 3 5 = 60 is the LCM of 3,4,5 3, 4,5.
What is the least common denominator of 3 4 in this case?
The lowest number that various numbers are factorsof is called the Least Common Multiple (LCM). Because 12 is the lowest number for which 3 and 4 are both components, the LCM of 3 and 4 equals 12.
One can also wonder what the LCM of 3 and 5 is. 3 and 5 have an LCM (least common multiple) of 15. There are no variables in common between 3 and 5. As a result, the LCM of 3 and 5 is simply 3
multiplied by 5, or 3*5, which is 15.
People also wonder how to discover the least common denominator.
The steps are as follows:
1. Find the denominators’ Least Common Multiple (whichiscalled the Least Common Denominator).
2. Make each fraction’s denominators the same as the least common denominator (using comparable fractions).
3. Then, as desired, add (or subtract) the fractions!
Which of the following is the least frequent multiple of 4 and 5?
Let’s say we’ve compiled a list of the first several multiples of 4and5: Multiples that appear in bothlists are called common multiples:
4,8,12,16,20,24,28,32,36,40,44,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,48 5,10,15,20,25,30,35,40,45,50, 55,10,15,20,25,30,35,40,45,50, 55,10,15,20,25,30,35,40,45,50,
Answers to Related Questions
What are the LCD values for 4 and 8?
If you wish to add or subtract two fractions with the denominators 4 and 8, you’ll need to know the least common denominator (LCD). Between 4 and 8, the least common denominator, often known as the
least common denominator (LCD), is 8.
What is the LCM for the numbers 9 and 12?
The least frequent multiple of 9 and 12 is to be found. We can achieve this by listing the multiples: 9:9,18,27,36 ,45,54,63,72, 12:12,24,36 ,48,60,72,
What is the LCM of the numbers 5 and 7?
5 and 7 have a lcm of 35.
What is the difference between a 5 and a 10 LCD?
What is the Least Common Denominator (LCD) for the numbers 5 and 10? If you wish to add or subtract two fractions with 5and10 as denominators, you’ll need to know the least common denominator (LCD).
Between 5 and 10, the least common denominator, commonly known as the least common denominator (LCD), is 10.
What is the LCM of the numbers 5 and 8?
5 and 8 have a lcm of 40.
What is the LCM for the numbers 8 and 12?
8 and 12 are the least common multiples.
8,16,24, 32, 40, and 48 are the first few multiples of eight. Twelve, 24, 36, 48, 60, and 72 are the first multiples of twelve.
What is the LCM for the numbers 4 and 12?
12 and 4 have the same lcm.
What is the LCM of the numbers 4 and 8?
4 and 8 are the least common multiples. The leastcommonmultiple (LCM) of four and eight is eight.
In arithmetic, what is an LCD?
The denominator with the smallest common factor is the lowest common denominator. The lowest common denominator, also known as the least common multiple of the denominators of a collection of
fractions, is the lowest common multiple of the denominators of a set of fractions in mathematics. It makes addition, subtracting, and comparing fractions more easier.
What is the GCF for the numbers 4 and 5?
The factors and prime factorization of 4and5 were discovered. The GCFnumber is the largest common factor number. As a result, the highest common factor between 4 and 5 is 1.
What is the difference between a 12 and a 15-inch LCD?
Example of an LCM
The following are the multiples of 12: 12, 24, 36, 48, 60, 72, 84. The following are the multiples of 15: 15, 30, 45, 60, 75, 90. There are no lower common multiples than 60, which is a common
multiple (a multiple of both 12 and 15). As a result, 60 is the lowest common multiple of 12 and 15.
What is the procedure for calculating the lowest common multiple?
To get the least common multiple of two integers, start by listing each number’s prime factors. After that, multiply each factor by the number of times it appears in either number. If the same factor
appears in both numbers more than once, increase the factor by the highest number of times it appears.
What is the LCM of the numbers 9 and 7?
9 and 7 have a lcm of 63.
What is the LCM for a 5’9″ and a 15″ person?
The LCM of 9,15,5 is obtained by multiplying all primefactors by the highest number of times they appear in either number. The LCM for 9,15,5 9,15,5 equals 335=45 3 3 5 =45.
What is the LCM of the numbers 3 and 6?
LCM(2,3) = 6 and LCM(6,10) = 30 are two examples. The lowest number that is evenly divisible by all of the numbers in the collection is the LCM of two or more numbers.
What is the GCF for the numbers 5 and 3?
The greatest common factor (GCF) for the numbers 3 and 5 is 1. We’ll now compute the prime factors of 3 and 5, then match the largest common factor of 3 and 5 to obtain the greatest common divisor
(gcd) of the integers.
What is the LCM for the numbers 10 and 15?
10 and 15 are the Least Common Multiples. The leastcommonmultiple (LCM) of ten and fifteen is thirty.
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cycle_basis(G, root=None)[source]¶
Returns a list of cycles which form a basis for cycles of G.
A basis for cycles of a network is a minimal collection of cycles such that any cycle in the network can be written as a sum of cycles in the basis. Here summation of cycles is defined as
“exclusive or” of the edges. Cycle bases are useful, e.g. when deriving equations for electric circuits using Kirchhoff’s Laws.
• G (NetworkX Graph) –
Parameters: • root (node, optional) – Specify starting node for basis.
Returns: • A list of cycle lists. Each cycle list is a list of nodes
• which forms a cycle (loop) in G.
>>> G=nx.Graph()
>>> G.add_cycle([0,1,2,3])
>>> G.add_cycle([0,3,4,5])
>>> print(nx.cycle_basis(G,0))
[[3, 4, 5, 0], [1, 2, 3, 0]]
This is adapted from algorithm CACM 491 [1].
[1] Paton, K. An algorithm for finding a fundamental set of cycles of a graph. Comm. ACM 12, 9 (Sept 1969), 514-518.
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Functional Programming Nomenclature Intro | Thomas Ekström Hansen
Functional Programming Nomenclature Intro
Table of Contents
When starting out in functional programming (FP), you’ll likely see a lot of syntax in the languages that you have never come across before and therefore don’t know how to pronounce. Or vice-versa:
someone might keep referring to “the ‘bind’ operator” without you having a clue what that looks like in the language you’re working in. The aim of this post is to try to cover most of those things.
The motivation for creating this post was/is deeply personal. When I first started my Ph.D., I encountered a multitude of terms and concepts that everyone around me seemed to Just Know (TM), while I
was left feeling perplexed and out of the loop. Eventually, through numerous awkward questions and conversations where I felt quite foolish, I managed to unravel the meanings of these arcane terms
and symbols. Since, as far as I could tell, there wasn’t a simple nomenclature out there (“simple” meaning “answering ‘what is this called?’ without raising exponentially more questions”) I decided
to put together this reference to hopefully help others avoid the same confusion and headaches I faced.
You can read it in one go and try to remember the terms, but I believe it might work better as a lookup reference that you have open adjacent to whatever you’re currently working on. But that’s up to
you : )
In any case, the idea is that reading this should at least let you think about FP in words rather than mystical runes you don’t understand (which I’ve personally found can be difficult). It should
also, hopefully, make it easier to talk about FP things or understand the feedback you get given from other FP people.
(Note: Since I mainly do functional programming in Idris, some of these examples will be Idris-specific. However, the syntax should hopefully not differ too much compared to other FP languages (e.g.
This document is, by nature, a never-ending Work In Progress, and the explanations are only as good as my own understanding. Should you encounter any errors (or terms you would like added), please
reach out on this website’s GitHub repository.
• >>= — “bind”More details
Bind takes a result and a function which uses the result, and combines the two, e.g. a variable greeting containing the string "Hello World" and the function putStrLn could be combined as
which would print the greeting to the terminal when run.
• >> — “seq”, short for “sequence”More details
In Idris land, this behaves similar to “bind”, except it only works with things that produce a side-effect (e.g. printing to the terminal) without also returning a new value to handle.
• : — “has/of type” in Idris, and “cons” in Haskell (yes this is confusing)
• :: — “cons” in Idris, and “has/of type” in Haskell (yes this is confusing)
• [] — also referred to as “Nil”
• () — “unit”, sometimes also written as: Unit
• “first-class” — being able to use the concept everywhere in the language, e.g. functions are first-class if you can manipulate them just like you would any other term in the language (assign them
to variables, pass them to other functions, put them in data structures, etc).
• “a record” — a data structure containing named fields of data where those names are automagically also turned into functions (getters).
Typically, records also come with special syntax to update their fields.
• “a projection” — a getter for a record (functional programmers like to use maths terminology).
• “data” — things that you create by describing how to construct them, e.g. MkPair a b is a data constructor used to create a pair (a, b)More on pairs
Pairs typically also have the projections (getter functions) fst and snd to retrieve a and b respectively.
• “codata” — things that you create by defining their getters rather than their constructors; effectively describing how to deconstruct the thing. For example, we could define a ‘CoPair’ as “a
thing that has the projections fst and snd such that I can call fst on it to get the thing’s first element and snd to get the thing’s second element”.
□ Why? Mainly because it allows us to define infinite things: a Stream is a thing you deconstruct by taking the first element, followed by a stream of potentially infinitely more things. As
long as we can take at least one element, which is the projection (getter function), everything is fine and we can reason about it.
• “mapping” — synonymous with “function”
□ Usually found in combination with “from a to b”, e.g. “a mapping from Int to Nat” means “a function that takes an Int and returns a Nat”.
• “pure” — something which does not produce any side effects (like printing to the terminal) and returns the same output given the same input
• “covering” — a function where you have defined what to return for all its possible inputs.
• “total” — in Idris: a function which is either covering or productive (what is considered total varies from language to language).
• “partial” — a function where you have not defined what to do for every possible input, or a function which may never terminate. In opposition to “total”.
• “bottom” (also written _|_ or $\bot$) — something which is provably false or absurd. If you can “construct bottom”, you should have a contradiction somewhere.
• “top” (also written $\top$) — something which is trivially true; a tautology.
• “eta-expansion” ($\eta$-expansion) — TODO
• “binders” — TODO
□ “lambda binders” ($\lambda$-binders) — TODO
□ “Pi binders” ($\Pi$-binders) — TODO
• “rig” — in Idris: the accessibility, i.e. 0 for erased, 1 for linear, and w (the default, usually not written) for unrestricted.Show technical definition
A rig is a mathematical concept. It is a set of elements with two binary operations “add” and “multiply” such that both operations have an identity (e.g. 0 and 1 respectively for natural
numbers), multiplication distributes over addition, both operations are commutative (a + b = b + a), and multiplying by the additive identity (0 for natural numbers) always returns that identity
(so for natural numbers: a * 0 = 0 * a = 0).
The name “rig” comes from “ring without negation” (Get it? We remove the ’n' from “ring” to get “rig” because it doesn’t have Negation. Mathematicians truly are funny people…)
The accessibilities in Idris form a rig, hence we refer to them as such.
• “rig 0” — in Idris: erased / runtime inaccessible.
• “rig 1” — in Idris: linear / must be used exactly once.
• “rig w” — rig omega; in Idris: unlimited use.
• $\cong$ — TODO
• $\preceq$ — TODO
• $\vdash$ — TODO
Category Theory terms
Category Theory (CT) is a whole area of maths, explaining it is beyond the scope of this blog entry. Unfortunately for us, a lot of FP is built on ideas and concepts from CT so we have to at least be
aware of it and know some of the terminology…
Category Theory for Programmers
Bartosz Milewski has a blog series called “Category Theory for Programmers” (also available as a book), which covers the “basics” of CT. I have put ‘basics’ in quotes here, because I own the book and
read the first section, about 120 dense pages, after which Bartosz concludes “We’ve learned the basic vocabulary of category theory.” At this point, I gave up.
To be clear, it is many times better than any of the more formal literature at explaining these things! It just also, in my opinion, goes way above and beyond what the everyday functional programmer
will ever need, and the explanations have on multiple occasions hurt my brain (I’m still not sure if I actually understand contravariant functors or if my brain has tricked itself into thinking it
understands it to avoid further headaches…). The first couple of entries/chapters might be worth a read, but beyond that you’re doing it out of [DEL:masochistic tendencies:DEL] personal
• “category”
□ don’t think “a category of what?”
□ it’s a mathematical way of describing things which have a common structure
□ examples: TODO
□ “My understanding is that a category describes a whole category of systems or the theories that share the same structure.” – Bartosz Milewski
• “functor” — a mapping from one category to another
• “endofunctor” — a mapping from a category to itself
□ since Idris (and Haskell) may be considered a category, all functors in them are technically endofunctors (and mathematicians like to be precise).
• TODO: eventually get to the infamous “A monad is just a monoid in the category of endofunctors, what’s the problem?”
□ actual quote:
“All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of endofunctors and unit set by the identity endofunctor.” – Saunders Mac
Lane, Categories for the Working Mathematician.
Would You Like To Know More?
And as always, thanks for reading : )
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Online Recurring Deposit (RD) Calculator 2025 - RD Interest Rate Calculation
What are Recurring Deposits?
A recurring deposit (RD) is an investment instrument similar to fixed deposits. However, one has to make monthly fixed deposits in RDs, unlike a lump sum amount in FDs. Recurring deposits are offered
by various banks and financial institutions in India. The RD returns calculation could be pretty complicated for a common person to grasp accurately every time. Hence, this is where Fintra's RD
calculator proves to be immensely beneficial.
How can Fintra's RD calculator assist you?
Since the RDs are constant investments, their returns on the deposits could be challenging to track. Interests are compounded quarterly, and there are various variables involved that make these
calculations multipart. Therefore, an RD deposit calculator reduces the hassles of computing the returns manually and enables the individual to know the exact amount the deposits will accrue after
the relevant period.
However, while calculating, the individual has to manually do the TDS deduction, and the new RBI norms state that RDs are also liable for TDS deduction. However, there's no concord in the
implementation across the financial institutions, which is why RD calculators don’t take it into account.
Fintra's RD calculator provides the following advantages:
1. Our RD calculator enables you to plan your future finances with greater clarity by presenting you with the exact amount your investment will accrue.
2. The calculator is convenient to use and will save you time.
3. The calculators' accuracy can never be in question. Accurate estimates are crucial for doing prudent financial planning.
Factors Affecting RD Interest Rates
The few major factors that are taken into consideration when deciding how much rate of interest should be given to the depositor are:
Tenure: Tenure is the duration of the money that is invested in a recurring deposit. Tenure is one of those factors on which the interest depends on, and the RD interest rates vary all tenure
Age of the Applicant: As an example, financial institutions such as banks provide higher rates of interest to senior citizens, it may range from 0.50% to 0.75% additional interest on the regular
deposit rates.
Current Economic Environment: Banks that provide recurring deposits keep updating their interest rates as per the economic conditions. For example, a few of the reasons for the changes in economic
conditions might be the change in RBI's repo rate, inflation and so on. Therefore, it's the prevailing conditions that play a vital role in determining the RD rates.
How Does Fintra's RD Calculator Work?
While deciding which RD to opt for, it is vital to know what combination of instalment, rate of interest, and deposit tenure would lead to better returns. Fintra's RD Calculator aims to do just that.
The four fields to fill in for ascertaining your potential earnings while investing in a recurring deposit are:
1. Monthly Deposit Amount
2. RD Frequency
3. Annual Interest Rate
4. Deposit Tenure in Years
Hit the 'Submit' button and the results will appear on the side of the calculator. Please note the amount calculated is applicable if the interest is compounded quarterly.
How is Interest on RD Calculated?
Most banks which offer recurring deposits compound the interest quarterly. In India or the maturity value of RD, banks use the following formula for doing the RD interest calculation:
(Maturity value of RD; based on quarterly compounding)
M =R[(1+i)n – 1]/1-(1+i)(-1/3)
The letters stand for:
M = Maturity value of the RD
R = Monthly RD instalment to be paid
n = Number of quarters (tenure)
i = Rate of Interest / 400
Instead of manually completing this calculation, you can easily use the Fintra's Recurring Deposit Calculator, and narrow down the best combination to earn better returns.
FAQs About Recurring Deposits (RD)
What are the benefits of RD?
• RDs are used as collateral for obtaining loans- Up to 80-90% loans you can take on your RD amount
• Under RD, premature withdrawals are allowed, but it may come with a small penalty
• RD schemes offer a slightly higher rate of interest, 0.25% to 0.75% more for senior citizens
• Even minors can open RD accounts under the supervision and guardianship of their parents
• RDs tenure is flexible; you can pick from 7 days to 10 years depending on convenience
• RD schemes enable you to save money regularly, and the minimum deposit amount is RS. 10
Tax Benefits on RD
• Like other personal tax-saving and investment instruments, Recurring Deposit schemes also attract taxes. On the returns accrued from RD, TDS of 10% deducts, if the total interest exceeds Rs.
10,000 in a single financial year.
• Compare RD to the SIP scheme, you'll notice SIPs are more beneficial for the long term. Long-term gains from equity are tax-free, any SIP that invests in ELSS (Equity Linked Mutual Funds) is also
tax-free after one year.
Who Can Invest in Recurring Deposits?
It is a viable option, investing in recurring deposits, for those who don't have a huge amount of funds but seeking for low-risk investment options where higher returns are obtained with regular
Any Indian resident and Hindu Undivided Families (HUFs) can open a Recurring Deposit with their bank. A few banks even allow minors to open an RD to build the habit of saving. However, guardians to
supervision are required over the minor's finances. Banks sometimes request their customers to set an amount in multiples of 100.
What are the minimum and maximum tenures for which you can open an account in RD?
The minimum period of deposit starts from 6 months and can be extended to the maximum period of 10 years in most RD Accounts.
Is TDS applicable on interest earned through RD?
As per the Income Tax Law, TDS of 10% is applicable on the interest gained on Recurring Deposits. TDS will deduct if the interest earned on the Recurring Deposits is higher than Rs. 10,000.
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Michal Luria – Project 11 – Composition
/* Submitted by: Michal Luria
Section A: 9:00AM - 10:20AM
var originalImage;
var px = 0;
var py = 0;
var currentColor;
var pBrightness;
function preload() {
var myImageURL = "http://i.imgur.com/cU9Kpic.jpg"; //image link
originalImage = loadImage(myImageURL); //load image
function setup() {
createCanvas(500, 667);
originalImage.loadPixels(); //load pixels
function draw() {
//create new turtle
var turtle = makeTurtle(px, py);
//settings for turtle
while (px < width) { //check first line
currentColor = originalImage.get(px,py); //fetch color of photo
if (isColor(currentColor)) { //check that it is a color
pBrightness = brightness(currentColor); //what is the brightness
if (pBrightness < 50) { //if the brightness is less than 50
//draw a short black line
px +=6; //adjust px location
py += 4; //move to next line
px = 0; //start over on the x axis
//check the get function fetched the color
function isColor(c) {
return (c instanceof Array);
//---Turtle Code---//
function turtleLeft(d){this.angle-=d;}function turtleRight(d){this.angle+=d;}
function turtleForward(p){var rad=radians(this.angle);var newx=this.x+cos(rad)*p;
var newy=this.y+sin(rad)*p;this.goto(newx,newy);}function turtleBack(p){
this.forward(-p);}function turtlePenDown(){this.penIsDown=true;}
function turtlePenUp(){this.penIsDown = false;}function turtleGoTo(x,y){
line(this.x,this.y,x,y);}this.x = x;this.y = y;}function turtleDistTo(x,y){
return sqrt(sq(this.x-x)+sq(this.y-y));}function turtleAngleTo(x,y){
var absAngle=degrees(atan2(y-this.y,x-this.x));
var angle=((absAngle-this.angle)+360)%360.0;return angle;}
function turtleTurnToward(x,y,d){var angle = this.angleTo(x,y);if(angle< 180){
this.angle+=d;}else{this.angle-=d;}}function turtleSetColor(c){this.color=c;}
function turtleSetWeight(w){this.weight=w;}function turtleFace(angle){
this.angle = angle;}function makeTurtle(tx,ty){var turtle={x:tx,y:ty,
right:turtleRight,forward:turtleForward, back:turtleBack,penDown:turtlePenDown,
penUp:turtlePenUp,goto:turtleGoTo, angleto:turtleAngleTo,
turnToward:turtleTurnToward,distanceTo:turtleDistTo, angleTo:turtleAngleTo,
setColor:turtleSetColor, setWeight:turtleSetWeight,face:turtleFace, drawPent: turtle};
return turtle;}
This week for my project I wanted to use a turtle to create a graphics project that would look different using a turtle than other methods. I decided to compute a “cartoon” version, a solid black and
white picture, from a regular photo. Using this method, the photo can be recreated by computing it in slightly different ways – each small variation would create a very different and new composition
from the same source. Furthermore, I liked how any photo can be uploaded into the code and by only changing the photo link – instantly create a new image. The one presented is the version that
appealed most to me.
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The Stacks project
Lemma 72.10.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $G$ be a finite group acting freely on $X$. Set $Y = X/G$ as in Properties of Spaces, Lemma 66.34.1. For $y \in |Y|$
the following are equivalent
1. $y$ is in the schematic locus of $Y$, and
2. there exists an affine open $U \subset X$ containing the preimage of $y$.
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Harmonic and Melodic Equivalence V14H Trichord Pair
The hexatonic scale formed by these six notes is 1, b2, 2, 3, 5, b6 and works over 7 and 7sus4 chords. The usual exercises are included in the course but the most important aspect of this course is
the discussion of these topics:
• Examples of how to divide two 3 notes groups rhythmically to form modern sounding melodies.
• How interval vectors inform you about the sound of any group of notes.
• Trichord distribution within the 1, b2, 2, 3, 5, b6 scale.
• All possible types of trichords and triads found in the 1, b2, 2, 3, 5, b6 scale.
• Chord progression ideas using the the 016-026 trichords found in the 1, b2, 2, 3, 5, b6 scale.
• Which chords can work over the C, Db, G and D, E, Ab trichords.
• Understanding the six note compliment to any trichord pair.
All the topics listed above are crucial for developing a deeper understanding of music and how the Harmonic and Melodic Equivalence Series is organized.
Two Trichords
Two 3 note pairs and also be called “two trichords.” Two trichords which unlike a two triad pair are not build in thirds. The term was coined by Milton Babbitt to distinguish a three note collection
from a triad built in 3rds. Trichords form a sound that is very useful to a modern improviser both as melodic and harmonic content. Through the Harmonic and Melodic Equivalence Series these trichords
are studied in-depth. I think you will find them to be a welcome addition to your improvisational palette.
Harmonic and Melodic Equivalence V14H Trichord Pair
Status: In stock, Digital book is available for immediate access.
Background Information on a Two Triad Pair or Two Trichords
“Harmonic and Melodic Equivalence Trichord Pair” is a series of books that will help you to develop many different musical skills simultaneously. The source materials for this book are exercises that
contain two 3 note groupings which are not built in thirds. First a little background, “two triad pairs” consist of two 3 note groupings that are built in 3rds. These combinations typically use a
major, minor, diminished or augmented triad and when grouped into a collection of two are called a “two triad pair.”
A trichord pair on the other hand takes any of the 9 other possible 3 note combinations and builds pairs using these pitch class sets. These include 012, 013, 014, 015, 016, 024, 025, 026 and 027.
Any three note group can be referred to as a “trichord” but is more commonly used when speaking of a three note grouping not built in thirds. Trichords can also be referred to as non-tertial two
triad pairs.
The exercises found in this course use many different types of harmonic and melodic ideas that can be superimposed over common chord progressions, scales and other musical situations. This course
concentrates on the 026-016 combinations. An example of that would be C, Db, G and D, E, Ab. It works over 7 and 7sus4 chords and extremely useful for an improvising musician looking for some new
fresh sounds. There are also many charts included in this course to show you how these notes function in all 12 keys.
Unique Aspect of These Two Trichord Books
The two trichord books are unique to the Muse Eek Publishing Inc. catalog. The various non-tertial two triad pairs found in this collection are both beautiful and highly applicable to modern
improvisation. They work well as a melodic and harmonic device and Mr. Arnold has written a body of work through both recordings, videos and books dedicated to these non-tertial combinations. As with
all of Mr. Arnold’s books there is a sharp focus on the ear training and the “Harmonic and Melodic Equivalence Two Trichord” Series is no different. Each exercise or chart is always relating back to
the idea of how you would hear these notes within a key center. This book includes a section where the two trichord pair are put into common chord progressions but more importantly shows you how
these progressions relate to the overall key center. Learning music based on how your hear it rather than relating everything to a chord by chord approach is the rosetta stone of music. This is the
secret to the previously undecipherable mystery of understanding music from an aural perspective.
Harmonic and Melodic Equivalence Series
This course is part of the Harmonic and Melodic Equivalence Series which explores over 50 different either trichord pairs or two triad pairs. To see all volumes follow the link above to explore each
volume and hear examples from each course as well as finding links to compositions that I’ve written using each combination.
Harmonic and Melodic Equivalence Exercises
This course is divided up into two sets of exercises written in treble and bass clef. The 1st set of exercises gets gradually harder but also more musical. Depending upon your musical skills you can
start anywhere you want but for beginners I would recommend starting from the 1st exercise of the five. The 2nd set of exercises are called “Atomic Scales.” These exercises are a technical exercise
that really helps you to learn these ideas but also sound great as a melody right off the bat. There are 6 different types of “Atomic Scales” exercises in this course. You don’t have to play every
exercise in every key. But doing this will greatly increase the likelihood of you using it in real music in the future. Below is a listing of the exercises found in this course:
• Closed position studies.
• 1st inversion studies.
• 2nd inversion studies.
• Random combinations of closed position along with 1st and 2nd inversion.
• Random combinations of closed position along with 1st and 2nd inversion with rhythmic displacement.
• Atomic Scales Exercise 1
• Atomic Scales Exercise 2
• Atomic Scales Exercise 3
• Atomic Scales Exercise 4
• Atomic Scales Exercise 5
• Atomic Scales Exercise 6
Explanation of 2nd Set of Exercises in Harmonic and Melodic Equivalence V14H Trichord Pair Course
Below is an explanation for each set of the 6 different atomic scale exercises found in this course. Three octave sequences that move back and forth between the two 3 note groups are presented in six
different configurations. These exercises are highly melodic and can be used verbatim as melodies when soloing. If we thought of the three notes as A,B,C then there would be six different ways to
combine these notes. i.e. ABC, ACB, BAC, BCA, CAB and CBA. All exercises include MP3s as well as midi files so that you can hear and play these exercises at any tempo as well as versions in all 12
• Three octave sequences that move back and forth between the two 3 note groups in the ABC sequence
• Three octave sequences that move back and forth between the two 3 note groups in the ACB sequence
• Three octave sequences that move back and forth between the two 3 note groups in the BAC sequence
• Three octave sequences that move back and forth between the two 3 note groups in the BCA sequence
• Three octave sequences that move back and forth between the two 3 note groups in the CAB sequence
• Three octave sequences that move back and forth between the two 3 note groups in the CBA sequence
1st Set of Exercises in Harmonic and Melodic Equivalence V14H Trichord Pair Course
Here are a few examples from the 1st set of exercises. A complete list of the different types of exercises can also be found below.
Closed Position Exercise
MP3 example
1st Inversion Exercise
MP3 example
2nd Inversion Exercise
MP3 example
Random combinations of closed position along with 1st and 2nd inversion.
MP3 example
Random combinations of closed position along with 1st and 2nd inversion and rhythm permutation
MP3 example
2nd Set of Exercises in Harmonic and Melodic Equivalence V14H Course
Here are a few examples from the 2nd set of exercises.
Atomic Scales 1st Rotation
MP3 example
Atomic Scales 2nd Rotation
MP3 example
Atomic Scales 3rd Rotation
MP3 example
Atomic Scales 4th Rotation
MP3 example
Atomic Scales 5th Rotation
MP3 example
Atomic Scales 6th Rotation
MP3 example
TOC in the Harmonic and Melodic Equivalence V14H Trichord Pair Course:
• How to Use This Course
• Harmonic/Melodic Possibilities of 026-016 combination.
• Chord Possibilities of this 026-016 combination
• Two triad pairs rotations Starting on every eighth note
• How to think of these an 026-016 combinations that are used in this course
• The 026-016 combination Daily Exercise-Atomic Scales
• Thinking of these two triad pairs as applications to Modes
• Thinking of these two triad pairs as one scale
• C, Db, G and D, E, Ab as one scale in all keys
• Forming chord progressions with two triad pairs
• Forming extended chord progressions with two triad pairs
• Choosing chord progressions from two triad pairs
• Additional practice ideas
Get Harmonic and Melodic Equivalence V14H Trichord Pair Today!
Status: In stock, Digital book is available for immediate access.
Additional Information for Harmonic and Melodic Equivalence V14H Trichord Pair:
• Digital Edition 978-1-59489-275-2
• One 14 page PDF explaining exercises, 5 different types of exercises, 328 pages of exercises in PDF format in treble and bass clef.
• Information and examples of forming extended chord progressions with this two triad pair.
• MP3’s and Midi files for all exercises.
• 12 MP3s from Tuba MetroDrone®
What people are saying:
Thanks for including the information on how to divide up the trichords rhythmically to form modern melodies. I remember you did this in your My Music book way back when. But this really reminded me
of the importance that rhythm plays in uses these ideas in music. J. Ogland
Hey Bruce, just wanted to let you know how much I enjoyed the 026-016 series. I can really see know how these two trichords can give you a whole universe of sound that is very useful on many chord
types and scales. I now feeling much more conmfortable with the shapes and ready to dig into some more ideas so keep’em coming! J. Deiter
In my opinion this is one of the more important books in the Harmonic and Melodic Equivalence Series. Using two triad pairs or two trichord pairs gives you a cool sound if you just move back and
forth in groups of 3 notes. I’ve found this information all over the internet. What I haven’t found is the rhythm principles that Mr. Arnold discusses in this book. If there is one book to get in
this series this is it. If you miss this whole rhythm concept you are really missing out on the beauty of this whole concept in my humble opinion! A. Edgars
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LTL Equivalence (Before relation)
+ General Questions (11)
I have a problem regarding the following exercise:
I'm trying to find 2 formulas which satisfy S1 but not S2.
Therefore, the formulas must imply ![b WB a] and !c.
I tried G(a & !b & !c) and G(!b & !c) & X(a).
In both formulas, [b WB a] is false since when a is true, b wasn't true before. Also, c is always false. Therefore those formulas should satisfy S1 but not S2. However, the online tool does not
accept this as a correct solution. Where am I wrong?
I checked your two formulas and the second one is not correct. We can prove that
(G(a & !b & !c)) -> [[b WB a] WB c] & ![[b WB a] SB c]
is valid, but (G(!b & !c) & X(a)) -> [[b WB a] WB c] & ![[b WB a] SB c] is not valid.
To find your formulas, I suggest to first disprove [[b WB a] WB c] -> [[b WB a] SB c] so that the tool gives you a counterexample that satisfies [[b WB a] WB c] & ![[b WB a] SB c]. When I do that, I
obtain a single path that is described by your first formula G(a & !b & !c).
To find the second formula, try a proof for !(G(a & !b & !c)) & [[b WB a] WB c] -> [[b WB a] SB c]. Since that is not valid, a counterexample will satisfy !(G(a & !b & !c)) & [[b WB a] WB c] & ![[b
WB a] SB c] and will give you a second formula.
Ok I think I understand why formula 2 is not valid. S2 might be true for G(!b & !c) & X(a).
We evaluate [b WB a] not just at the start, but for each "step" of the automata. And once a occured (since it just has to occur at least in the next state due to X(a)) [b WB a] might evaluate to true
if a never appears again. And then [[b WB a] SB c] would be true.
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natural numbers in SEAR
Natural numbers in SEAR
This page is a spin-off of the structural set theory described at SEAR. The aim is to define (individual) natural numbers in (a fragment of) SEAR without assuming the axiom of infinity. As a result,
there should be a set with $n$ elements for all $n$, but we will not have a set of all natural numbers unless we assume the axiom of infinity.
The aim is to use as few axioms/results as possible, in particular, not the result that the category of SEAR sets is a topos. What we define in the first instance is along the lines of natural
numbers a la von Neumann.
Zero and one
Barring opinions whether zero should be a natural number, in SEAR we have sets $\empty$ and $\mathbf{1}$ with zero and one element respectively. These follow from axioms 0, 1 and 2 at SEAR.
Two and above
To define $\mathbf{2}$ we consider $P\mathbf{1}$. In a classical setting, this would be a two-element set, but not so for intuitionistic SEAR. We know that $\mathbf{1}$ has at least two subsets,
namely $\empty$ and $\mathbf{1}$, so we let $\mathbf{2}$ be the subset of $P\mathbf{1}$ consisting of the corresponding elements. Continuing to higher numbers, we know that $\mathbf{n}$ has $n$
elements, so $n$ one-element subsets. Together with another 'obvious' subset, this gives a set with $n+1$ elements.
More formally, let $\phi_2:\mathbf{1} \looparrowright P\mathbf{1}$ be the relation such that $\phi_2(*,empty)$ and $\phi_2(*,\mathbf{1})$. Then a tabulation $|\phi_2|$ has two elements. Let us fix
one of these and call it $\mathbf{2}$.
Now assume we have defined a set $\mathbf{n}$ with $n$ elements, $1,\ldots,n$, where $n \geq 2$. From axiom 3 we have a power set $P\mathbf{n}$. Let $\phi_{n+1}:\mathbf{1} \looparrowright P\mathbf{n}
$ be a relation such that $\phi_{n+1}(*,u)$ whenever the subset $\{ i | \epsilon(i,u)\}$ of $\mathbf{n}$ has either exactly one element or is equal to all of $\mathbf{n}$.
Mike Shulman: I think it’s fine, though AN might object. A more formal thing to say would be that $\phi_{n+1}(*,u)$ whenever the subset $\{ i | \epsilon(i,u)\}$ of $\mathbf{n}$ has either exactly one
element or is equal to all of $\mathbf{n}$.
However, I don’t think your definition works when $n=1$, since in that case, $\underline{1} = \mathbf{1}$! Perhaps instead of $\phi(*,\mathbf{n})$ you want $\phi(*,\emptyset)$?
David Roberts: I’ve fixed the problem with $\mathbf{2}$, using Toby’s original suggestion, incorporated the more formal suggestion you made for $\phi_{n+1}$ and added a new definition below.
Then a tabulation of $\phi_{n+1}$ will have $n+1$ elements and we fix one of these as $\mathbf{n+1}$.
The construction on $\mathbf{1},\mathbf{2},\ldots$ only requires a fragment of SEAR, namely axioms 0,1,2 and 3, and even holds in the analogous fragment of bounded SEAR.
Alternative definition
To avoid having to treat $\mathbf{2}$ as a special case, we can use another definition, again starting from $\mathbf{0},\mathbf{1}$ as before.
Assume we have defined $\mathbf{n}$ for $n \geq 1$. Then let $\psi_{n+1}:\mathbf{1} \looparrowright P\mathbf{n}$ be the relation such that $\psi_{n+1}(*,u)$ whenever the subset $\{ i | \epsilon(i,u)
\}$ of $\mathbf{n}$ has either exactly one element or no elements. A tabulation $|\psi_{n+1}|$ has $n+1$ elements, and fixing one of these we denote it by $\mathbf{n+1}$.
This definition holds in the same fragment of (bounded) SEAR as described above.
David Roberts: Does this remark (about collection/power sets) belong here or in the next section? Certainly, I haven’t gotten rid of power sets and I hope I haven’t included collection.
Mike Shulman: I think it belongs in the next section.
Toby: Sorry, I put my remarks in the wrong place! It was late …
Replacing the power set axiom by something else
As suggested by Toby, one could take as an axiom the existence of $\mathbf{2}$, together with axioms 0,1,2 and 5 (collection) of SEAR, instead of powersets (axiom 3). From Collection and $\mathbf{2}$
, we get binary coproducts, so we could define $\mathbf{n}$ as $\mathbf{1}\coprod(\mathbf{1} \coprod ( \ldots\coprod \mathbf{1})\ldots)$ ($n$ times). (DR: this needs spelling out better, with
tabulations, but that’s the general idea).
This definition also holds in the bounded fragment of SEAR.
With the axiom of infinity
based on this blog post and its predecessors
To construe elements $n \in \mathbb{N}$ as giving actual finite sets, we construct a “family” $\phi\colon F \to \mathbb{N}$ where each fiber $F_n$ is a set of cardinality $n$. For example, consider
the function
$\phi\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}\colon (m,n) \mapsto m + n + 1$
Then, for each $n \geq 0$, the fiber $\phi^{-1}(n)$ is a set of cardinality $n$.
If you use the Axiom of Collection, then it will do this for you automatically; you don't need to come up with an ad hoc representation of the family.
Mike Shulman: If you don’t want to use powersets, then I’m pretty sure you’ll need an axiom of coproducts. Unless I’m mistaken, the collection of subsingleton sets satisfies Axioms 0, 1, and 2.
David Roberts: Actually what I probably mean is not use the result that $Set$ is a topos. I think you are correct on the subsingleton front, because it looks like we have no way of obtaining sets
with more than one element without more axioms.
Toby: If you want to be cute, you can make the existence of $\mathbf{2}$ an axiom and then get arbitrary binary coproducts using Collection. On the other hand, if you're happy using power sets, then
don't bother proving that $P\mathbf{1} = \mathbf{2}$ (which won't generalise to the intuitionistic case anyway); instead use Separation to carve out $\{ x \in P\mathbf{1} \;|\; x = \empty \;\vee\; x
= \mathbf{1} \}$. Then $\mathbf{3}$ is a subset of $P\mathbf{2}$, etc; or make $\mathbf{3}$ and $\mathbf{4}$ both subsets of $P\mathbf{2}$, with the general rule going logarithmically.
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NCERT Class 12th Physics - Electric Charges and Field | a point charge 10 mu c is a distance 5 cm direct Answer (02 Nov) | SaralStudy
This page offers a step-by-step solution to the specific question NCERT Class 12th Physics - Electric Charges and Field | a point charge 10 mu c is a distance 5 cm direct Answer from NCERT Class 12th
Physics, Chapter Electric Charges and Field.
Question 18
A point charge +10 μC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in Fig. 1.34. What is the magnitude of the electric flux through the square? (Hint: Think of the
square as one face of a cube with edge 10 cm.)
The square can be considered as one face of a cube of edge 10 cm with a centre where charge q is placed. According to Gauss’s theorem for a cube, total electric flux is through all its six faces.
Hence, electric flux through one face of the cube i.e., through the square,
Where, ∈0 = Permittivity of free space
= 8.854 × 10^−12 N^−1C^2 m^−2 q = 10 μC = 10 × 10^−6 C
= 1.88 × 10^5 N m^2 C^−1
Therefore, electric flux through the square is 1.88 × 10^5 N m^2 C^−1.
10 Comment(s) on this Question
2021-12-13 01:11:42
Nice ð
2020-07-30 20:28:32
2020-07-30 20:27:56
2019-11-24 15:34:39
Thnq ð
2019-06-18 22:30:02
Can we solve this without using Gauss law?? ( By using EdScos0 alone)
Arvendra dhakar
2018-12-25 07:29:22
Thanks a lot
Shivangi srivastava
2018-07-01 12:04:54
2017-06-16 13:41:37
Two charges q1= 10uc and q2= -12uc are within a spherical surface of radius 10cm. What is the total flux through the surface ?
2017-04-22 13:09:50
2017-04-16 11:13:19
Thanks a lot
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Matrix factorization
From Encyclopedia of Mathematics
2020 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]
factorization of matrices
Factorizations of matrices over a field are useful in quite a number of problems, both analytical and numerical; for example, in the (numerical) solution of linear equations and eigenvalue problems.
A few well-known factorizations are listed below.
Let $A$ be an $(m\times n)$-matrix with $m\ge n$ over $\C$. Then there exist a unitary $(m\times m)$-matrix $Q$ and a right-triangular $m\times n$-matrix $R$ such that $A=QR$. Here, a
right-triangular $(m\times n)$-matrix, $m\ge n$, is of the form
$$\begin{pmatrix} r_{11} & r_{12} & \cdots & r_{1n} \\ 0 & r_{22} & \cdots & r_{2n} \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & r_{nn} \\ 0 & 0 & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ 0 &
0 & \cdots & 0 \\ \end{pmatrix}.$$ A real (respectively, complex) non-singular matrix $A$ has a factorization $QR$ with $Q$ orthogonal (respectively, unitary) and $R$ having all elements positive.
Such a factorization is unique and given by the Gram–Schmidt orthogonalization process (cf. Orthogonalization method). The frequently used $QR$-algorithm for eigenvalue problems (cf. Iteration
methods) is based on repeated $QR$-factorization.
Singular value factorization.
Let $A$ be an $(m\times n)$-matrix over $\C$ of rank $k$. Then it can be written as $A=U\def\S{ {\Sigma}}\S V$, with $U$ a unitary $(m\times n)$-matrix, $V$ a unitary $(n\times n)$-matrix and $\S$ of
the form
$$\S = \begin{pmatrix} \def\cD{ {\mathcal D}} \cD & 0\\ 0 & 0\end{pmatrix},$$ where $\cD$ is diagonal with as entries the singular values $s_1,\dots,s_k$ of $A$, i.e. the positive square roots of the
eigenvalues of $AA^*$ (equivalently, of $A^* A$).
An $(n\times n)$-matrix $A$ (over a field) such that the leading principal minors are non-zero,
$$\det\begin{pmatrix} a_{11} & \cdots & a_{1i}\\ \vdots & & \vdots\\ a_{i1} & \cdots & a_{ii} \end{pmatrix} \ne 0,\quad i=1,\dots,n,$$ can be written as a product $A=LU$ with $L$ a lower-triangular
matrix and $U$ an upper-triangular matrix. This is also known as triangular factorization. This factorization is unique if the diagonal elements of $L$ (respectively, $U$) are specified (e.g., all
equal to $1$); see, e.g., [YoGr], p. 821. Conversely, if $A$ is invertible and $A=LU$, then all leading principal minors are non-zero.
In general, permutations of rows (or columns) are needed to obtain a triangular factorization. For any $(m\times n)$-matrix there are a permutation matrix $P$, a lower-triangular matrix $L$ with unit
diagonal and an $(m\times n)$ echelon matrix $U$ such that $PA=LU$. Here, an echelon matrix can be described as follows:
i) the non-zero rows come first (the first non-zero entry in a row is sometimes called a pivot);
ii) below each pivot is a column of zeros;
iii) each pivot lies to the right of the pivot in the row above. For example,
$$\def\bu{\bullet}\begin{pmatrix} 0 & \bu & * & * & * & * & * & * & * & * \\ 0 & 0 &\bu& * & * & * & * & * & * & * \\ 0 & 0 & 0 & 0 & 0 &\bu& * & * & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\bu\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}$$ where the pivots are denoted by $\bu$. $LU$-factorization is tightly connected with Gaussian elimination, see Gauss method and [St].
Iwasawa decomposition.
The $QR$-factorization for real non-singular matrices immediately leads to the Iwasawa factorization $A=QDR$ with $Q$ orthogonal, $D$ diagonal and $R$ an upper (or lower) triangular matrix with $1$s
on the diagonal, giving an Iwasawa decomposition for any non-compact real semi-simple Lie group.
Choleski factorization.
For each Hermitean positive-definite matrix $A$ over $\C$ (i.e., $A = A^*$, $x^* Ax > 0$ for all $0\ne x\in\C^n$) there is a unique lower-triangular matrix $L$ with positive diagonal entries such
that $A=LL^*$. If $A$ is real, so is $L$. See, e.g., [StBu], p. 180ff. This $L$ is called a Choleski factor. An incomplete Choleski factorization of a real positive-definite symmetric $A$ is a
factorization of $A$ as $A=LDL^T$ with $D$ a positive-definite diagonal matrix and $L$ lower-triangular.
Decomposition of matrices.
Instead of "factorization" , the word "decomposition" is also used: Choleski decomposition, $LU$-decomposition, $QR$-decomposition, triangular decomposition.
However, decomposition of matrices can also mean, e.g., block decomposition in block-triangular form:
$$A=\begin{pmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{pmatrix}$$ and a decomposable matrix is generally understood to mean a matrix that, via a similarity transformation, can be brought to the form
$$SAS^{-1}=\begin{pmatrix} A_{1} & 0 \\ 0 & A_{2} \end{pmatrix}.$$ Still other notions of decomposable matrix exist, cf., e.g., [MaMi].
Matrices over function fields.
For matrices over function fields there are (in addition) other types of factorizations that are important. E.g., let $W$ be an $(m\times m)$-matrix with coefficients in the field of rational
functions $\def\l{ {\lambda}}\C(\l)$ and without poles in $\R$ or at $\infty$. Assume also that $\det W(\l)\ne 0$ for $\l\in\R\cup\{\infty\}$. Then there are rational function matrices $W_+$ and
$W_-$, also without poles in $\R$ or at $\infty$, and integers $k_1\le \cdots \le k_m$ such that
$$W(\l) = W_-(\l)\begin{pmatrix} \Big(\frac{\l-i}{\l+i}\Big)^{k_1} & & \\ &\ddots& \\ & & \Big(\frac{\l-i}{\l+i}\Big)^{k_m} \end{pmatrix}W_+(\l)$$ and
a) $W_+$ has no poles in $\def\Im{ {\rm Im}\;}\Im \l \ge 0$ and $\det W_+(\l)\ne 0$ for $\Im \l \ge 0$;
b) $W_-$ has no poles in $\Im \l \le 0$ and $\det W_-(\l)\ne 0$ for $\Im \l \le 0$;
c) $\det W_+(\infty)\ne 0$; $\det W_-(\infty)\ne 0$. This is called a Wiener–Hopf factorization; more precisely, a right Wiener–Hopf factorization with respect to the real line. There are also left
Wiener–Hopf factorizations (with $W_+$ and $W_-$ interchanged) and Wiener–Hopf factorizations with respect to the circle (or any other contour in $\C$).
In the scalar case, $m=1$, the factors $W_-(\l)$ and $W_+(\l)$ are unique. This is no longer the case for $m\ge 2$ (however, the indices $k_1,\dots,k_m $ are still unique). See also Integral equation
of convolution type.
If all indices in the decomposition are zero, one speaks of a right canonical factorization. For more, and also about spectral factorization and minimal factorization, and applications, see [BaGoKa],
[ClGo], [GoGoKa].
Matrix polynomials.
The factorization of matrix polynomials, i.e., the study of the division structure of the ring of $(m\times m)$-matrices with polynomial entries, is a quite different matter. See [Ma], [Ro] for
results in this direction.
[BaGoKa] H. Bart, I. Gohberg, M.A. Kaashoek, "Minimal factorization of matrix and operator functions", Birkhäuser (1979) MR0560504 Zbl 0424.47001
[ClGo] K. Clancey, I. Gohberg, "Factorization of matrix functions and singular integral operators", Birkhäuser (1981) MR0657762 Zbl 0474.47023
[GoGoKa] I. Gohberg, S. Goldberg, M.A. Kaashoek, "Classes of linear operators", I–II, Birkhäuser (1990–1993) MR1130394 MR1246332 Zbl 1065.47001 Zbl 0789.47001
[Ma] A.N. Malyshev, "Matrix equations: Factorization of matrix polynomials" M. Hazewinkel (ed.), Handbook of Algebra, I, Elsevier (1995) pp. 79–116 MR1421799 Zbl 0857.15003
[MaMi] M. Marcus, H. Minc, "A survey of matrix theory and matrix inequalities", Dover (1992) pp. 122ff MR1215484 Zbl 0126.02404
[NoDa] B. Noble, J.W. Daniel, "Applied linear algebra", Prentice-Hall (1969) pp. Sect. 9.4–9.5 MR0572995 Zbl 0203.33201
[Ro] L. Rodman, "Matrix functions" M. Hazewinkel (ed.), Handbook of Algebra, I, Elsevier (1995) pp. 117–154 MR1421800 MR2035093 Zbl 0858.15011 Zbl 1068.15035
[St] G. Strang, "Linear algebra and its applications", Harcourt–Brace–Jovanovich (1976) MR0384823 Zbl 0338.15001
[StBu] J. Stoer, R. Bulirsch, "Introduction to numerical analysis", Springer (1993) MR1295246 Zbl 0771.65002
[YoGr] D.M. Young, R.T. Gregory, "A survey of numerical mathematics", II, Dover (1988) MR1102901 Zbl 0732.65003
How to Cite This Entry:
Singular value decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_value_decomposition&oldid=40289
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Merkle Mountain Range Multi Proofs
Merkle mountain ranges are an improvement over conventional merkle trees for growing, potentially unbounded lists. Where conventional merkle tree constructions over growing lists prove very
inefficient to compute, as all nodes in the tree must be recomputed. Merkle mountain ranges amortise this cost by growing subtrees incrementally and merging subtrees at the same height, rather than
growing the full tree.
Merkle mountain ranges are an improvement over conventional merkle trees for growing, potentially unbounded lists. Where conventional merkle tree constructions over growing lists prove very
inefficient to compute, as all nodes in the tree must be recomputed. Merkle mountain ranges amortise this cost by growing subtrees incrementally and merging subtrees at the same height, rather than
growing the full tree.
You can observe how the tree grows by looking at the order of the nodes.
Merkle mountain ranges, true to it’s name, are a type of merkle tree that is composed of perfectly balanced merkle trees (ie each sub tree has leaves = a power of 2) such that it does indeed look
like a mountain range. This construction provides a few benefits:
• Smaller proof sizes when compared to conventional merkle trees for very large sets of data (especially for more recent leaves in the tree).
• Better insertion complexity, where the insertion complexity of conventional merkle trees is .
First we define the termial function as the sum:
Where can be evaluated through the function:
Let’s recall the properties of merkle trees:
height of the tree
number of leaves in the tree.
For merkle trees, the number of proof nodes for a single item proof is defined as . In order to understand the improvements that mmr’s bring to the table, lets consider the number of leaves present
in both trees to get a maximum proof size of nodes. Note that the number of proof items needed for an item in a merkle tree = height of the merkle tree.
Since an mmr is itself composed of perfectly balanced binary trees with decreasing heights. The total number of leaves in an mmr can be expressed as , where is the total number of leaves in the first
subtree. We can therefore derive the maximum number of leaves in an mmr where the height of the first subtree is using the function:
Whereas for a conventional merkle tree, whose height is the total number of leaves is:
So now, we see how mmrs enable more efficient for merkle proof sizes on much larger data sets than conventional merkle trees.
MMR Multi Proofs
Our approach to verifying mmr multi proofs will be to regard each sub tree as an isolated merkle tree, using the -index model defined in my
previous article
. The -index of each node in an mmr, will be the distance from the left-most node in each subtree.
We can describe each sub tree as a standalone merkle tree.
Given this model, we can re-use the calculate_merkle_multi_root function defined in my previous article to verify mmr multi proofs using the algorithm:
pub fn calculate_mmr_root(
mut leaves: Vec<(u64, usize, Hash)>, // mmr_index, k_index, node_hash
mmr_size: u64,
mut proof_iter: Vec<Hash>,
) -> Hash {
let peaks = get_peaks(mmr_size);
let mut peak_roots = vec![];
for peak in peaks {
let mut leaves: Vec<_> = take_while(&mut leaves, |(pos, _, _)| *pos <= peak);
match leaves.len() {
1 if leaves[0].0 == peak => {
// this is a peak root.
0 => {
// the next proof item is a peak
if let Some(peak) = proof_iter.pop() {
} else {
_ => {
let leaves = leaves
.map(|(_, index, leaf)| {
(index, leaf)
let height = pos_to_height(peak);
let mut current_layer: Vec<_> = leaves.iter().map(|(i, _)| *i).collect();
let mut sub_tree: Vec<Vec<_>> = vec![];
for i in 0..height {
let siblings = sibling_indices(current_layer.clone());
let diff = difference(&siblings, ¤t_layer);
if diff.len() == 0 {
// fill the remaining layers
sub_tree.extend((i..height).map(|_| vec![]));
let len = diff.len();
let layer = diff.into_iter().zip(proof_iter.drain(..len)).collect();
current_layer = parent_indices(siblings);
// insert the leaves at the base layer.
sub_tree[0].sort_by(|a, b| a.0.cmp(&b.0));
let peak_root = calculate_merkle_multi_root(sub_tree);
// bagging peaks from right to left via hash(right, left).
while peak_roots.len() > 1 {
let right_peak = peak_roots.pop().unwrap();
let left_peak = peak_roots.pop().unwrap();
let mut buf = vec![];
P. Todd. Merkle Mountain Ranges. GitHub, 2012.
Donald Knuth in The Art of Computer Programming.
(Third edition, Volume 1, page 48).
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Am I Reading This Right?
04 Dec 2013
Dang, I think the chickens are coming home to roost from all the times I called Scott Sumner “insane.” In an update to his post on Williamson Scott writes:
Update: Kudos to Bob Murphy for getting there before I did. I wonder if the quickness someone gets to the problem with Williamson’s argument is inversely related to the technical skills that
person possesses.
Taken literally, that’s a super backhanded compliment. But Scott usually makes self-deprecating jokes about how he can’t do math. So did he really mean “I wonder if the time it takes someone to get
to the problem is inversely related…” ?
13 Responses to “Am I Reading This Right?”
1. Scott Sumner
Not sure I follow that. I just meant that I’m pretty non-technical, and you are even more so (I assume).
□ Bob Murphy says:
I actually am feigning outrage, Scott. But strictly speaking, it sounds like you’re saying I was quicker than you, because you have more technical prowess.
□ Ken B says:
“you don’t have to be crazy to understand this but it helps.”
□ Bob Murphy says:
Oh so you DID word it correctly.
I don’t think there’s any way I come out ahead on this. If I break out my technical street cred I look like a jerk, and if I just ignore it I’m seen to be admitting I’m just a verbal hack.
2. Mule Rider
I view Scott Sumner as being on intellectual par with Mike Norman. That sums up just about how seriously I take him.
3. Peter
Just curious (I am an engineer): In the field of economics, what would be considered good “technical skills”?
I assume it is similar to “fundamentals” versus “technicals” (cup and handle nonsense and all that) in stock trading, i.e. “keynes” (technicals, C + I + G + X – M = Y and all that) versus “mises”
(fundamentals) ?
And no worries, not looking for a career change…
□ Ken B says:
Judging from this result, the ability to parse Paul Krugman correctly.
☆ Matt Tanous
There is no ‘correct’ parsing of Krugman. There is only whatever parsing works for the point you are making. Krugman says too many contradictory things for there to be anything else.
■ Matt M (Dude Where's My Freedom) says:
Krugman is the modern day Rorschach test for economics and political pundits. Everyone looks into the same random mess and what they claim to decipher tells you something
meaningful about the reader, but not about Krugman himself…
□ Martin
Mankiw, did a post on what sort of mathematics – I presume this is what you mean by technical? – you need for economics these days and a typical undergraduate aiming to be placed at a good
school will need to take
Linear Algebra
Multivariable Calculus
Real Analysis
Probability Theory
Mathematical Statistics
Game Theory
Differential Equations
At a fancy graduate school you will then learn some more and will learn to apply it to some problems. I am not very fancy – and not even doing purely econ – so the fanciest thing I have done
is setting up a problem as an optimal control problem.
4. Cody S
Whatever it is you think about what Krugman said, you’re a moron.
Krugman is the only non-moron, because he clearly doesn’t think about what he says.
Mostly, he just thinks about how tragic it is that other people are allowed to earn money, or own things.
Also, (pure conjecture) probably beards.
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Java Math.min() method
How to find min() of two numbers in Java?
Java provides a system library known as “
” for extensive handy operations. From trigonometry to logarithmic functions, you can find min/max or even absolute of a number using the methods provided by this library due to its diverse
Math.min() Method
Here’s a regular representation of the method.
Math.min(a, b)
Kindly note that this function accepts two parameters of same types
. Let’s look at an executable example of the
method to understand an effective use of it. Make sure you run the script in your IDE to validate the outputs.
Example 1
package com.math.min.core
public class MathMinMethod {
public static void main(String[] args) {
int leenasAge = 10;
int tiffanysAge = 15;
// example of min () function
int min = Math.min(leenasAge, tiffanysAge);
System.out.print("Who's the younger sister? ");
if (min < tiffanysAge)
System.out.print("Leena ------- Age " + leenasAge);
System.out.print("Tiffany ------- Age " + tiffanysAge);
Who's the younger sister? Leena ------- Age 10
At line 8,
int min = Math.min(leenasAge, tiffanysAge);
int min stores the minimum number returned by the
function. Later we use that result to find the age of the smaller sibling.
Example 2
package com.math.min.core;
public class MathMinMethod {
public static void main(String[] args) {
double berriesSoldInKg = 15.6;
double cherriesSoldInKg = 21.3;
// example of min () function
double min = Math.min(berriesSoldInKg, cherriesSoldInKg);
System.out.println("What's the minimum weight sold?");
if (min != cherriesSoldInKg)
System.out.print("Berries: " + berriesSoldInKg + " kg");
System.out.print("Cherries: " + cherriesSoldInKg +"kg");
What's the minimum weight sold? Berries: 15.6 kg
At line 8,
double min = Math.min(berriesSoldInKg, cherriesSoldInKg);
the double “min” stores the lowest of both weights. Later, we compare two doubles (amount in kgs) to check the minimum of the two fruits. That result can be used according to your requirements in any
By now you’d be able to understand the need and efficiency of
method. However, in case of any query or confusion feel free to consult this article again. Keep growing and practising!
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Economic model predictive control without terminal constraints : optimal periodic operation
Title data
Müller, Matthias A. ; Grüne, Lars:
Economic model predictive control without terminal constraints : optimal periodic operation.
In: Proceedings of the 54th IEEE Conference on Decision and Control. - Piscataway, NJ : IEEE , 2015 . - pp. 4946-4951
DOI: https://doi.org/10.1109/CDC.2015.7402992
This is the latest version of this item.
Project information
Project title: Project's official title
Project's id
DFG-Project "Performance Analysis for Distributed and Multiobjective Model Predictive Control"
Cluster of Excellence in Simulation Technology (SimTech)
EXC 310/1
Project financing: Deutsche Forschungsgemeinschaft
Abstract in another language
In this paper, we analyze economic model predictive control schemes without terminal constraints, where the optimal operating regime is not steady-state operation, but periodic behavior. We first
show by means of two counterexamples, that a classical receding horizon control scheme does not necessarily result in an optimal closed-loop performance. Instead, a multi-step MPC scheme may be
needed in order to establish near optimal performance of the closed-loop system. This behavior is analyzed in detail, and we derive checkable dissipativity-like conditions in order to obtain
closed-loop performance guarantees.
Further data
Available Versions of this Item
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Using the Pythagorean Theorem to Determine If a Triangle is a Right Triangle
Question Video: Using the Pythagorean Theorem to Determine If a Triangle is a Right Triangle Mathematics • Third Year of Preparatory School
A triangle has vertices of the points ๐ ด(4, 1), ๐ ต(6, 2), and ๐ ถ(2, 5). Work out the lengths of the sides of the triangle. Give your answers as surds in their simplest form. Is this triangle a
right triangle?
Video Transcript
A triangle has vertices of the points ๐ ด four, one; ๐ ต six, two; and ๐ ถ two, five. Work out the lengths of the sides of the triangle. Give your answers as surds in their simplest form. And
secondly, is this triangle a right triangle?
Letโ s begin by sketching this triangle on a coordinate grid. We absolutely donโ t need to plot this triangle accurately. We arenโ t going to be measuring the lengths of any of the lines. We just
want to sketch it using the approximate position of these three points relative to one another.
So the triangle looks a little something like this. Now, from our sketch, it looks possible that this could be a right triangle with the right angle at ๐ ด. But we canโ t confirm this from our
sketch. Letโ s consider the first part of the question. We need to find the lengths of the three sides of the triangle. And weโ ll begin by finding the length of the side ๐ ด๐ ต.
We can sketch in a right triangle below this line using ๐ ด๐ ต as its hypotenuse. We can also work out the lengths of the other two sides in this triangle. The horizontal side will be the
difference between the ๐ ฅ-values at its endpoints. Thatโ s the difference between six and four, which is two. And the vertical side will be the difference between the ๐ ฆ-values at its endpoints.
Thatโ s the difference between two and one, which is one.
As we now have the lengths of two sides in a right triangle and we wish to calculate the length of the third side, we can apply the Pythagorean theorem, which tells us that, in a right triangle, the
sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Remember, ๐ ด๐ ต is the hypotenuse. So we have that ๐ ด๐ ต squared is equal to one squared plus two squared.
One squared is one and two squared is four. So adding these values together, we have that ๐ ด๐ ต squared is equal to five.
To find the length of ๐ ด๐ ต, we need to square root each side of this equation. And remember at this point, weโ ve been told to give our answer as a surd. So we have that ๐ ด๐ ต is equal to
root five. We can find the lengths of the other two sides of the triangle in the same way. We sketch in a right triangle below the line ๐ ต๐ ถ. And we see that it has a horizontal side of four
units and a vertical side of three units.
๐ ต๐ ถ is the hypotenuse of this triangle. So applying the Pythagorean theorem, we have that ๐ ต๐ ถ squared is equal to three squared plus four squared. Thatโ s nine plus 16, which is equal to
25. ๐ ต๐ ถ is therefore equal to the square root of 25, which is simply the integer five. In the same way, ๐ ด๐ ถ is the hypotenuse of a right triangle with shorter sides of two and four units.
So ๐ ด๐ ถ is equal to the square root of 20, which simplifies to two root five.
So weโ ve answered the first part of the question. And now we need to determine whether this triangle is a right triangle. Well, if it is, then the Pythagorean theorem will hold for its three side
lengths. Now we suspect it that the right angle was at ๐ ด, which would make ๐ ต๐ ถ the hypotenuse of the triangle if it is indeed a right triangle.
We therefore want to know whether ๐ ต๐ ถ squared is equal to ๐ ด๐ ต squared plus ๐ ด๐ ถ squared. Well, we can in fact use the squared side lengths. We know that ๐ ต๐ ถ squared is 25. We know
that ๐ ด๐ ต squared is five. And we know that ๐ ด๐ ถ squared is 20. So is it true that 25 is equal to five plus 20? Yes, of course, itโ s true, which means that the Pythagorean theorem holds for
this triangle. And therefore, it is indeed a right triangle. So weโ ve completed the problem. We have the three side lengths. ๐ ด๐ ต equals root five, ๐ ต๐ ถ equals five, and ๐ ด๐ ถ equals two
root five. And weโ ve determined that the triangle is a right triangle.
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How To Calculate Antilog Easily - Science Trends
If you’re wondering what an antilog is and how to calculate it, you actually probably already know a little about antilogs. Antilogs are just another term for exponents. You can think of an antilog
as being the inverse of a logarithm, and to calculate the antilog, you’ll just invert the regular logarithm you have.
Let’s take a closer look at antilogs and how they relate to exponents and logarithms.
Since it is difficult to discuss something without a common definition of the terms involved, let’s start out by defining all of our terms.
Definition of Exponents
Exponents are just shorthand/stand for the repeated multiplication of a variable by itself. When we speak of something being to the “Nth power” (N standing in for any number), we mean that number/
variable is being multiplied by itself that many times. For instance, 4^4 means 4 to the fourth power, or 4 x 4 x 4 x 4. Remember that the term “raising to a power” just means multiplying a given
variable by itself to the specified number of times.
A Note On Squaring and Cubing
There are specific terms for exponents that multiply something by itself once and those that multiply something by itself twice. Multiplying a value by itself is known as “squaring” something,
represented as Y^2 or Y to the second power. Multiplying something by itself twice (Y x Y x Y) is known as cubing something, or raising it to the third power.
Inverse Functions
An inverse function means taking a function (such as an exponent) and doing the inverse of it. So if the given function A take X as an input and gives an output of Y, the function A^-1 (the negative
denoting the inverse), would take Y as an input and give X as an output.
The Relationship Between Exponents And Logarithms
Remember that exponents are just the inverse function of a logarithm, and logarithms the inverse function of exponents. Knowing this, you might guess that Inverse Log or AntiLog is just another term
for exponents. Logarithms, therefore, are just another way to conceive of exponents. If you know that 8 to the second power, or eight squared equals 64, you can represent that as: 8^2 = 64.
Yet as a logarithm, the variable you are trying to find is the exponent, so you are attempting to find the exponent as the missing value. Here’s an example question: 8 raised to some power (X) equals
64. You can express this question with the logarithmic equation: log8(64) = 2. So to make this comparison explicit: log8(64) = 2 is just the inverse of 8^2 = 64. Logarithms and exponents work with
the same basic variables, the primary difference is that logarithms isolate the exponent while exponential equations isolate the power. Knowing this, it’s easy to see why a logarithm’s base will be
exactly the same as an exponent’s base.
Now that we have gone over all the definitions, let’s look at an example of converting logs to antilogs(exponents). If log(10)100 = 2, then the conversion is quite easy, just swap the position of the
middle and end values to get antilog. So the antilog is: Antilog(10) 2 = 100. Now to express this as an exponent formula, just make the 2 an exponent: 10^2 = 100.
Examples Of Converting Between Logs And Antilogs
Let’s take a look at some examples of converting between antilogs and logs.
If given the logarithmic expression log2(⅛) = -3: The equivalent exponential form is 1 / 8 = 2^-3
If given the logarithmic expresssion log3(27) = 3: The equivalent exponential form is 27 = 3^3
If given the logarithmic expression log36(6) = ½: The equivalent exponential form is 6 = 36^1/2
If given the logarithmic expression log8(2) = ⅓: The equivalent exponential form is 1 / 8 = 2^-3
The Use Of Antilogs
When calculations were still done with slide rules, instead of calculators, it was more common to see the term antilog used. Most people nowadays will use the term exponent instead of antilog,
largely because of the fact that computers now perform most calculations for us. The term antilog is primarily used in the field of electronics, in reference to electrical circuits and the operators/
amplifiers that affect the flow/output of a circuit.
An amplifier is an electrical device that is capable of altering/increasing the power, current, or voltage of a signal. Amplifiers increase the magnitude of a signal. Different sizes and types of
amplifiers are used to carry out different tasks. Smaller, weaker amplifiers are usually used in wireless receivers and to amplify sound in CD players and headsets. Power amplifiers are often used in
broadcasting transmitters, wireless transmitters, and hi-fi audio devices.
Logarithmic amplifiers are nodes found in a circuit where the output to a terminal needs to be altered. A logarithmic electrical amplifier will produce an output that is scaled in proportion to the
input log of the signal. The input terminal takes in the signal, the amplifier alters it and then the signal is sent out to the next terminal.
As you might guess, an antilog amplifier operates very similarly to a logarithmic amplifier, just inverting the signal. An antilog amplifier would take the output signal and run it, with negative
feedback, to the inverting terminal.
To Sum Up:
Antilogs are just another term for exponents, and exponents are the inverse of logarithms. When you want to convert between logarithms and antilogs, swap the position of the middle and end values to
get antilog. So now you know that antilogs are the same thing as exponents, and that exponents are just the inverse of logarithmic functions. As long as you know how to convert between exponents and
logarithms, you’ll be fine.
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Big O notation
Today I will discuss an important topic that is useful to know for your program but also more importantly it is useful to know for your job interview. According to Wikipedia big-o notation in
computer science is used to classify algorithms according to how their run time or space requirements grow as the input size grows. It is essentially how you will talk about and explain your
algorithms and how efficient they are as they grow with time “time complexity”. For example how much an algorithm will slow down if you gave it a list of 10 versus a list of 10000. Below I will talk
about the common types of big-o notation and exampls.
Constant time “O(1)”
This means that the algorithm’s run time will be constant no matter the size of inputs. A good example would be setting bookmarks in your browser, this is because no matter the size of the page that
you are going to the bookmark will take you directly to that page in a single step. Other great examples of constant time big-o are math operations, accessing a hash using a key, push/pop from a
stack, returning a value from a function. As you will see in the function below it is a big-0 of O(1) complexity because no matter how many numbers are in the list you are always just returning the
number at first index.
Linear Time “O(n)”
This means the run time will increase as input increases, you can think about this like working out. Let us say the time to do one rep of an exercise takes 15 seconds, but as you, if you were to do 3
sets then it would take you 45 seconds. One of the most common operations you will see in programming is transversing an array, this would be methods like forEach, map, and any other method that
loops through an array from start to finish. As the example below.
Quadratic Time “O(n²)”
This means that the size of the input data is squared, some examples of algorithms that have quadratic time are bubble sort, insertion sort, and selection sort. Below we will look at an example.
As we look at this example, we see in the function we have two nested loops that increment after each iteration. If we call the function on smallList our n will be 16 operations. We can see how
quickly that can add up so imagine what n will be on the function bigList or something even bigger. An array with only one thousand elements ends up creating one million operations, so that gives you
an idea.
Logarithmic Time”O(log n)”
This means that as the input size increase the running time grows along with it in proportion. This makes run time barely increase as you increase input. You can think of this as taking a book full
of names in alaphabetical order and looking for the name Samuel, so you take the book and open it half way and you are at Mary. You say to yourself that S is after M so now you can go from half of
that and so on and so on till you find the name Samuel. You can see how this is much quicker than going from the start of the book page by page till you get to the name Samuel or what ever other name
you are looking for. The most used examle of logarithmic time is a binary search operation, we also see it in merge sort, time sort, and heap sort. Below we will see an example.
Just to give you an idea of how fast logarathmic big-o is if we have a n size of one million we will only return 2 operations thats a big difference from the quadratic big-o.
So at the end of the day you see how important big-o can be, do you nessicaraly need it ot be able to code your project at first no. But if you want to refactor or make it scalable and design
someting great from scratch you can see that knowing the big-o can be a very useful tool and help make your app run quicker and smoother.
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English National Curriculum, Programme Of Study For Key Stage 3 Mathematics
\( \DeclareMathOperator{cosec}{cosec} \)
English National Curriculum, Programme Of Study For Key Stage 3 Mathematics
[ << Main Page ]
Subject Content:
Pupils should be taught to recognise geometric sequences and appreciate other sequences that arise.
Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.
Here are some exam-style questions on this statement:
See all these questions
Here are some Advanced Starters on this statement:
Click on a topic below for suggested lesson Starters, resources and activities from Transum.
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Saturation Pressure
For a mixture at given temperature, the pressure at which a new infinitesimal (“incipient”) phase appears upon slight change in pressure. The mixture and its incipient phase are in thermodynamic
equilibrium. If the incipient phase is lighter than the mixture phase, the saturation pressure is a bubblepoint and the incipient phase is a bubble appearing from the oil mixture. If the incipient
phase is heavier than the mixture phase, the saturation pressure is a dewpoint and the incipient phase is a liquid (“dew”) appearing from the gas mixture. Petroleum mixtures will always exhibit lower
dewpoints for the entire range of temperatures exhibiting two phases (i.e. less than the cricondentherm), while upper saturation pressures of both bubblepoint and dewpoint type are usually found in
the range of relevant operational temperatures.
The state of a fluid mixture characterized by the co-existence of a liquid phase saturated with an infinitesimal quantity of equilibrium gas phase.
The state of a fluid mixture characterized by the co-existence of a vapor phase saturated with an infinitesimal quantity of equilibrium liquid (condensate) phase.
The state of a fluid mixture at which all properties of all coexisting vapor and liquid phases become identical (densities, viscosities, etc.), and the equilibrium ratios \(K_i=1\) for all
components. The mixture is called a saturated critical fluid at its critical state (not a saturated bubblepoint oil or a saturated dewpoint gas).
Vapor Pressure
For a compound at a temperature below the critical temperature (\(T_c\)), down to the triple point \(T_t\), and further down to 0 degrees absolute (\(T_0\)), the vapor pressure defines where the
compound exists in a multi-phase thermodynamic equilibrium with (a) saturated vapor and saturated liquid (\(T_t<T<T_c\)), (b) saturated vapor and saturated solid (\(0<T<T_t\)), or (c) saturated
vapor, saturated liquid, and saturated solid (\(T=T_t\)). The collection of vapor pressures is called the vapor pressure curve. As temperature increases from the triple point to the critical point,
the difference in equilibrium phase properties will decrease monotonically until the phase properties show no difference, and the two phases become identical at the critical point. The Gibbs chemical
energy (or fugacities) of saturated phases at the vapor pressure for a given temperature will always be equal, independent of the amount of each equilibrium phase. The system volume will determine
how much of each equilibrium phase exists. For a temperature on the saturated vapor-liquid curve (\(T_t<T<T_c\))), the volume changes from its minimum value with 100% saturated liquid to a maximum
value with 100% saturated vapor. The pressure will remain completely constant, equal to the vapor pressure, as volume changes from 100% saturated liquid to 100% saturated vapor. A plot of volume
versus pressure will, therefore, lead to a horizontal shock line connecting the minimum and maximum saturated volumes. Interestingly, no equation of state functional form reproduces this fundamental
(horizontal shock) pressure-volume behavior for any point on the vapor pressure curve (except at the critical point).
Convergence Pressure
The pressure of a fluid mixture at a given temperature where the K-values of all components appear to converge to unity when the isothermal (\(\log (K_i) - \log (p)\)) curves are extrapolated to
pressures above the upper saturation pressure (i.e., into the undersaturated pressure region). The convergence pressure of a mixture can be calculated by an EOS using the negative flash, where the
computed equilibrium compositions \(y_i\) and \(x_i\) are identical, fall on a tie line with the original mixture composition, and represent a composition \(z_{ci}=y_i=x_i\) with critical pressure
equal to the convergence pressure of \(z_i\) at system temperature.
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The fifteenth meeting of the scientific workshop "BMSTU mathematical colloquium"
On April 9, 2020 the Faculty of Fundamental Sciences of Bauman Moscow State Technical University will held the fourth meeting of the Scientific Workshop "BMSTU Mathematical Colloquium". It addresses
a broad mathematical audience including keen students. The workshop aims to give listeners a general view of various areas of modern mathematics
The workshop will be held on Thursday at 4:00 pm, ZOOM, meeting id: 975 725 677, password: 031691
The workshop topic: An Upper Bound for Weak B_k – Sets
The report will be given by PhD, Correspondent Member of Russian Academy of Sciences, Main Researcher of Steklov Mathematical Institute, Professor Shkredov Ilya Dmitrievich
The abstract of his report in PDF file.
If you are going to come or with any questions, please write to e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
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Dan Halperin, Michal Meyerovitch, Ron Wein, and Baruch Zukerman
A continuous surface \( S\) in \( {\mathbb R}^3\) is called \( xy\)-monotone, if every line parallel to the \( z\)-axis intersects it at a single point at most. For example, the sphere \( x^2 + y^2 +
z^2 = 1\) is not \( xy\)-monotone as the \( z\)-axis intersects it at \( (0, 0, -1)\) and at \( (0, 0, 1)\); however, if we use the \( xy\)-plane to split it to an upper hemisphere and a lower
hemisphere, these two hemispheres are \( xy\)-monotone.
An \( xy\)-monotone surface can therefore be represented as a bivariate function \( z = S(x,y)\), defined over some continuous range \( R_S \subseteq {\mathbb R}^2\). Given a set \( {\cal S} = \{
S_1, S_2, \ldots, S_n \}\) of \( xy\)-monotone surfaces, their lower envelope is defined as the point-wise minimum of all surfaces. Namely, the lower envelope of the set \( {\cal S}\) can be defined
as the following function:
\begin{eqnarray*} {\cal L}_{{\cal S}} (x,y) = \min_{1 \leq k \leq n}{\overline{S}_k (x,y)} \ , \end{eqnarray*}
where we define \(\overline{S}_k(x,y) = S_k(x,y)\) for \((x,y) \in R_{S_k}\), and \(\overline{S}_k(x,y) = \infty\) otherwise.
Similarly, the upper envelope of \({\cal S}\) is the point-wise maximum of the \(xy\)-monotone surfaces in the set:
\begin{eqnarray*} {\cal U}_{{\cal S}} (x,y) = \max_{1 \leq k \leq n}{\underline{S}_k (x,y)} \ , \end{eqnarray*}
where in this case \( \underline{S}_k(x,y) = -\infty\) for \( (x,y) \not\in R_{S_k}\).
Given a set of \( xy\)-monotone surfaces \( {\cal S}\), the minimization diagram of \( {\cal S}\) is a subdivision of the \( xy\)-plane into cells, such that the identity of the surfaces that induce
the lower diagram over a specific cell of the subdivision (be it a face, an edge, or a vertex) is the same. In non-degenerate situation, a face is induced by a single surface (or by no surfaces at
all, if there are no \( xy\)-monotone surfaces defined over it), an edge is induced by a single surface and corresponds to its projected boundary, or by two surfaces and corresponds to their
projected intersection curve, and a vertex is induced by a single surface and corresponds to its projected boundary point, or by three surfaces and corresponds to their projected intersection point.
The maximization diagram is symmetrically defined for upper envelopes. In the rest of this chapter, we refer to both these diagrams as envelope diagrams.
It is easy to see that an envelope diagram is no more than a planar arrangement (see Chapter 2D Arrangements), represented using an extended DCEL structure, such that every DCEL record (namely each
face, halfedge and vertex) stores an additional container of it originators: the \( xy\)-monotone surfaces that induce this feature.
Lower and upper envelopes can be efficiently computed using a divide-and-conquer approach. First note that the envelope diagram for a single \( xy\)-monotone curve \( S_k\) is trivial to compute: we
project the boundary of its range of definition \( R_{S_k}\) onto the \( xy\)-plane, and label the faces it induces accordingly. Given a set \( {\cal D}\) of (non necessarily \( xy\)-monotone)
surfaces in \( {\mathbb R}^3\), we subdivide each surface into a finite number of weakly \( xy\)-monotone surfaces, and obtain the set \( {\cal S}\). Then, we split the set into two disjoint subsets
\( {\cal S}_1\) and \( {\cal S}_2\), and we compute their envelope diagrams recursively. Finally, we merge the diagrams, and we do this by overlaying them and then applying some post-processing on
the resulting diagram. The post-processing stage is non-trivial and involves the projection of intersection curves onto the \( xy\)-plane - see [1] for more details.
The Envelope-Traits Concept
The implementation of the envelope-computation algorithm is generic and can handle arbitrary surfaces. It is parameterized with a traits class, which defines the geometry of surfaces it handles, and
supports all the necessary functionality on these surfaces, and on their projections onto the \( xy\)-plane. The traits class must model the EnvelopeTraits_3 concept, the details of which are given
As the representation of envelope diagrams is based on 2D arrangements, and the envelop-computation algorithm employs overlay of planar arrangements, the EnvelopeTraits_3 refines the
ArrangementXMonotoneTraits_2 concept. Namely, a model of this concept must define the planar types Point_2 and X_monotone_curve_2 and support basic operations on them, as listed in Section The
Geometry Traits. Moreover, it must define the spatial types Surface_3 and Xy_monotone_surface_3 (in practice, these two types may be the same). Any model of the envelope-traits concept must also
support the following operations on these spatial types:
• Subdivide a given surface into continuous \( xy\)-monotone surfaces. It is possible to disregard \( xy\)-monotone surfaces that do not contribute to the surface envelope at this stage (for
example, if we are given a sphere, it is possible to return just its lower hemisphere if we are interested in the lower envelope; the upper hemisphere is obviously redundant).
• Given an \( xy\)-monotone surface \( S\), construct all planar curves that form the boundary of the vertical projection \( S\)'s boundary onto the \( xy\)-plane.
This operation is used at the bottom of the recursion to build the minimization diagram of a single \( xy\)-monotone surface.
• Construct all geometric entities that comprise the projection (onto the \( xy\)-plane) of the intersection between two \( xy\)-monotone surfaces \( S_1\) and \( S_2\). These entities may be:
□ A planar curve, which is the projection of an 3D intersection curve of \( S_1\) and \( S_2\) (for example, the intersection curve between two spheres is a 3D circle, which becomes an ellipse
when projected onto the \( xy\)-plane). In many cases it is also possible to indicate the multiplicity of the intersection: if it is odd, the two surfaces intersect transversely and change
their relative \( z\)-positions on either side of the intersection curve; if it the multiplicity is even, they maintain their relative \( z\)-position. Providing the multiplicity information
is optional. When provided, it is used by the algorithm to determine the relative order of \( S_1\) and \( S_2\) on one side of their intersection curve when their order on the other side of
that curve is known, thus improving the performance of the algorithm.
□ A point, induces by the projection of a tangency point of \( S_1\) and \( S_2\), or by the projection of a vertical intersection curve onto the \( xy\)-plane.
Needless to say, the set of intersection entities may be empty in case \( S_1\) and \( S_2\) do not intersect.
• Given two \( xy\)-monotone surfaces \( S_1\) and \( S_2\), and a planar point \( p = (x_0,y_0)\) that lies in their common \( xy\)-definition range, determine the \( z\)-order of \( S_1\) and \(
S_2\) over \( p\), namely compare \( S_1(x_0,y_0)\) and \( S_2(x_0,y_0)\). This operation is used only in degenerate situations, in order to determine the surface inducing the envelope over a
vertex (see Figure 39.1 (a) for an illustration of a situation when this operation is used).
• Given two \( xy\)-monotone surfaces \( S_1\) and \( S_2\), and a planar \( x\)-monotone curve \( c\), which is a part of their projected intersection, determine the \( z\)-order of \( S_1\) and \
( S_2\) immediately above (or, similarly, immediately below) the curve \( c\). Note that \( c\) is a planar \( x\)-monotone curve, and we refer to the region above (or below) it in the plane. If
\( c\) is a vertical curve, we regard the region to its left as lying above it, and the region to its right as lying below it.
This operation is used by the algorithm to determine the surface that induce the envelope over a face incident to \( c\).
• Given two \( xy\)-monotone surfaces \( S_1\) and \( S_2\), and a planar \( x\)-monotone curve \( c\), which fully lies in their common \( xy\)-definition range, and such that \( S_1\) and \( S_2
\) do not intersect over the interior of \( c\), determine the relative \( z\)-order of \( s_1\) and \( s_2\) over the interior of \( c\). Namely, we compare \( S_1(x_0,y_0)\) and \( S_2(x_0,y_0)
\) for some point \( (x_0, y_0)\) on \( c\).
This operation is used by the algorithm to determine which surface induce the envelope over an edge associated with the \( x\)-monotone curve \( c\), or of a face incident to \( c\), in
situations where the previous predicate cannot be used, as \( c\) is not an intersection curve of \( S_1\) and \( S_2\) (see Figure 39.1 (b) for an illustration of a situation where this
operation is used).
The package currently contains a traits class for named Env_triangle_traits_3<Kernel> handling 3D triangles, and another named Env_sphere_traits_3<ConicTraits> for 3D spheres, based on geometric
operations on conic curves (ellipses). In addition, the package includes a traits-class decorator that enables users to attach external (non-geometric) data to surfaces. The usage of the various
traits classes is demonstrated in the next section.
Example for Envelope of Triangles
The following example shows how to use the envelope-traits class for 3D triangles and how to traverse the envelope diagram. It constructs the lower and upper envelopes of the two triangles, as
depicted in Figure 39.2 (a) and prints the triangles that induce each face and each edge in the output diagrams. For convenience, we use the traits-class decorator Env_surface_data_traits_3 to label
the triangles. When printing the diagrams, we just output the labels of the triangles:
File Envelope_3/envelope_triangles.cpp
// Constructing the lower and the upper envelope of a set of triangles.
#include <iostream>
#include <list>
#include <CGAL/Exact_rational.h>
#include <CGAL/Cartesian.h>
#include <CGAL/Env_triangle_traits_3.h>
#include <CGAL/Env_surface_data_traits_3.h>
#include <CGAL/envelope_3.h>
using Number_type = CGAL::Exact_rational;
using Data_triangle_3 = Data_traits_3::Surface_3;
/* Auxiliary function - print the features of the given envelope diagram. */
void print_diagram(const Envelope_diagram_2& diag) {
// Go over all arrangement faces.
for (auto fit = diag.faces_begin(); fit != diag.faces_end(); ++fit) {
// Print the face boundary.
if (fit->is_unbounded()) std::cout << "[Unbounded face]";
else {
// Print the vertices along the outer boundary of the face.
auto ccb = fit->outer_ccb();
std::cout << "[Face] ";
do std::cout << '(' << ccb->target()->point() << ") ";
while (++ccb != fit->outer_ccb());
// Print the labels of the triangles that induce the envelope on this face.
std::cout << "--> " << fit->number_of_surfaces() << " triangles:";
for (auto sit = fit->surfaces_begin(); sit != fit->surfaces_end(); ++sit)
std::cout << ' ' << sit->data();
std::cout << std::endl;
// Go over all arrangement edges.
for (auto eit = diag.edges_begin(); eit != diag.edges_end(); ++eit) {
// Print the labels of the triangles that induce the envelope on this edge.
std::cout << "[Edge] (" << eit->source()->point()
<< ") (" << eit->target()->point()
<< ") --> " << eit->number_of_surfaces()
<< " triangles:";
for (auto sit = eit->surfaces_begin(); sit != eit->surfaces_end(); ++sit)
std::cout << ' ' << sit->data();
std::cout << std::endl;
/* The main program: */
int main() {
// Construct the input triangles, makred A and B.
std::list<Data_triangle_3> triangles;
triangles.push_back(Data_triangle_3(t1, 'A'));
triangles.push_back(Data_triangle_3(t2, 'B'));
// Compute and print the minimization diagram.
Envelope_diagram_2 min_diag;
std::cout << std::endl << "The minimization diagram:" << std::endl;
// Compute and print the maximization diagram.
Envelope_diagram_2 max_diag;
std::cout << std::endl << "The maximization diagram:" << std::endl;
print_diagram (max_diag);
return 0;
Base::Edge_iterator Edge_const_iterator
The class Env_surface_data_traits_3 is a model of the EnvelopeTraits_3 concept and serves as a decora...
Definition: Env_surface_data_traits_3.h:39
The traits class template Env_triangle_traits_3 models the EnvelopeTraits_3 concept,...
Definition: Env_triangle_traits_3.h:41
The class-template Envelope_diagram_2 represents the minimization diagram that corresponds to the low...
Definition: envelope_3.h:26
void upper_envelope_3(InputIterator begin, InputIterator end, Envelope_diagram_2< Traits > &diag)
Computes the upper envelope of a set of surfaces in , as given by the range [begin,...
void lower_envelope_3(InputIterator begin, InputIterator end, Envelope_diagram_2< Traits > &diag)
Computes the lower envelope of a set of surfaces in , as given by the range [begin,...
Example for Envelope of Spheres
The next example demonstrates how to instantiate and use the envelope-traits class for spheres, based on the Arr_conic_traits_2 class that handles the projected intersection curves. The program reads
a set of spheres from an input file and constructs their lower envelope:
File Envelope_3/envelope_spheres.cpp
// Constructing the lower envelope of a set of spheres read from a file.
#include <CGAL/config.h>
#ifndef CGAL_USE_CORE
#include <iostream>
int main() {
std::cout << "Sorry, this example needs CORE ..." << std::endl;
return 0;
#include <iostream>
#include <list>
#include <chrono>
#include <CGAL/Cartesian.h>
#include <CGAL/CORE_algebraic_number_traits.h>
#include <CGAL/Arr_conic_traits_2.h>
#include <CGAL/Env_sphere_traits_3.h>
#include <CGAL/envelope_3.h>
using Rational = Nt_traits::Rational;
using Algebraic = Nt_traits::Algebraic;
using Rat_point_3 = Rat_kernel::Point_3;
using Conic_traits_2 =
int main(int argc, char* argv[]) {
// Get the name of the input file from the command line, or use the default
// fan_grids.dat file if no command-line parameters are given.
const char* filename = (argc > 1) ? argv[1] : "spheres.dat";
// Open the input file.
std::ifstream in_file(filename);
if (! in_file.is_open()) {
std::cerr << "Failed to open " << filename << " ..." << std::endl;
return 1;
// Read the spheres from the file.
// The input file format should be (all coordinate values are integers):
// <n> // number of spheres.
// <x_1> <y_1> <x_1> <R_1> // center and squared radious of sphere #1.
// <x_2> <y_2> <x_2> <R_2> // center and squared radious of sphere #2.
// : : : :
// <x_n> <y_n> <x_n> <R_n> // center and squared radious of sphere #n.
int n = 0;
std::list<Sphere_3> spheres;
int x = 0, y = 0, z = 0, sqr_r = 0;
in_file >> n;
for (int i = 0; i < n; ++i) {
in_file >> x >> y >> z >> sqr_r;
(Rat_point_3(x, y, z), Rational(sqr_r)));
std::cout << "Constructing the lower envelope of " << n << " spheres.\n";
// Compute the lower envelope.
Envelope_diagram_2 min_diag;
auto start = std::chrono::system_clock::now();
std::chrono::duration<double> secs = std::chrono::system_clock::now() - start;
// Print the dimensions of the minimization diagram.
std::cout << "V = " << min_diag.number_of_vertices()
<< ", E = " << min_diag.number_of_edges()
<< ", F = " << min_diag.number_of_faces() << std::endl;
std::cout << "Construction took " << secs.count() << " seconds.\n";
return 0;
The traits class Env_sphere_traits_3 models the EnvelopeTraits_3 concept, and is used for the constru...
Definition: Env_sphere_traits_3.h:32
Example for Envelope of Planes
The next example demonstrates how to instantiate and use the envelope-traits class for planes, based on the Arr_linear_traits_2 class that handles infinite linear objects such as lines and rays.
File Envelope_3/envelope_planes.cpp
// Constructing the lower and the upper envelope of a set of planes.
#include <iostream>
#include <list>
#include <CGAL/Exact_rational.h>
#include <CGAL/Cartesian.h>
#include <CGAL/Env_plane_traits_3.h>
#include <CGAL/envelope_3.h>
using Number_type = CGAL::Exact_rational;
using Surface_3 = Traits_3::Surface_3;
/* Auxiliary function - print the features of the given envelope diagram. */
void print_diagram(const Envelope_diagram_2& diag) {
// Go over all arrangement faces.
for (auto fit = diag.faces_begin(); fit != diag.faces_end(); ++fit) {
// Print the face boundary.
// Print the vertices along the outer boundary of the face.
auto ccb = fit->outer_ccb();
std::cout << "[Face] ";
do if (!ccb->is_fictitious()) std::cout << '(' << ccb->curve() << ") ";
while (++ccb != fit->outer_ccb());
// Print the planes that induce the envelope on this face.
std::cout << "--> " << fit->number_of_surfaces() << " planes:";
for (auto sit = fit->surfaces_begin(); sit != fit->surfaces_end(); ++sit)
std::cout << ' ' << sit->plane();
std::cout << std::endl;
/* The main program: */
int main() {
// Construct the input planes.
std::list<Surface_3> planes;
(0, -1, 1, 0)));
(-1, 0, 1, 0)));
(0, 1 , 1, 0)));
(1, 0, 1, 0)));
// Compute and print the minimization diagram.
Envelope_diagram_2 min_diag;
std::cout << std::endl << "The minimization diagram:" << std::endl;
// Compute and print the maximization diagram.
Envelope_diagram_2 max_diag;
std::cout << std::endl << "The maximization diagram:" << std::endl;
print_diagram (max_diag);
return 0;
The traits class template Env_plane_traits_3 models the EnvelopeTraits_3 concept, and is used for the...
Definition: Env_plane_traits_3.h:38
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SQL代写 - CSC343数据库|学霸联盟
1. Consider a relation R with attributes ABCDEF GHI with functional dependencies S: S = { AEG → F, B → AD, AG → HI, BG → D } (a) State which of the given FDs violate BCNF. (b) Employ the BCNF
decomposition algorithm to obtain a lossless and redundancy-preventing decomposition of relation R into a collection of relations that are in BCNF. Make sure it is clear which relations are in the
final decomposition, and don’t forget to project the dependencies onto each relation in that final decomposition. Because there are choice points in the algorithm, there may be more than one correct
answer. List the final relations in alphabetical order (order the attributes alphabetically within a relation, and order the relations alphabetically). (c) Does your schema preserve dependencies?
Explain how you know that it does or does not. (d) Use the Chase Test to show that your schema is a lossless-join decomposition. (This us guaranteed by the BCNF algorithm, but it’s a good exercise.)
2. Consider a relation A with attributes LMNOP QRS and functional dependencies B. B = { LNOP → M, M → NQ, NO → LQ, MNQ → LO, LMQ → NOS } (a) Compute a minimal basis for T. In your final answer, put
the FDs into alphabetical order. Within a single FD, this means stating an FD as XY → A, not as Y X → A. Also, list the FDs in alphabetical order ascending according to the left-hand side, then by
the right-hand side. This means, W X → A comes before W XZ → A which comes before W XZ → B. (b) Using your minimal basis from the last subquestion, compute all keys for P. (c) Employ the 3NF
synthesis algorithm to obtain a lossless and dependency-preserving decomposition of relation P into a collection of relations that are in 3NF. Do not “over normalize”. This means that you should
combine all FDs with the same left-hand side to create a single relation. If your schema includes one relation that is a subset of another, remove the smaller one. (d) Does your schema allow
redundancy? Explain how you know that it does or does not. Show all of your steps so that we can give part marks where appropriate. There are no marks for simply a correct answer. You must justify
every shortcut that you take. What to hand in Type your answers up using LaTeX or Word. Hand in your typed answers, in a single pdf file called A3.pdf. Final Thoughts Declare your group now: Well
before the due date, declare your team (even if you are working solo) on MarkUs. It is impossible to do so during the late-with-penalty period. Submission: Check that you have submitted the correct
version of your files by downloading it from MarkUs; new files will not be accepted after the due date. Some parting advice: It will be tempting to divide the assignment up with your partner.
Remember that both of you probably want to answer all the questions on the final test.
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A speeder passes a parked police car at 29.6 m/s. Instantaneously, the police car starts f...
A speeder passes a parked police car at 29.6 m/s. Instantaneously, the police car starts from rest with a uniform acceleration of 2.49 m/s². How much time passes before the speeder is overtaken by
the police car? Answer in units of s. How far does the speeder get before being overtaken by the police car? Answer in units of m
STEP 1
Assumptions1. We assume that the motion of both the speeder and the police car is along a straight line. . We assume that the speeder maintains a constant speed of29.6 m/s.
3. We assume that the police car accelerates uniformly from rest at.49 m/s².
STEP 2
We need to find the time it takes for the police car to catch up to the speeder. This is the point where both the speeder and the police car have traveled the same distance.
STEP 3
We can use the equation for motion to describe the distance traveled by the speeder and the police car. For the speeder, since the speed is constant, the distance (d1) is given by$d1 = v \cdot t\]
where v is the speed of the speeder and t is the time.
STEP 4
For the police car, since it starts from rest and accelerates uniformly, the distance (d2) is given by$d2 = \frac{1}{2} a \cdot t^2\]
where a is the acceleration of the police car and t is the time.
STEP 5
The police car catches up to the speeder when they have both traveled the same distance. Therefore, we can set d1 equal to d2 and solve for t.
$v \cdot t = \frac{1}{2} a \cdot t^2\]
STEP 6
To solve for t, we can rearrange the equation to$t = \frac{2v}{a}$
STEP 7
Substituting the given values into the equation, we get$t = \frac{2 \cdot29.6 m/s}{2.49 m/s^2}$
STEP 8
olving this gives the time it takes for the police car to catch up to the speeder$t = \frac{2 \cdot29.6 m/s}{2.49 m/s^2} =23.7 s\]
STEP 9
Next, we need to find how far the speeder gets before being overtaken by the police car. We can use the equation for the distance traveled by the speeder, d = v \cdot t.
STEP 10
Substituting the given values into the equation, we get$d =29.6 m/s \cdot23.7 s\]
olving this gives the distance the speeder travels before being overtaken$d =29.6 m/s \cdot23.7 s =702 m\]
So, the police car overtakes the speeder after23.7 seconds and the speeder travels702 meters before being overtaken.
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An object with a mass of 8 kg is on a plane with an incline of pi/8 . If the object is being pushed up the plane with 3 N of force, what is the net force on the object? | HIX Tutor
An object with a mass of #8 kg# is on a plane with an incline of #pi/8 #. If the object is being pushed up the plane with # 3 N # of force, what is the net force on the object?
Answer 1
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Answer 2
The net force on the object can be calculated using the formula: ( F_{\text{net}} = F_{\text{applied}} - F_{\text{gravity}} ). First, find the force of gravity acting on the object using the formula:
( F_{\text{gravity}} = m \cdot g ), where ( m ) is the mass of the object (8 kg) and ( g ) is the acceleration due to gravity (approximately 9.8 m/s²). Then, calculate the component of the applied
force acting against gravity along the incline using ( F_{\text{applied_parallel}} = F_{\text{applied}} \cdot \sin(\text{inclined angle}) ). Finally, subtract the force of gravity from the component
of the applied force along the incline to find the net force on the object.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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composition of functions
07-22-2014, 01:23 AM
Post: #1
Alberto Candel Posts: 169
Member Joined: Dec 2013
composition of functions
Is there a command for the composition of two functions, f and g, in CAS? There is f@@g in xcas, but I do not seem to get it to work in the Prime.
07-22-2014, 02:17 AM
(This post was last modified: 07-22-2014 02:19 AM by Mark Hardman.)
Post: #2
Mark Hardman Posts: 525
Senior Member Joined: Dec 2013
RE: composition of functions
(07-22-2014 01:23 AM)Alberto Candel Wrote: Is there a command for the composition of two functions, f and g, in CAS? There is f@@g in xcas, but I do not seem to get it to work in the Prime.
Try something along the lines of:
The composition of the two functions is simply:
Ceci n'est pas une signature.
07-22-2014, 03:54 AM
Post: #3
Alberto Candel Posts: 169
Member Joined: Dec 2013
RE: composition of functions
Thank you Mark. But I was looking for something like that on page 12 of this
Xcas/giac tutorial
07-22-2014, 09:51 AM
Post: #4
parisse Posts: 1,336
Senior Member Joined: Dec 2013
RE: composition of functions
In Xcas, @ does function composition, not @@, @@ is for composition power.
07-22-2014, 03:44 PM
Post: #5
Alberto Candel Posts: 169
Member Joined: Dec 2013
RE: composition of functions
(07-22-2014 09:51 AM)parisse Wrote: In Xcas, @ does function composition, not @@, @@ is for composition power.
Yes, thanks, I should have written f@g for the composition and f@@n for the composite of f with itself n times. The prime accepts f@g, but it seems to return a function like (x,y)->(f(x),g(y)) (if f
and g are 1 variable functions).
07-22-2014, 04:07 PM
Post: #6
parisse Posts: 1,336
Senior Member Joined: Dec 2013
RE: composition of functions
Please give an example.
07-22-2014, 09:14 PM
Post: #7
Alberto Candel Posts: 169
Member Joined: Dec 2013
RE: composition of functions
(07-22-2014 04:07 PM)parisse Wrote: Please give an example.
For instance
f@g returns ((x)->x^2)@((x)->x+1)
I do not know the meaning of @
07-22-2014, 11:11 PM
Post: #8
Mark Hardman Posts: 525
Senior Member Joined: Dec 2013
RE: composition of functions
(07-22-2014 09:14 PM)Alberto Candel Wrote:
(07-22-2014 04:07 PM)parisse Wrote: Please give an example.
For instance
f@g returns ((x)->x^2)@((x)->x+1)
I do not know the meaning of @
The CAS is providing an intermediate solution to the composition.
If you execute:
You get the expected result:
Ceci n'est pas une signature.
07-23-2014, 02:41 PM
(This post was last modified: 07-23-2014 03:32 PM by Alberto Candel.)
Post: #9
Alberto Candel Posts: 169
Member Joined: Dec 2013
RE: composition of functions
That works, thanks.
But it would be better without the simplify part.
[edit] Actually, if you define k:=f@g, then k(x)=x^2+2*x+1, without the simplify.[/edit]
07-23-2014, 06:23 PM
Post: #10
parisse Posts: 1,336
Senior Member Joined: Dec 2013
RE: composition of functions
You can also type (f@g)(x). Note that f@g is a function, not an expression.
07-23-2014, 09:21 PM
Post: #11
Alberto Candel Posts: 169
Member Joined: Dec 2013
RE: composition of functions
(07-23-2014 06:23 PM)parisse Wrote: You can also type (f@g)(x). Note that f@g is a function, not an expression.
Yes, thanks. I think my original mistake was to write f@g(x) instead of (f@g)(x).
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Equation solution using minimization
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2 Replies
1 Total Likes
Equation solution using minimization
I have an equation F[x,y]=0. I need to plot y as a function of x. Due to the properties of the function F[x,y], it is impossible to just use ContourPlot for this problem. Also, FindRoot (I don't know
why) doesn't work in this situation.
However, NMinimize[...,Method -> "NelderMead"] works well. I need to find all yi for every xi starting from some point {x0,y0} which I know a priori. I can do it step by step by hand and it takes a
lot of time, but due to my low knowledge in Mathematica to date I can't automotize this process.
So, please, could anyone help me to to write such a code in Mathematica:
1. put starting point {x0,y0}.
2. for i from 1 to N
NMinimize[{F[xi, yi] y{i-1}-deltaY < yi < y{i-1}+deltaY}, {yi}, Method -> "NelderMead"]
(*so, for every xi we find yi and look for y{i+1} in the vicinity of yi; deltaX is just a step, deltaY is a small constant (much less then y_i)*)
1. Plot the points {yi,xi}.
For example, two steps of this algorithm:
NMinimize[{F[x1, y1] 0-0.1 < yi < 0+0.1}, {yi}, Method -> "NelderMead"]
we get y_1=0.05.
NMinimize[{F[x2, y2] 0.05-0.1 < yi < 0.05+0.1}, {yi}, Method -> "NelderMead"]
we get y_1=0.75.
2 Replies
Dear Daniel, thank you for your response!
Firstly, the most interesting area for me is 0<y<4, 0<x<5.
Secondly, there is a physical reason to expect real-valued zeros because this equation describes waves in a dispersive media WITHOUT any absorption, damping. X is a wave vector, y is frequency.
Moreover, I know asymptotic behavior of the solution of this equation when parameters a->inb, b is finite and when b->inf a is finite. These solutions are described by other equations which are
simply solved in Mathematica (using ContourPlot).
But I can't solve this equation using ContourPlot (in some cases ContourPlot gives a solution, but it is wrong). Also, when I use FindRoot I receive very strange (sometimes even unphysical)
solutions. As I know after looking through literature, it is possible to find solutions by Nelder-Mead minimization procedure "While no exact solution exists for the dispersion relations... in the
real frequency domain...To obtain the complex wave vectors, numerical solution of ... was accomplished through implementation of a two-dimensional unconstrained Nelder-Mead minimization algorithm".
In that article authors have considered absorptive media, but I firstly would like to consider non absorptive media, that's why I expect real-valued zeros.
It seems to be effectively impossible to evaluate in those ranges without either huge error (at machine precision) or else internal overflows. I think this is from negative radicals giving
imaginaries making trigs into hyperbolics. Behaves better for Abs[y]>3. Those ranges do not likely have roots though. Is there some reason to expect real-valued zeros?
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Berlin 2005 – wissenschaftliches Programm
Q 42.5: Vortrag
Dienstag, 8. März 2005, 15:00–15:15, HU Audimax
Localizable Entanglement — •Markus Popp^1, Frank Verstraete^1, Ignacio Cirac^1, and Miguel-Angel Martín-Delgado^2 — ^1Max-Planck-Institut für Quantenoptik, 85748 Garching — ^2Departamento de Física
Teórica I, Universidad Complutense de Madrid, E-28040, Spain
We consider systems of interacting spins and study the entanglement that can be localized, on average, between two separated spins by performing local measurements on the remaining spins. This
concept of Localizable Entanglement (LE) [1] leads naturally to notions like entanglement length and entanglement fluctuations. For both spin-1/2 and spin-1 systems we prove that the LE of a pure
quantum state can be lower bounded by classical correlation functions. We further propose a scheme, based on matrix-product states and the Monte Carlo method, to efficiently calculate the LE for
quantum states of a large number of spins. The virtues of LE are illustrated for various spin models. In particular, characteristic features of a quantum phase transition such as a diverging
entanglement length can be observed. We also give examples for pure quantum states exhibiting a diverging entanglement length but finite correlation length [2]. We have numerical evidence that the
ground state of the antiferromagnetic spin-1 Heisenberg chain can serve as a perfect quantum channel. Furthermore we apply the numerical method to mixed states and study the entanglement as a
function of temperature.
References: [1] F. Verstraete, M. Popp, and J.I. Cirac, Phys. Rev. Lett. 92, 027901 (2004). [2] F. Verstraete, M.A. Martin-Delgado, and J.I. Cirac, Phys. Rev. Lett. 92, 087201 (2004).
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Kierownik projektu: Jakub Zakrzewski
Fundator: Quant-ERA21
Realizacja: 2022-2025
As of today, Quantum Simulators (QS) are the systems that can address, deepen our understanding of, and ultimately solve some of the most challenging problems of contemporary science: from quantum
many body dynamics, through static and transient high Tc superconductivity, to the design of new materials. In DYNAMITE, we will design, realize in the labs, and characterize a new generation of QS
with ultracold atoms and beyond. With ascending degree of experimental complexity this involves: (WP1) systems with statistical gauge fields, i.e. “single-component” lattice or continuum systems with
density-dependent gauge fields changing the effective quantum statistics of the particles and realizing topological gauge theories; (WP2) systems in dynamical lattices, with “matter” living on the
sites, and additional dynamical fields/particles living on the bonds; (WP3) lattice gauge theory models (LGT), from systems with Abelian (Z2, U(1)) to non-Abelian local gauge symmetry. Such systems
address questions from condensed matter physics, nuclear physics, high energy physics, and material science: In particular, WP1 allows one to engineer topological gauge theories in the continuum, and
to design and control novel types of topological and chiral order, with possible applications to quantum computing and quantum memories. The theoretical and experimental goal here is to tailor the
proper matter dependence for the gauge fields, which corresponds to correctly imposing the local symmetry constraint of the gauge theory. WP2 allows one to design and study the interplay between
topological order and symmetry breaking. Simpler systems without gauge invariance that are already simulated in the labs, permit us indeed to study the fundamental question of how gauge theory
phenomena translate into systems of coupled degrees of freedom without explicit gauge symmetry and how gauge symmetry can emerge. WP3 allows us to study statics of the confinement-deconfinement
transition, and more importantly its dynamics, relation to absence/presence of thermalization, the dynamical role of many body localization and quantum scars. While experimental work will focus on
Abelian LGTs, theory will design as well scalable implementations of nonAbelian symmetries. In DYNAMITE, experiment and theory will be inseparably entangled. Its results will provide unprecedented
control over salient phenomena at the frontier of quantum many-body physics.
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Euclid's Elements Reference Page
Book III
(III.1) To find the center of a given circle.
(III.2) If two points are taken at random on the circumference of a circle, then the straight line joining the points falls within the circle.
(III.3) If a straight line passing through the center of a circle bisects a straight line not passing through the center, then it also cuts it at right angles; and if it cuts it at right angles, then
it also bisects it.
(III.4) If in a circle two straight lines which do not pass through the center cut one another, then they do not bisect one another.
(III.5) If two circles cut one another, then they do not have the same center.
(III.6) If two circles touch one another, then they do not have the same center.
(III.7) If on the diameter of a circle a point is taken which is not the center of the circle, and from the point straight lines fall upon the circle, then that is greatest on which passes through
the center, the remainder of the same diameter is least, and of the rest the nearer to the straight line through the center is always greater than the more remote; and only two equal straight lines
fall from the point on the circle, one on each side of the least straight line.
(III.8) If a point is taken outside a circle and from the point straight lines are drawn through to the circle, one of which is through the center and the others are drawn at random, then, of the
straight lines which fall on the concave circumference, that through the center is greatest, while of the rest the nearer to that through the center is always greater than the more remote, but, of
the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote; and only two
equal straight lines fall on the circle from the point, one on each side of the least..
(III.9) If a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, then the point taken is the center of the circle.
(III.10) A circle does not cut a circle at more than two points.
(III.11) If two circles touch one another internally, and their centers are taken, then the straight line joining their centers, being produced, falls on the point of contact of the circles.
(III.12) If two circles touch one another externally, then the straight line joining their centers passes through the point of contact.
(III.13) A circle does not touch another circle at more than one point whether it touches it internally or externally.
(III.14) Equal straight lines in a circle are equally distant from the center, and those which are equally distant from the center equal one another.
(III.15) Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote.
(III.16) The straight line drawn at right angles to the diameter of a circle from its end will fall outside the circle, and into the space between the straight line and the circumference another
straight line cannot be interposed, further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle.
(III.17) From a given point to draw a straight line touching a given circle.
(III.18) If a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent.
(III.19) If a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn.
(III.20) In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base.
(III.21) In a circle the angles in the same segment equal one another.
(III.22) The sum of the opposite angles of quadrilaterals in circles equals two right angles.
(III.23) On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.
(III.24) Similar segments of circles on equal straight lines equal one another.
(III.25) Given a segment of a circle, to describe the complete circle of which it is a segment.
(III.26) In equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences.
(III.27) In equal circles angles standing on equal circumferences equal one another whether they stand at the centers or at the circumferences.
(III.28) In equal circles equal straight lines cut off equal circumferences, the greater circumference equals the greater and the less equals the less.
(III.29) In equal circles straight lines that cut off equal circumferences are equal.
(III.30) To bisect a given circumference.
(III.31) In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater
segment is greater than a right angle, and the angle of the less segment is less than a right angle.
(III.32) If a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent
equal the angles in the alternate segments of the circle.
(III.33) On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilinear angle.
(III.34) From a given circle to cut off a segment admitting an angle equal to a given rectilinear angle.
(III.35) If in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals the rectangle contained by the segments of the other.
(III.36) If a point is taken outside a circle and two straight lines fall from it on the circle, and if one of them cuts the circle and the other touches it, then the rectangle contained by the whole
of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the tangent.
(III.37) If a point is taken outside a circle and from the point there fall on the circle two straight lines, if one of them cuts the circle, and the other falls on it, and if further the rectangle
contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the straight line
which falls on the circle, then the straight line which falls on it touches the circle.
1. T. L. Heath, Euclid: The Thirteen Books of The Elements, Dover, 1956
2. R. Simson, The Elements of Euclid, Books I-VI, XI, XII + Euclid's Data, Elibron Classics, 2005
|Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018
Alexander Bogomolny
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Transition state theory or Activated complex theory
Transition state theory or Activated complex theory
It is more modern theory proposed in 1932 by Pelzer & Manger later developed by Erying and colleagues.
1. Rate equation formation using the statistical mechanics to describe equilibrium between activated complex and reactant.
2. Rate equation formulation as per thermodynamical state function to describe transition state complex and reactant.
3. And rate equation 's statistical mechanical derivation.
Various postulates of Transition state theory or Activated complex theory are......
1. For any chemistry reaction to take place, it requires reactant have sufficient minimum energy. So it forms activated complex and reactant & complex are in equilibrium.
Reactant <~> [ activated complex ]
2. Activated complex have normal molecule with 4th degree of freedom along chemistry reaction coordinate.
3. And activated complex decompose along this 4th degree of freedom to yield products.
[Activated complex ] <~> Product
Transition state theory or Activated complex theory postulates says...
Reactant <~> [activated complex] <~> Product
Rate of reaction = decomposition rate of activated complex
rate of reaction = probability of crossing energy barriers * concentration of activated complex at top of energy barrier * frequency of crossing energy barrier
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Riccardo Grazzi
Mar 28, 2024
Abstract:We study the problem of efficiently computing the derivative of the fixed-point of a parametric nondifferentiable contraction map. This problem has wide applications in machine learning,
including hyperparameter optimization, meta-learning and data poisoning attacks. We analyze two popular approaches: iterative differentiation (ITD) and approximate implicit differentiation (AID). A
key challenge behind the nonsmooth setting is that the chain rule does not hold anymore. Building upon the recent work by Bolte et al. (2022), who proved linear convergence of nondifferentiable ITD,
we provide an improved linear rate for ITD and a slightly better rate for AID, both in the deterministic case. We further introduce NSID, a new stochastic method to compute the implicit derivative
when the fixed point is defined as the composition of an outer map and an inner map which is accessible only through a stochastic unbiased estimator. We establish rates for the convergence of NSID,
encompassing the best available rates in the smooth setting. We present illustrative experiments confirming our analysis.
* Removed the assumption on the conservative derivative of the fixed point map having a product structure: the product of partial conservative derivatives is not conservative in general
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Net Revenue - What Is It, Formula, Vs Gross Revenue, Examples
Net Revenue
Last Updated :
21 Aug, 2024
What is Net Revenue?
Net revenue is a company's sales from which returns, discounts, and other items are subtracted. In accounting, Net refers to adjustments made to the original. Therefore, it can be calculated
after adjusting gross revenue with the discounts, returned products, or other direct selling expenses.
Net revenue helps firms recognize their profit structure and also check how much money it is making after utilizing resources and making other operational expenditure. However, considering this
revenue figure alone may not provide the accurate status of a business's progress. Hence, other metrics must also be studied for a reliable assessment.
Table of contents
Net Revenue Explained
Net revenue sales account for the sales revenue generated through daily operations. This is the amount that is obtained when expenses are deducted from the gross income. These expenses include
marketing costs, office supplies, taxes, total cost of goods sold (COGS), employee wages and salaries, legal and administrative costs, rent and utilities, etc.
Though net revenue tries figuring out the actual revenue that the firm is left with after applicable deductions, it cannot be the sole metric for firms to depend on to assess its status and
performance in the market.
In finance, no single metric can provide essential elements of investment. Net revenue alone cannot help a person decide where to put his money, what to do with his business, and how to enhance his
business. But it does provide an important metric to help in making a decision. There will never be a single metric that will help in entire decision-making. Net revenue is a metric that, in
augmentation with profits and other basic financial metrics, will help in investing in a company. It is not just the author of this article that thinks so, Warrant Buffet and his guru Benjamin Graham
think so too.
The question of why to calculate net revenue instead of revenue is the one we shall answer first. Revenue has all sorts of inclusions in it. Let us assume we own an electronics company that produces
laptops, and during Black Friday, we offer huge discounts on our laptops. Now, in our revenue, we include the total amount – because that is the selling price of the laptop. However, using those
numbers for financial calculations will mislead us into thinking that the revenue is more than what we got. So, we removed such discounts and also returned products.
The equation that helps calculate net revenue is mentioned below:
Net Revenue Formula = Gross Revenue – Directly Related Selling Expenses
Let us consider the following examples to understand the concept and net revenue calculation better:
Example #1
Let us take the same example above and put some numbers to it. Let us assume our annual turnover last year was 1,000,000 USD. That originated from selling 2,000 laptops for 500 USD each. Now, of
those 2000 laptops, 200 of them were sold during Black Friday at a discount of 20%. And then, 20 laptops in total were returned because of faulty parts. Since we have part of the revenue, let us put
some numbers on the cost too. Let us assume that each laptop costs us 250 USD to make. So, the Cost of Goods Sold (COGS) is 250*2000, which is 500,000 USD.
If we use the above numbers for financial analysis, our profits will be 500,000 USD. Now, let us look into why this overstates the actual profit numbers. To be true, we haven’t got 1,000,000 USD in
total. People returned 20 laptops, which is 10,000, and we have given a discount of 20% on 200 laptops – That comes out to 40,000 USD. SO, in total, we have 50,000 USD under discount schemes.
If we use these numbers, we can see that our profit numbers are different when we calculate net revenue and gross revenue.
Example #2
Let us take the example of Warren Buffet. In an era where quantitative hedge funds make billions of calculations a second to invest and companies build straight-lined optical fibers from Chicago to
New York to get data faster and better, Buffet is one last triumph of traditional investing.
And he pays very close attention to "Profit Margins." He can tear through the financial industry's witchcraft by looking at Profit Margins. How does he calculate them? That is where we will use Net
Profit Margins = Net Income/Net Sales.
Keep an eye on 'net income.' Every investor looks at multiple numbers and makes a decision. Because of how the financial world works, it is impossible to look at one number and take it as gospel for
investing. When people started looking at gross profit, many companies started selling their products at a discount and boosting sales.
Now, everything is overstated. In such situations – Net Revenue is truer than the original numbers. A high number indicates that the company is doing well and vice versa.
Net Revenue vs Gross Revenue
While considering revenues generated, there are two categories of them that one comes across – gross and net revenue. Let us have a look at the differences between the two:
• Gross revenue is the total revenue generated from the sales, while net revenue is the revenue calculated after the expenses and liabilities are deducted. In short, the latter is the actual
revenue figure that stays with the firm.
• Most of the time, investors are more bothered with gross revenue than with net revenue – because it shows one’s ability to conduct business and progress into a growth structure. On the contrary,
the net revenue gives the figure that the firm retains after the liabilities or expenses are deducted.
• When selling in a new location, it makes more sense to use gross revenue – because it shows us the potential growth rate at the new locations. However, net revenue is the number that matters for
all the financial aspects. To see where the profits are high and where they are low, which parts have to be cut and which parts have to be grown, and to make a strategic decision on what to do
for more profits.
Recommended Articles
This has been a guide to What net revenue is and its Definition. Here we discuss the formula to calculate Net Revenue, examples, advantages, and disadvantages. You can learn more about accounting
from the following articles –
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Dynamic response of various von-Kármán non-linear plate models and their 3-D counterparts
Dynamic von-Kármán plate models consist of three coupled non-linear, time-dependent partial differential equations. These equations have been recently solved numerically [Kirby, R., Yosibash, Z.,
2004. Solution of von-Kármán dynamic non-linear plate equations using a pseudo-spectral method. Comp. Meth. Appl. Mech. Eng. 193 (6-8) 575-599 and Yosibash, Z., Kirby, R., Gottlieb, D., 2004.
Pseudo-spectral methods for the solution of the von-Kármán dynamic non-linear plate system. J. Comp. Phys. 200, 432-461] by the Legendre-collocation method in space and the implicit Newmark-β scheme
in time, where highly accurate approximations were realized. Due to their complexity, these equations are often reduced by discarding some of the terms associated with time derivatives which are
multiplied by the plate thickness squared (being a small parameter). Because of the non-linearities in the system of equations we herein quantitatively investigate the influence of these a-priori
assumption on the solution for different plate thicknesses. As shown, the dynamic solutions of the so called "simplified von-Kármán" system do not differ much from the complete von-Kármán system for
thin plates, but may have differences of few percent for plates with thicknesses to length ratio of about 1/20. Nevertheless, when investigating the modeling errors, i.e. the difference between the
various von-Kármán models and the fully three-dimensional non-linear elastic plate solution, one realizes that for relatively thin plates (thickness is 1/20 of other typical dimensions), this
difference is much larger. This implies that the simplified von-Kármán plate model used frequently in the literature is as good as an approximation as the complete (and more complicated) model. As a
side note, it is shown that the dynamic response of any of the von-Kármán plate models, is completely different compared to the linearized plate model of Kirchhoff-Love for deflections of an order of
magnitude as the plate thickness.
• Finite elements
• Modal analysis
• Modeling errors
• Von-Kármán plate model
ASJC Scopus subject areas
• Modeling and Simulation
• General Materials Science
• Condensed Matter Physics
• Mechanics of Materials
• Mechanical Engineering
• Applied Mathematics
Dive into the research topics of 'Dynamic response of various von-Kármán non-linear plate models and their 3-D counterparts'. Together they form a unique fingerprint.
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NCERT Solutions for Class 12 Physics Chapter 2 Electrostatic Potential and Capacitance
NCERT Solutions for Class 12 Physics Chapter 2 – Free PDF Download
NCERT Solutions for Class 12 Physics Chapter 2 Electrostatic Potential and Capacitance includes the usage of many complicated equations and formulas that students learn in their Class 12. Also, the
PDF file of the NCERT Solutions for Class 12 Physics Electrostatic Potential and Capacitance is available here for free download. The PDF includes important questions, answers to questions from the
textbook, worksheets and other assignments.
The NCERT Solutions for Class 12 are essential to score good marks in the Class 12 board examination. These solutions are curated by individual subject matter experts according to the latest CBSE
Syllabus (2023-24) and the guidelines. Further, the NCERT Solutions for Class 12 Physics Chapter 2 Electrostatic Potential and Capacitance provided here can be used by students to understand the
concepts discussed in the chapter in detail.
Class 12 Physics NCERT Solutions Chapter 2 Electrostatic Potential and Capacitance Important Questions
Q 2.1) Two charges 5 × 10^-8 C and -3 x 10^-8 C are located 16 cm apart from each other. At what point (s) on the line joining the two charges is the electric potential zero? Take the potential at
infinity to be zero.
q[1] = 5 x 10^-8 C
q[2 ]= -3 x 10^-8 C
The two charges are at a distance, d = 16 cm = 0.16 m from each other.
Let us consider a point “P” over the line joining charges q[1] and q[2].
Let the distance of the considered point P from q[1 ] be ‘r’
Let us consider point P to have zero electric potential (V).
The electric potential at point P is the summation of potentials due to charges q[1] and q[2].
\(\begin{array}{l} V = \frac{1}{4\pi \epsilon _{o}} . \frac{q_{1}}{r} + \frac{1}{4\pi \epsilon _{o}}.\frac{q_{2}}{d – r} \end{array} \)
\(\begin{array}{l} \epsilon _{o} \end{array} \)
= permittivity of free space.
Putting V = 0, in equation (1), we get,
0 =
\(\begin{array}{l} \frac{1}{4\pi \epsilon _{o}} . \frac{q_{1}}{r} + \frac{1}{4\pi \epsilon _{o}}.\frac{q_{2}}{d – r} \end{array} \)
\(\begin{array}{l}\frac{1}{4\pi \epsilon _{o}}.\frac{q_{1}}{r} = -\frac{1}{4\pi \epsilon _{o}}.\frac{q_{2}}{d – r}\end{array} \)
\(\begin{array}{l}\frac{q_{1}}{r} = – \frac{q_{2}}{d – r}\end{array} \)
\(\begin{array}{l}\frac{5 \times 10^{-8}}{r} = – \frac{(- 3 \times 10^{-8})}{0.16 – r}\end{array} \)
5(0.16 – r) = 3r
0.8 = 8r
r = 0.1 m = 10 cm.
Therefore, at a distance of 10 cm from the positive charge, the potential is zero between the two charges.
Let us assume a point P at a distance ‘s’ from the negative charge is outside the system, having a potential zero.
So, for the above condition, the potential is given by
\(\begin{array}{l} V = \frac{1}{4\pi \epsilon _{o}} . \frac{q_{1}}{s} + \frac{1}{4\pi \epsilon _{o}}.\frac{q_{2}}{s – d} \end{array} \)
\(\begin{array}{l} \epsilon _{o} \end{array} \)
= permittivity of free space.
For V = 0, equation (2) can be written as :
0 =
\(\begin{array}{l} \frac{1}{4\pi \epsilon _{o}} . \frac{q_{1}}{s} + \frac{1}{4\pi \epsilon _{o}}.\frac{q_{2}}{s – d} \end{array} \)
\(\begin{array}{l} \frac{1}{4\pi \epsilon _{o}}.\frac{q_{1}}{s} = -\frac{1}{4\pi \epsilon _{o}}.\frac{q_{2}}{s – d} \end{array} \)
\(\begin{array}{l} \frac{q_{1}}{s} = – \frac{q_{2}}{s – d} \end{array} \)
\(\begin{array}{l} \frac{5 \times 10^{-8}}{s} = – \frac{(- 3 \times 10^{-8})}{s – 0.16} \end{array} \)
5(s – 0.16) = 3s
0.8 = 2s
S = 0.4 m = 40 cm.
Therefore, the potential is zero at a distance of 40 cm from the positive charge outside the system of charges.
Q 2.2) A regular hexagon of side 10 cm has a charge of 5 µC at each of its vertices. Calculate the potential at the centre of the hexagon.
The figure shows a regular hexagon containing charges q at each of its vertices.
q = 5 µC = 5 × 10^-6 C.
Length of each side of the hexagon, AB =BC = CD = DE = EF = FA = 10 cm.
The distance of the vertices from the centre O, d = 10 cm.
The electric potential at point O,
The electric potential, V =
\(\begin{array}{l} \frac{1}{4\pi \epsilon _{o}}.\frac{6q}{d} \end{array} \)
\(\begin{array}{l} \epsilon _{o} \end{array} \)
= Permittivity of free space and
\(\begin{array}{l} \frac{1}{4\pi \epsilon _{o}} \end{array} \)
= 9 x 10^9 Nm^2C^-2
V =
\(\begin{array}{l} \frac{9 × 10^{9} × 6 × 5 × 10^{-6}}{0.1} \end{array} \)
= 2.7 × 10^6 V.
Q 2.3) Two charges, 2 µC and -2 µC, are placed at points A and B, 6 cm apart.
(1) Identify the equipotential surface of the system.
(2) What is the direction of the electric field at every point on this surface?
(1) An equipotential surface is defined as the surface over which the total potential is zero. In the given question, the plane is normal to line AB. The plane is located at the mid-point of line AB
as the magnitude of the charges is the same.
(2) At every point on this surface, the direction of the electric field is normal to the plane in the direction of AB.
Q 2.4) A spherical conductor of radius 12 cm has a charge of 1.6 x 10^-7C distributed uniformly on its surface. What is the electric field
(1) inside the sphere?
(2) just outside the sphere?
(3) at a point 18 cm from the centre of the sphere?
(1) Given,
Radius of spherical conductor, r = 12cm = 0.12m
The charge is distributed uniformly over the surface, q = 1.6 x 10^-7 C.
The electric field inside a spherical conductor is zero.
(2) Electric field E, just outside the conductor, is given by the relation,
E =
\(\begin{array}{l} \frac{1}{4\pi \epsilon _{o}}.\frac{q}{r^{2}} \end{array} \)
ε[0] = permittivity of free space and
\(\begin{array}{l} \frac{1}{4\pi \epsilon _{o}} \end{array} \)
= 9 x 10^9 Nm^2C^-2
E =
\(\begin{array}{l} \frac{9 \times 10^{9} \times 1.6 \times 10^{-7}}{(0.12)^{2}}=10^5\,NC^{-1} \end{array} \)
Therefore, just outside the sphere, the electric field is 10^5 NC^-1.
(3) From the centre of the sphere, the electric field at a point 18m = E[1].
From the centre of the sphere, the distance of point d = 18 cm = 0.18m.
E[1] =
\(\begin{array}{l} \frac{1}{4\pi \epsilon _{o}} . \frac{q}{d^{2}} \end{array} \)
\(\begin{array}{l} \frac{9 \times 10^{9} \times 1.6 \times 10^{-7}}{(1.8 \times 10^{-2})^{2}} \end{array} \)
= 4.4 x 10^4 NC^-1
So, from the centre of the sphere, the electric field at a point 18 cm away is 4.4 x 10^4 NC^-1.
Q 2.5) A parallel plate capacitor with air between the plates has a capacitance of 8pF (1pF = 10^-12 F. What will be the capacitance if the distance between the plates is reduced by half and the
space between them is filled with a substance of dielectric constant 6?
Capacitance, C = 8pF.
In the first case, the parallel plates are at a distance ‘d’ and are filled with air.
Air has a dielectric constant, k = 1
Capacitance, C =
\(\begin{array}{l} \frac{k \times \epsilon _{o} \times A}{d} \end{array} \)
\(\begin{array}{l} \frac{\epsilon _{o} \times A}{d} \end{array} \)
… eq(1)
A = area of each plate
\(\begin{array}{l} \epsilon _{o} \end{array} \)
= permittivity of free space.
Now, if the distance between the parallel plates is reduced to half, then d[1] = d/2
Given the dielectric constant of the substance, k[1] = 6
Hence, the capacitance of the capacitor,
C[1] =
\(\begin{array}{l} \frac{k_{1} \times \epsilon _{o} \times A}{d_{1}} \end{array} \)
\(\begin{array}{l} \frac{6 \epsilon _{o} \times A}{d/2} \end{array} \)
\(\begin{array}{l} \frac{12 \epsilon _{o} A}{d} \end{array} \)
… (2)
Taking ratios of equations (1) and (2), we get,
C[1] = 2 x 6 C = 12 C = 12 x 8 pF = 96pF.
Hence, the capacitance between the plates is 96pF.
Q 2.6) Three capacitors connected in series have a capacitance of 9pF each.
(1) What is the total capacitance of the combination?
(2) What is the potential difference across each capacitor if the combination is connected to a 120 V supply?
(1) Given,
The capacitance of the three capacitors, C = 9 pF
Equivalent capacitance (c[eq]) is the capacitance of the combination of the capacitors given by
\(\begin{array}{l} \frac{1}{C_{eq}} = \frac{1}{C} + \frac{1}{C} + \frac{1}{C} = \frac{3}{C} = \frac{3}{9} =\frac{1}{3} \end{array} \)
\(\begin{array}{l} \frac{1}{C_{eq}} = \frac{1}{3} \end{array} \)
= C[eq] = 3 pF
Therefore, the total capacitance = 3pF.
(2) Given supply voltage, V = 100V
The potential difference (V[1]) across the capacitors will be equal to one-third of the supply voltage.
Therefore, V[1] =
\(\begin{array}{l} \frac{V}{3} \end{array} \)
\(\begin{array}{l} \frac{120}{3} \end{array} \)
= 40V.
Hence, the potential difference across each capacitor is 40V.
Q 2.7) Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel.
(1) What is the total capacitance of the combination?
(2) Determine the charge on each capacitor if the combination is connected to a 100 V supply.
(1) Given, C[1 ]= 2pF, C[2] = 3pF and C[3] = 4pF.
Equivalent capacitance for the parallel combination is given by C[eq].
Therefore, C[eq] = C[1] + C[2] + C[3] = 2 + 3 + 4 = 9pF
Hence, the total capacitance of the combination is 9pF.
(2) Supply voltage, V = 100V
The three capacitors have the same voltage, V = 100v
q = VC
q = charge
C = capacitance of the capacitor
V = potential difference
for capacitance, c = 2pF
q = 100 x 2 = 200pC = 2 x 10^-10C
for capacitance, c = 3pF
q = 100 x 3 = 300pC = 3 x 10^-10C
for capacitance, c = 4pF
q = 100 x 4 = 400pC = 4 x 10^-10 C
Q 2.8) In a parallel plate capacitor with air between the plates, each plate has an area of 6 x 10^-3 m ^2 and the distance between the plates is 3 mm. Calculate the capacitance of the capacitor. If
this capacitor is connected to a 100 V supply, what is the charge on each plate of the capacitor?
Solution: Given,
The area of the plate of the capacitor, A = 6 x 10^-3 m^2
Distances between the plates, d = 3mm = 3 x 10^-3 m
The voltage supplied, V = 100V
Capacitance of a parallel plate capacitor is given by, C =
\(\begin{array}{l} \frac{\epsilon \times A}{d} \end{array} \)
ε[o] = permittivity of free space = 8.854 × 10^-12 N^-1 m ^-2 C^-2
C =
\(\begin{array}{l}\frac{8.854 \times 10^{-12} \times 6 \times 10^{-3}}{3 \times 10^{-3}}\end{array} \)
= 17.71 x 10^-12 F = 17.71 pF.
Therefore, each plate of the capacitor has a charge of
q = VC = 100 × 17.71 x 10^-12 C = 1.771 x 10^-9 C
Q 2.9: Explain what would happen if, in the capacitor given in Exercise 2.8, a 3 mm thick mica sheet (of dielectric constant = 6) were inserted between the plates,
(a) while the voltage supply remained connected.
(b) after the supply was disconnected.
Answer 2.9:
(a) Dielectric constant of the mica sheet, k = 6
If the voltage supply remains connected, the voltage between the two plates will be constant.
Supply voltage, V = 100 V
Initial capacitance, C = 1.771 × 10^−11 F
New capacitance, C[1 ]= kC = 6 × 1.771 × 10^−11 F = 106 pF
New charge, q[1] = C[1]V = 106 × 100 pC = 1.06 × 10^–8 C
Potential across the plates remains 100 V.
(b) Dielectric constant, k = 6
Initial capacitance, C = 1.771 × 10^−11 F
New capacitance, C[1] = kC = 6 × 1.771 × 10^−11 F = 106 pF
If the supply voltage is removed, then there will be a constant amount of charge in the plates.
Charge = 1.771 × 10^−9 C
Potential across the plates is given by,
V[1] = q/C[1] =
\(\begin{array}{l}\frac{1.771 \times 10^{-9}}{106 \times 10^{-12}}\end{array} \)
= 16.7 V
Q 2.10) A 12pF capacitor is connected to a 50V battery. How much electrostatic energy is stored in the capacitor?
Solution: Given,
Capacitance of the capacitor, C = 12pF = 12 x 10^-12 F
Potential difference, V = 50 V
Electrostatic energy stored in the capacitor is given by the relation,
E =
\(\begin{array}{l} \frac{1}{2}\end{array} \)
CV^2 =
\(\begin{array}{l} \frac{1}{2}\end{array} \)
x 12 x 10^-12 x (50)^2 J = 1.5 x 10^-8 J
Therefore, the electrostatic energy stored in the capacitor is 1.5 x 10^-8 J.
Q 2.11) A 600pF capacitor is charged by a 200V supply. It is then disconnected from the supply and connected to another uncharged 600 pF capacitor. How much electrostatic energy is lost in the
Solution: Given,
Capacitance, C = 600pF
The potential difference, V = 200v
Electrostatic energy stored in the capacitor is given by :
E[1] =
\(\begin{array}{l} \frac{1}{2}\end{array} \)
CV^2 =
\(\begin{array}{l} \frac{1}{2}\end{array} \)
x (600 x 10^-12) x (200)^2J = 1.2 x 10^-5 J
According to the question, the source is disconnected from the 600pF and connected to another capacitor of 600pF, then equivalent capacitance (C[eq]) of the combination is given by,
\(\begin{array}{l} \frac{1}{C_{eq}} \end{array} \)
\(\begin{array}{l} \frac{1}{C} \end{array} \)
\(\begin{array}{l} \frac{1}{C} \end{array} \)
\(\begin{array}{l} \frac{1}{C_{eq}} \end{array} \)
\(\begin{array}{l} \frac{1}{600} \end{array} \)
\(\begin{array}{l} \frac{1}{600} \end{array} \)
\(\begin{array}{l} \frac{2}{600} \end{array} \)
\(\begin{array}{l} \frac{1}{300} \end{array} \)
C[eq] = 300pF
New electrostatic energy can be calculated by:
E[2] =
\(\begin{array}{l} \frac{1}{2}\end{array} \)
CV^2 =
\(\begin{array}{l} \frac{1}{2}\end{array} \)
x 300 x 10^-12 x (200)^2 J = 0.6 x 10^-5 J
Loss in electrostatic energy,
E = E[1] – E[2]
E = 1.2 x 10^-5 – 0.6 x 10^-5 J = 0.6 x 10^-5 J = 6 x 10^-6 J
Therefore, the electrostatic energy lost in the process is 6 x 10^-6 J.
Q 2.12) A charge of 8 mC is located at the origin. Calculate the work done in taking a small charge of –2 × 10^–9 C from a point P (0, 0, 3 cm) to a point Q (0, 4 cm, 0) via a point R (0, 6 cm, 9
Charge located at the origin, q = 8 mC = 8 x 10^-3 C
The magnitude of the charge taken from the point P to R and then to Q, q[1] = – 2 x 10^-9 C
Here, OP= d[1]= 3 cm = 3 x 10^-2 m
OQ = d[2]= 4 cm = 4 x 10^-2 m
Potential at the point P,
\(\begin{array}{l}V_{1}=\frac{q}{4\pi \epsilon _{0}d_{1}}\end{array} \)
Potential at the point Q,
\(\begin{array}{l}V_{2}=\frac{q}{4\pi \epsilon _{0}d_{2}}\end{array} \)
The work done (W) is independent of the path
Therefore, W = q[1][V[1] – V[2]]
\(\begin{array}{l}W=q_{1}\left [ \frac{q}{4\pi \epsilon _{0}d_{2}} – \frac{q}{4\pi \epsilon _{0}d_{1}} \right ]\end{array} \)
\(\begin{array}{l}W=\frac{qq_{1}}{4\pi \epsilon _{0}}\left [ \frac{1}{d_{2}} – \frac{1}{d_{1}} \right ]\end{array} \)
\(\begin{array}{l}\frac{1}{4\pi \epsilon _{0}}=9 \times 10^{9}Nm^{2}C^{-2}\end{array} \)
\(\begin{array}{l}W = 9 \times 10^{9}\times 8\times 10^{-3}\times (-2\times 10^{-9})\left [ \frac{1}{4\times 10^{-2}}-\frac{1}{3\times 10^{-2}} \right ]\end{array} \)
= -144 x 10^-3 x (-100/12)
= 1.2 ^ Joule
Therefore, the work done during the process is 1.2 J
Q 2.13) A cube of side b has a charge q at each of its vertices. Determine the potential and electric field due to this charge array at the centre of the cube.
Sides of the cube = b
Charge at the vertices = q
Diagonal of one of the sides of the cube
\(\begin{array}{l}d^{2}= \sqrt{b^{2}+b^{2}}= \sqrt{2b^{2}}\end{array} \)
d = b√2
Length of the diagonal of the cube
\(\begin{array}{l}l^{2}= \sqrt{d^{2}+b^{2}}= \sqrt{2b^{2}+b^{2}}=\sqrt{3b^{2}}\end{array} \)
l = b√3
The distance between one of the vertices and the centre of the cube is
r = l/2 =(b√3/2)
The electric potential (V) at the centre of the cube is due to the eight charges at the vertices
V = 8q/4πε[0]
\(\begin{array}{l}V = \frac{8q}{4\pi \epsilon _{0}\left ( b\frac{\sqrt{3}}{2} \right )}\end{array} \)
\(\begin{array}{l}= \frac{4q}{\sqrt{3}\pi \epsilon _{0}b}\end{array} \)
Therefore, the potential at the centre of the cube is
\(\begin{array}{l}\frac{4q}{\sqrt{3}\pi \epsilon _{0}b}\end{array} \)
The electric field intensity at the centre of the cube, due to the eight charges, is zero. The charges are distributed symmetrically with respect to the centre of the cube. Therefore, they get
Q 2.14) Two tiny spheres carrying charges 1.5 µC and 2.5 µC are located 30 cm apart. Find the potential and electric field:
(a) at the mid-point of the line joining the two charges
(b) at a point 10 cm from this midpoint in a plane normal to the line and passing through the mid-point.
Two tiny spheres carrying charges are located at points A and B
The charge at point A, q[1]= 1.5 µC
The charge at point B, q[2] = 2.5 µC
The distance between the two charges = 30 cm = 0.3 m
(a) Let O be the midpoint. Let V[1] and E[1] be the potential and electric fields, respectively, at the midpoint.
V[1] = Potential due to charge at A + Potential due to charge at B
\(\begin{array}{l}V_{1}=\frac{q_{1}}{4\pi \epsilon _{0}(\frac{d}{2})}+ \frac{q_{2}}{4\pi \epsilon _{0}(\frac{d}{2})}\end{array} \)
\(\begin{array}{l}V_{1}=\frac{1}{4\pi \epsilon _{0}(\frac{d}{2})}(q_{1}+q_{2})\end{array} \)
ε[0] = Permittivity of the free space
\(\begin{array}{l}\frac{1}{4\pi \epsilon _{0}} = 9 \times 10^{9}NC^{2}m^{-2}\end{array} \)
\(\begin{array}{l}V_{1}=\frac{9\times 10^{9}\times 10^{-6}}{(\frac{0.30}{2})}(2.5+ 1.5)= 2.4\times 10^{5}V\end{array} \)
Electric field at O, E[1] = Electric field due to q[2] – Electric field due to q[1]
\(\begin{array}{l}=\frac{q_{2}}{4\pi \epsilon _{0}(\frac{d}{2})^{2}}- \frac{q_{1}}{4\pi \epsilon _{0}(\frac{d}{2})^{2}}\end{array} \)
\(\begin{array}{l}=\frac{9\times 10^{9}}{(\frac{0.30}{2})^{2}}\times 10^{-6}\times (2.5-1.5)\end{array} \)
= 400 x 10^3 V/m
Therefore, the potential at the midpoint is 2.4 x 10^5 V, and the electric field at the midpoint is 400 x 10^3 V/m.
(b) Consider a point Z such that the distance OZ = 10 cm = 0.1 m, as shown in the figure.
Let V[2] and E[2] be the potential and electric field, respectively, at point Z. The distance
\(\begin{array}{l}BZ = AZ = \sqrt{(0.1)^{2}+(0.15)^{2}}= 0.18 m\end{array} \)
The potential at V[2] = Potential due to the charge at A + Potential due to the charge at B
\(\begin{array}{l}= \frac{q_{1}}{4\pi \epsilon _{0}(AZ)}+\frac{q_{2}}{4\pi \epsilon _{0}(BZ)}\end{array} \)
\(\begin{array}{l}=\frac{9\times 10^{9}\times 10^{-6}}{0.18}(1.5+2.5)\end{array} \)
= 2 x 10^5 V
The electric field due to q[1] at Z
\(\begin{array}{l}E_{A}=\frac{q_{1}}{4\pi \epsilon _{0}(AZ)^{2}}\end{array} \)
\(\begin{array}{l}= \frac{9\times 10^{9}\times 1.5\times 10^{-6}}{(0.18)^{2}}\end{array} \)
= 416 x 10^3 V/m
The electric field due to q[2 ]at Z
\(\begin{array}{l}E_{B}=\frac{q_{2}}{4\pi \epsilon _{0}(BZ)^{2}}\end{array} \)
\(\begin{array}{l}= \frac{9\times 10^{9}\times 2.5\times 10^{-6}}{(0.18)^{2}}\end{array} \)
= 694 x 10^3 V/m
The resultant field intensity at Z
\(\begin{array}{l}E=\sqrt{E_{A}^{2}+E_{B}^{2}+2E_{A}E_{B}cos2\theta }\end{array} \)
From the figure we get cos θ = (0.10/0.18) = 5/9 = 0.5556
θ = cos ^-1 (0.5556) = 56.25
2θ = 2 x 56.25 = 112.5^0
cos 2θ = – 0.38
\(\begin{array}{l}E=\sqrt{(0.416\times 10^{6})^{2}+(0.69\times 10^{6})^{2}+2\times 0.416\times 0.69\times 10^{12}\times (-0.38)}\end{array} \)
= 6.6 x 10^5 V/m
Therefore the potential at the point Z is 694 x 10^3 V/m, and the electric field is 6.6 x 10^5 V/m.
Q 2.15) A spherical conducting shell of inner radius r[1] and outer radius r[2] has a charge Q.
(a) A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell?
(b) Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical but has any irregular shape? Explain.
(a) If a charge +q is placed at the centre of the shell, a charge of magnitude -q is induced in the inner surface of the shell. Therefore, the surface charge density at the inner surface of the shell
is given by the relation,
σ[1] = Total charge / Inner surface area = – q/4πr[1]^2——–(1)
A charge +q is induced on the outer surface of the shell. The total charge on the outer surface of the shell is Q+q. Surface charge density at the outer surface of the shell,
σ[2] = Total charge / Outer surface area = (Q+q)/4πr[1]^2————(2)
(b) Yes, the electric field inside a cavity (with no charge) will be zero, even if the shell is not spherical but has an irregular shape. Take a closed loop, part of which is inside the cavity along
a field line and the rest inside the conductor. Since the field inside the conductor is zero, this gives a net work done by the field in carrying a test charge over a closed loop. We know this is
impossible for an electrostatic field. Hence, there are no field lines inside the cavity (i.e., no field) and no charge on the inner surface of the conductor, whatever its shape.
Q 2.16) (a) Show that the normal component of the electrostatic field has a discontinuity from one side of a charged surface to another given by
\(\begin{array}{l}\left ( E_{2}-E_{1} \right ).\hat{n}=\frac{\sigma }{\epsilon _{0}}\end{array} \)
\(\begin{array}{l}\hat{n}\end{array} \)
is a unit vector normal to the surface at a point, and σ is the surface charge density at that point. (The direction of
\(\begin{array}{l}\hat{n}\end{array} \)
is from side 1 to side 2.) Hence, show that just outside a conductor, the electric field is
\(\begin{array}{l}\sigma \hat{n}/\epsilon _{0}\end{array} \)
(b) Show that the tangential component of the electrostatic field is continuous from one side of a charged surface to another. [Hint: For (a), use Gauss’s law. For (b), use the fact that work done by
the electrostatic field on a closed loop is zero.].
(a) Let E[1] be the electric field on one side of the charged body and E[2] is the electric field on the other side of the charged body. If the infinite plane charged body has a uniform thickness,
the electric field due to one of the surfaces of the charged body is
\(\begin{array}{l}\vec{E_{1}}=-\frac{\sigma }{2\epsilon _{0}}\hat{n}\end{array} \)
\(\begin{array}{l}\hat{n}\end{array} \)
= unit vector normal to the surface at a point
σ = surface charge density at that point
The electric field due to the other surface of the charged body is
\(\begin{array}{l}\vec{E_{2}}=\frac{\sigma }{2\epsilon _{0}}\hat{n}\end{array} \)
The electric field at any point due to the charge surfaces
\(\begin{array}{l}\vec{E_{2}}-\vec{E_{1}}=\frac{\sigma }{2\epsilon _{0}}\hat{n} + \frac{\sigma }{2\epsilon _{0}}\hat{n} = \frac{\sigma }{\epsilon _{0}}\hat{n}\end{array} \)
Since inside the conductor,
\(\begin{array}{l}\vec{E_{1}}= 0\end{array} \)
\(\begin{array}{l}\vec{E_{2}}-\vec{E_{1}}= \frac{\sigma }{\epsilon _{0}}\hat{n}\end{array} \)
Therefore, the electric field just outside the conductor is
\(\begin{array}{l}\frac{\sigma }{\epsilon _{0}}\hat{n}\end{array} \)
(b) When a charged particle is moved from one point to the other on a closed loop, the work done by the electrostatic field is zero. Hence, the tangential component of the electrostatic field is
continuous from one side of a charged surface to the other.
Q 2.17) A long charged cylinder of linear charged density λ is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders?
Let the length of the charged cylinder and the hollow co-axial conducting cylinder be L
The charge density of the long-charged cylinder is λ
Let E be the electric field in the space between the two cylinders
According to Gauss’s theorem, the electric flux through the Gaussian surface is given as Φ = E (2πd)L
d is the distance between the common axis of the cylinders
Therefore , Φ = E (2πd)L = q/ε[0]
Here, q is the charge on the inner surface of the outer cylinder
ε[0 ]is the permittivity of the free space
E (2πd)L = λL/ε[0]
E = λ/2πdε[0]
Therefore, the electric field in the space between the two cylinders is λ/2πdε[0]
Q 2.18) In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å:
(a) Estimate the potential energy of the system in eV, taking the zero of the potential energy at the infinite separation of the electron from the proton.
(b) What is the minimum work required to free the electron, given that its kinetic energy in orbit is half the magnitude of potential energy obtained in (a)?
(c) What are the answers to (a) and (b) above if the zero of potential
energy is taken at 1.06 Å separation?
The distance between the proton and electron of the hydrogen atom, d = 0.53 Å
Charge of the electron, q[1] = -1.6 x 10^-19 C
Charge of the proton, q[2] = +1.6 x 10^-19 C
(a) At an infinite separation of electron and proton, potential energy is zero
The potential energy of the system = Potential energy at infinity – Potential energy at distance d
= 0 –
\(\begin{array}{l}\frac{q_{1}q_{2}}{4\pi \epsilon _{0}d}\end{array} \)
ε[0] is the permittivity of the free space
\(\begin{array}{l}\frac{1}{4\pi \epsilon _{0}}=9×10^{9}Nm^{2}C^{-2}\end{array} \)
Potential energy= 0 –
\(\begin{array}{l}\frac{9\times 10^{9}\times (1.6 \times 10^{-19})^{2}}{0.53\times 10^{10}} = -43.7 x 10^{-19}J\end{array} \)
Potential energy = -43.7 x 10^-19/1.6 x 10^-19 = -27.2 eV [Since 1.6 x 10^-19 J = 1 eV]
Therefore, the potential energy of the system is -27.2 eV
(b) Half of the magnitude of the potential energy is equal to the kinetic energy
Kinetic energy = |V|/2 = (1/2) x (27.2) = 13. 6 eV
Total energy = Kinetic energy + potential energy
= 13.6 eV – 27.2 eV
Total energy = – 13.6 eV
Therefore, the minimum work required to free an electron is – 13.6 eV
(c) When the zero of the potential energy is taken as d[1]= 1.06 Å
The potential energy of the system = Potential energy at d[1] – Potential energy at d
\(\begin{array}{l}\frac{q_{1}q_{2}}{4\pi \epsilon _{0}d_{1}}\end{array} \)
– 27.2 eV
\(\begin{array}{l}\frac{9\times 10^{9}\times (1.6 \times 10^{-19})^{2}}{1.06\times 10^{10}}\end{array} \)
-27.2 eV
= 21.73 x 10^-19 J – 27.2 eV
= 13.58 eV -27.2 eV [Since 1.6 x 10^-19 J = 1 eV]
= -13.6 eV
Q 2.19) If one of the two electrons of the H[2] molecule is removed, we get a hydrogen molecular ion H[2^+]. In the ground state of an H[2^+], the two protons are separated by roughly 1.5 Å, and the
electron is roughly 1 Å from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy.
Charge of the 1^st proton, q[1] = 1.6 x 10^-19 C
Charge of the 2^nd proton, q[2] = 1.6 x 10^-19 C
Charge of the electron, q[3] = -1.6 x 10^-19 C
Distance between the 1^st and the 2^nd proton, d[1] = 1.5 x 10^-10 m
Distance between the 1^st proton and the electron, d[2] = 1 x 10^-10 m
Distance between the 2nd proton and the electron, d[3] = 1 x 10^-10 m
The potential energy at infinity is zero
Therefore, the potential energy of the system is
\(\begin{array}{l}V= \frac{q_{1}q_{2}}{4\pi \epsilon _{0}d_{1}}+\frac{q_{2}q_{3}}{4\pi \epsilon _{0}d_{3}}+ \frac{q_{3}q_{1}}{4\pi \epsilon _{0}d_{2}}\end{array} \)
\(\begin{array}{l}V = \frac{1}{4\pi \epsilon _{0}}\left [ \frac{q_{1}q_{2}}{d_{1}}+\frac{q_{2}q_{3}}{d_{3}}+\frac{q_{3}q_{1}}{d_{2}}\right ]\end{array} \)
Substituting (1/4πε[0]) = 9 x 10^9 Nm^2C^-2 we get
\(\begin{array}{l}V= 9\times 10^{9}\left [ \frac{1.6\times 10^{-19}\times1.6\times 10^{-19} }{1.5\times 10^{-10}}+\frac{1.6\times 10^{-19}\times (-1.6\times 10^{-19})}{1\times 10^{-10}}+\frac{(-1.6\
times 10^{-19})(1.6\times 10^{-19})}{1\times 10^{-10}}\right ]\end{array} \)
\(\begin{array}{l}V=\frac{9\times 10^{9}\times 10^{-19}\times 10^{-19}}{10^{-10}}\left [ \frac{(1.6)^{2}}{1.5}-(1.6)^{2}-(1.6)^{2} \right ]\end{array} \)
=-30.7 x 10^-19 J
=-19.2 eV (1eV = 1.6 x 10^-19 J)
Therefore, the potential energy of the system is -19.2 eV.
Q 2.20) Two charged conducting spheres of radii a and b are connected to each other by a wire. What is the ratio of electric fields at the surfaces of the two spheres? Use the result obtained to
explain why the charge density on the sharp and pointed ends of a conductor is higher than on its flatter portions.
Let A be the sphere of radius a, Charge Q[A] and capacitance C[A]
Let B be the sphere of radius b, Charge Q[B] and capacitance C[B]
The conducting spheres are connected by a wire; therefore, the potential of both capacitors will be V
The electric field due to a,
\(\begin{array}{l}E_{A}=\frac{Q_{A}}{4\pi \epsilon _{0}a^{2}}\end{array} \)
The electric field due to b,
\(\begin{array}{l}E_{B}=\frac{Q_{B}}{4\pi \epsilon _{0}b^{2}}\end{array} \)
The ratio of electric fields at the surface of the spheres is
\(\begin{array}{l}\frac{E_{A}}{E_{B}}=\frac{Q_{A}}{4\pi \epsilon _{0}a^{2}}\times \frac{b^{2}4\pi \epsilon _{0}}{Q_{B}}\end{array} \)
\(\begin{array}{l}\frac{E_{A}}{E_{B}}=\frac{Q_{A}}{Q_{B}}\times \frac{b^{2}}{a^{2}}\end{array} \)
\(\begin{array}{l}\frac{Q_{A}}{Q_{B}}=\frac{C_{A}V}{C_{B}V}\end{array} \)
\(\begin{array}{l}\frac{Q_{A}}{Q_{B}}=\frac{a}{b}\end{array} \)
Putting equation (2) in equation (1), we get
\(\begin{array}{l}\frac{E_{A}}{E_{B}}=\frac{a}{b}\times \frac{b^{2}}{a^{2}}\end{array} \)
Therefore, the ratio of the electric field at the surface is b/a
A sharp and pointed end is like a sphere of a very small radius, and the flat portion is like a sphere of a large radius. Therefore, the charge density of pointed ends is higher than the flat
Q 2.21) Two charges –q and +q, are located at points (0, 0, –a) and (0, 0, a), respectively.
(a) What is the electrostatic potential at the points (0, 0, z) and (x, y, 0)?
(b) Obtain the dependence of potential on the distance r of a point from the origin when r/a >> 1.
(c) How much work is done in moving a small test charge from the point (5,0,0) to (–7,0,0) along the x-axis? Does the answer change if the path of the test charge between the same points is not along
the x-axis?
(a) Two charges –q and +q, are located at points (0, 0, –a) and (0, 0, a), respectively. They will form a dipole. The point (0, 0, z) is on the axis of the dipole and (x,y,0) is normal to the
dipole. The electrostatic potential at (x,y,0) is zero. The electrostatic potential at (0,0,z) is given by
\(\begin{array}{l}V = \frac{1}{4\pi \epsilon _{0}}\left ( \frac{q}{z-a} \right )+\frac{1}{4\pi \epsilon _{0}}\left ( -\frac{q}{z+a} \right )\end{array} \)
\(\begin{array}{l}V = \frac{q(z+a-z+a)}{4\pi \epsilon _{0}(z^{2}-a^{2})}\end{array} \)
\(\begin{array}{l}V = \frac{q(2a)}{4\pi \epsilon _{0}(z^{2}-a^{2})}\end{array} \)
\(\begin{array}{l}= \frac{p}{4\pi \epsilon _{0}(z^{2}-a^{2})}\end{array} \)
ε[0] = Permittivity of free space
p = dipole moment of the system= q x 2a
(b) The distance “r” is much larger than half of the distance between the two charges. Therefore, the potential at the point r is inversely proportional to the square of the distance, i.e. V∝(1/r^2).
(c) x,y plane is a equipotential surface and x-axis is a equipotential line. Therefore, the change in potential (dV) along the x-axis will be zero. The work done in moving a small test charge from
the point (5,0,0) to (–7,0,0) along the x-axis is given by
Potential at (5,0,0)
\(\begin{array}{l}V_{1} = \frac{1}{4\pi \epsilon _{0}}\left ( \frac{q}{\sqrt{(5-0)^{2}-a^{2}}} \right )+\frac{1}{4\pi \epsilon _{0}}\left ( -\frac{q}{\sqrt{(5-0)^{2}-(-a)^{2}}} \right )=0\end{array}
Potential at (-7,0,0)
\(\begin{array}{l}V_{2} = \frac{1}{4\pi \epsilon _{0}}\left ( \frac{q}{\sqrt{((-7)-0)^{2}-a^{2}}} \right )+\frac{1}{4\pi \epsilon _{0}}\left ( -\frac{q}{\sqrt{((-7)-0)^{2}-(-a)^{2}}} \right )= 0\end
{array} \)
V[2] – V[1] = 0
Work done = Charge (q) x Change in Potential (V[2] – V[1])
Since the change in potential is zero, the work done is also zero.
The change in potential is independent of the path taken between the two points. Therefore, the work done in moving a point charge will remain zero.
Q 2.22) Figure below shows a charge array known as an electric quadrupole. For a point on the axis of the quadrupole, obtain the dependence of potential on r for r/a >> 1, and contrast your results
with that due to an electric dipole and an electric monopole (i.e., a single charge).
Four charges are placed at points A, B, B and C, respectively.
Let us consider point P, located at the axis of the quadrupole.
It can be considered that the electric quadrupole has three charges.
The charge +q is placed at A
The charge -2q is placed at B
The charge +q is placed at C
AB = BC = a
BP = r
PA = r + a
PZ = r – a
Therefore, the electrostatic potential due to the system of three charges is
\(\begin{array}{l}V =\frac{1}{4\pi \epsilon _{0}}\left [ \frac{q}{PA}-\frac{2q}{PB}+\frac{q}{PC} \right ]\end{array} \)
\(\begin{array}{l}\begin{array}{l} =\frac{1}{4 \pi \in_{0}}\left[\frac{q}{r+a}-\frac{2 q}{r}+\frac{q}{r-a}\right] \\ =\frac{q}{4 \pi \in_{0}}\left[\frac{r(r-a)-2(r+a)(r-a)+r(r+a)}{r(r+a)(r-a)}\right]
\\ =\frac{q}{4 \pi \in_{0}}\left[\frac{r^{2}-r a-2 r^{2}+2 a^{2}+r^{2}+r a}{r\left(r^{2}-a^{2}\right)}\right]=\frac{q}{4 \pi \in_{0}}\left[\frac{2 a^{2}}{r\left(r^{2}-a^{2}\right)}\right] \\ =\frac{2
q a^{2}}{4 \pi \in_{0} r^{3}\left(1-\frac{a^{2}}{r^{2}}\right)} \end{array}\end{array} \)
Since r/a >>1,
a/r<< 1
Therefore, a^2/r^2 is negligible
So we get,
\(\begin{array}{l}V = \frac{2qa^{2}}{4\pi \epsilon _{0}r^{3}}\end{array} \)
Therefore we get,
V∝ 1/r^3
However, for a dipole, V ∝ 1/r^2
And for a monopole, V ∝ 1/r
Q 2.23) An electrical technician requires a capacitance of 2 µF in a circuit across a potential difference of 1 kV. A large number of 1 µF capacitor are available to him, each of which can withstand
a potential difference of not more than 400 V. Suggest a possible arrangement that requires the minimum number of capacitors.
Required Capacitance, C= 2μF
Potential difference, V = 1 kV = 1000 V
The capacitance of each capacitor, C[1] = 1μF
Potential difference that the capacitors can withstand, V[1] = 400 V
Suppose a number of capacitors are connected in series and then connected parallel to each other. Then the number of capacitors in each row is given by
1000/400 = 2.5
Therefore, the number of capacitors connected in series is three.
So the capacitance of each row is
\(\begin{array}{l}\frac{1}{1+1+1}= \frac{1}{3}\mu F\end{array} \)
Let there be n parallel rows. Each of these rows will have 3 capacitors. Therefore, the equivalent capacitance of the circuit is given as
\(\begin{array}{l}\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+——-n \, terms\end{array} \)
The required capacitance of the circuit is 2μF
Therefore, n/3 = 2
n = 6
Therefore, there are 6 rows of three capacitors in the circuit. A minimum of 6 x 3 = 18 capacitors are required.
Q 2.24) What is the area of the plates of a 2 F parallel plate capacitor, given that the separation between the plates is 0.5 cm? [You will realise from your answer why ordinary capacitors are in the
range of µF or less. However, electrolytic capacitors do have a much larger capacitance (0.1 F) because of the very minute separation between the conductors.]
The capacitance of the parallel plate capacitor, C = ε[0]A/d
The capacitance of the capacitor, C=2 F
Separation between the plates, d= 0.5 cm = 0.5 x 10^-2 m
ε[0 ]= permittivity of the free space = 8.85 x 10^-12 C^2N^-1m^-2
Area of the plates, A = Cd/ε[0]
A = [2 x 0.5 x 10^-2]/8.85 x 10^-12
= 1130 x 10^6 m^2
Q 2.25) Obtain the equivalent capacitance of the network in Figure. For a 300 V supply, determine the charge and voltage across each capacitor.
The above figure can be redrawn as given below
The capacitors C[2] and C[3] are connected in series. The equivalent capacitance C’
\(\begin{array}{l}\frac{1}{C’}= \frac{1}{C_{2}}+\frac{1}{C_{3}} = \frac{1}{200}+\frac{1}{200}=\frac{2}{200}\end{array} \)
C’ = 100 pF
The capacitors C’ and C[1] are parallel. The equivalent capacitance is C” = C’ + C[1]
C” = 100 + 100 = 200 pF
C” and C[4] are connected in series. Let the equivalent capacitance be C
\(\begin{array}{l}\frac{1}{C} = \frac{1}{C^{”}}+\frac{1}{C_{4}} = \frac{1}{200}+\frac{1}{100}=\frac{3}{200}\end{array} \)
C = 200/3 pF
Hence, the equivalent capacitance of the circuit is 200/3 pF
Total charge, Q = CV =
\(\begin{array}{l}\frac{200}{3}\times 10^{-12}\times 300 = 2 \times 10^{-8}C\end{array} \)
Q = Q[4] = 2 x 10^-8 C
Potential difference across C[4, ]V[4] = Q/C[4]
= (2 x 10 ^-8)/(100 x 10^-12) = 200 V
Potential difference across C”, V” = 300 V – 200 V = 100 V
Potential difference across C[1, ]V[1] = V” = 100 V
Charge across C[1], Q[1] = C[1]V[1] = (100 x 10^-12) x 100 = 10^-8 C
The charge across C[2] and C[3], Q[2] = Q – Q[1] = 2 x 10^-8 – 10^-8
= 10^-8 C
Potential across C[2 , ]V[2] = Q[2]/C[2] = 10^-8/ (200 x 10^-12) = 50 V
Potential across C[3], V[3] = Q[2]/C[3] = 10^-8/(200 x 10^-12) = 50 V
Q[1] = 10^-8 C , V[1] = 100 V
Q[2] = 10^-8 C, V[2] = 50 V
Q[2] = Q[3] = 10^-8 C, V[3] = 50 V
Q[4] = 2 x 10^-8 C , V[4] = 200 V
Q 2.26) The plates of a parallel plate capacitor have an area of 90 cm^2 each and are separated by 2.5 mm. The capacitor is charged by connecting it to a 400 V supply.
(a) How much electrostatic energy is stored by the capacitor?
(b) View this energy as stored in the electrostatic field between the plates and obtain the energy per unit volume u. Hence, arrive at a relation between u and the magnitude of electric field E
between the plates.
Area of the plates of a parallel plate capacitor, A = 90 cm^2 = 90 x 10^-4 m^2
Separation between the plates, d = 2.5 mm = 2.5 x 10^-3 m
The potential difference across the plates, V = 400 V
The capacitance of the capacitor, C = ε[0]A/d
Electrostatic energy stored in the capacitor, E = (1/2) CV^2
\(\begin{array}{l}= \frac{1}{2}\frac{\epsilon _{0}A}{d}V^{2}\end{array} \)
\(\begin{array}{l}= \frac{1}{2}\frac{8.85 \times 10^{-12}\times 90\times 10^{-4}\times 400^{2}}{2\times 2.5\times 10^{-3}} = 2.55 \times 10^{-6}J\end{array} \)
Therefore, the electrostatic energy stored by the capacitor is 2.55 x 10^-6 J
The volume of the capacitor, V = A x d
= 90 x 10^-4 x 25 x 10^-3
= 2.25 x 10^-4 m^3
Energy stored in the capacitor per unit volume is
u = E/V =
\(\begin{array}{l}\frac{\frac{1}{2}CV^{2}}{Ad} = \frac{\frac{\epsilon _{0}A}{2d}V^{2}}{Ad}= \frac{1}{2}\epsilon _{0}(\frac{V}{d})^{2}\end{array} \)
Here, V/d = Electric intensity = E
\(\begin{array}{l}u = \frac{1}{2}\epsilon _{0}E^{2}\end{array} \)
Q 2.27) A 4 µF capacitor is charged by a 200 V supply. It is then disconnected from the supply and connected to another uncharged 2 µF capacitor. How much electrostatic energy of the first capacitor
is lost in the form of heat and electromagnetic radiation?
The capacitance of the capacitor, C[1]= 4 μF
Supply voltage, V[1] = 200 V
The capacitance of the uncharged capacitor, C[2]= 2 μF
Electrostatic energy stored in C[1] is given as
E[1] = (1/2)C[1]V[1]^2
= (1/2) x 4 x 10^-6 x (200)^2
= 8 x 10^-2 J
When C[1] is disconnected from the power supply and connected to C[2], the voltage acquired by it is V[2].
According to the law of conservation of energy, the initial charge on the capacitor C[1] is equal to the final charge on the capacitors C[1] and C[2].
V[2] (C[1] + C[2]) = C[1]V[1]
V[2] (4+ 2) x 10^-6 = 4 x 10^-6 x 200
V[2] = (400/3) V
The electrostatic energy of the combination is
E[2] = (1/2)(C[1]+C[2]) V[1]^2
= (1/2) x (2+4) x 10^-6 x (400/3)^2
= 5.33 x 10^-2 J
Hence, the amount of electrostatic energy lost by capacitor C[1] = E[1] – E[2]
= 0.08 – 0.0533 = 0.0267
= 2.67 x 10^-2 J
Q 2.28) Show that the force on each plate of a parallel plate capacitor has a magnitude equal to (½) QE, where Q is the charge on the capacitor, and E is the magnitude of the electric field between
the plates. Explain the origin of the factor ½.
Let F be the force required to separate the plates of the parallel plate capacitors.
Let Δx be the distance between the two plates.
Therefore, the work done to separate the plates, W = F Δx. As a result, the potential energy of the capacitor increases by an amount equal to uAΔx.
here, u = energy density
A = area of each plate
d = distance between the plates
V = potential difference across the plates
The work done will be equal to the increase in potential energy
FΔx = uAΔx
F = uA
\(\begin{array}{l}u = \frac{1}{2}\epsilon _{0}E^{2}\end{array} \)
in above equation
\(\begin{array}{l}F= uA = \frac{1}{2}\epsilon _{0}E^{2}A\end{array} \)
The electric intensity, E = V/d
\(\begin{array}{l}F= uA = \frac{1}{2}\epsilon _{0}\frac{V}{d}EA\end{array} \)
However, capacitance,
\(\begin{array}{l}C = \frac{\epsilon _{0}A}{d}\end{array} \)
Therefore, F = (1/2) (CV) E
Charge on the capacitor is given as Q = CV
Therefore, F = (1/2)QE
The electric field just outside the conductor is E and inside the conductor is zero. Hence, the average of the electric field E/2 is equal to the force.
Q 2.29) A spherical capacitor consists of two concentric spherical conductors held in position by suitable insulating supports (as shown in the figure). Show that the capacitance of a spherical
capacitor is given by
\(\begin{array}{l}C = \frac{4\pi \epsilon _{0}r_{1}r_{2}}{r_{1}-r_{2}}\end{array} \)
Where r[1] and r[2] are the radii of outer and inner spheres, respectively.
The radius of the outer shell = r[1]
The radius of the inner shell = r[2]
The charge on the inner surface of the outer shell = +Q
The charge on the outer surface of the inner shell = -Q
The potential difference between the two shells is given as
\(\begin{array}{l}V = \frac{Q}{4\pi \epsilon _{0}r_{2}}-\frac{Q}{4\pi \epsilon _{0}r_{1}}\end{array} \)
ε[0] = Permittivity of free space
\(\begin{array}{l}V = \frac{Q}{4\pi \epsilon _{0}}\left [ \frac{1}{r_{2}}-\frac{1}{r_{1}} \right ]\end{array} \)
\(\begin{array}{l}V = \frac{Q(r_{1}-r_{_{2}})}{4\pi \epsilon _{0}r_{1}r_{2}}\end{array} \)
The capacitance of the given system is
C = Charge (Q)/ Potential difference (V)
\(\begin{array}{l}C = \frac{4\pi \epsilon _{0}r_{1}r_{2}}{(r_{1}-r_{_{2}})}\end{array} \)
Q 2.30) A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed, and the inner sphere is given a charge of 2.5 µC. The space between
the concentric spheres is filled with a liquid of dielectric constant 32.
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere?
(c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller.
Radius of the inner sphere, r[1] = 12 cm = 0.12 m
Radius of the outer sphere, r[2] = 13 cm = 0.13 m
Charge on the inner sphere, q = 2.5 μC = 2.5 x 10^-6 C
The dielectric constant of the liquid, ε[r]= 32
(a) Capacitance of the capacitor is given by the relation,
\(\begin{array}{l}C = \frac{4\pi \epsilon _{0}\epsilon _{r}r_{1}r_{2}}{(r_{1}-r_{_{2}})}\end{array} \)
ε[0] = permittivity of the free space = 8.85 x 10^-12 C^2 N^-1 m^-2
\(\begin{array}{l}\frac{1}{4\pi \epsilon _{0}}= 9\times 10^{9} Nm^{2}C^{-2}\end{array} \)
\(\begin{array}{l}C = \frac{32\times 0.12\times 0.13 }{(9\times10^{9}\times (0.13-0.12) }= 5.5\times 10^{-9}F\end{array} \)
Therefore, the capacitance of the capacitor is approximately 5.5 x 10^-9 F.
(b) Potential of the inner sphere is
V = q/C
= (2.5 x 10^-6)/(5.5 x 10^-9) = 4.5 x 10^2 V
Hence the potential energy of the inner sphere is 4.5 x 10^2 V
(c) Radius of the isolated sphere, r = 12 x 10^-2 m
The capacitance of the isolated sphere is given by the relation,
C’ = 4πε[0]r
= 4 x 3.14 x 8.85 x 10^-12 x 12 x 10^-2
= 1.333 x 10^-11 F
The outer sphere of the concentric spheres is earthed. Therefore, the potential difference will be less for the concentric spheres, and the capacitance is more than for the isolated sphere.
Q 2.31) Answer carefully:
(a) Two large conducting spheres carrying charges Q[1] and Q[2] are brought close to each other. Is the magnitude of the electrostatic force between them exactly given by Q[1]Q[2]/4πε[0]r^2, where r
is the distance between their centres?
(b) If Coulomb’s law involved 1/r^3 dependence (instead of 1/r^2), would Gauss’s law still be true?
(c) A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line passing through that point?
(d) What is the work done by the field of a nucleus in a complete circular orbit of the electron? What if the orbit is elliptical?
(e) We know that the electric field is discontinuous across the surface of a charged conductor. Is electric potential also discontinuous there?
(f) What meaning would you give to the capacitance of a single conductor?
(g) Guess a possible reason why water has a much greater dielectric constant (= 80) than say, mica (= 6).
(a) No, because charge distributions on the spheres will not be uniform.
(b) No.
(c) Not necessarily. (True only if the field line is a straight line). The field line gives the direction of acceleration and not that of the velocity.
(d) Zero, no matter what the shape of the complete orbit is.
(e) No, the potential is continuous.
(f) A single conductor is a capacitor with one of the ‘plates’ at infinity.
(g) A water molecule has a permanent dipole moment. However, a detailed explanation of the value of the dielectric constant requires microscopic theory and is beyond the scope of the book.
Q 2.32) A cylindrical capacitor has two co-axial cylinders of length 15 cm and radii 1.5 cm and 1.4 cm. The outer cylinder is earthed, and the inner cylinder is given a charge of 3.5 µC. Determine
the capacitance of the system and the potential of the inner cylinder. Neglect end effects (i.e., bending of field lines at the ends).
Length of the coaxial cylinders,l = 15 cm = 0.15 m
Radius of the outer cylinder, r[1] = 1.5 cm = 0.015 m
Radius of the inner cylinder, r[2] = 1.4 cm = 0.014 m
Charge of the inner cylinder, q = 3.5 μC = 3.5 x 10^-6 C
The outer cylinder is earthed
The capacitance of the co-axial cylinder of radii r[1] and r[2] is given by the relation
\(\begin{array}{l}C = \frac{2\pi \epsilon _{0}l}{log_{e}\frac{r_{1}}{r_{2}}}\end{array} \)
ε[0] is the permittivity of free space = 8.85 x 10^-12 N^-1 m^-2 C^2
\(\begin{array}{l}C = \frac{2\times 3.14\times 8.85\times 10^{-12}\times 0.15}{2.3026log_{10}\frac{0.15}{0.14}}\end{array} \)
\(\begin{array}{l}C = \frac{2\times 3.14\times 8.85\times 10^{-12}\times 0.15}{2.3026\times 0.0299}=1.2 \times 10^{-10}F\end{array} \)
Therefore, the potential difference of the inner cylinder is given by
V = q/C
= 3.5 x 10^-6/1.2 x 10^-10 = 2.92 x 10^4 V
Q 2.33) A parallel plate capacitor is to be designed with a voltage rating of 1 kV, using a material of dielectric constant 3 and dielectric strength of about 10^7Vm^–1. (Dielectric strength is the
maximum electric field a material can tolerate without breakdown, i.e. without starting to conduct electricity through partial ionisation.) For safety, we should like the field never to exceed, say,
10% of the dielectric strength. What minimum area of the plates is required to have a capacitance of 50 pF?
Voltage rating of the parallel plate capacitor, V = 1kV= 1000 V.
Dielectric constant, ε = 3
Dielectric strength = 10^7 V/m
For safety, the field should never exceed, say, 10% of the dielectric strength, E = 10% of dielectric strength = (10/100) x 10^7 = 10^6 V/m
Capacitance of capacitor , C = 50pF = 50 × 10^-12 F
Distance between the plates, d = V/E = 10³/10^6 = 10^-3 m
\(\begin{array}{l}C = \frac{\epsilon_{0}\epsilon _{r}A}{d}\end{array} \)
Area of the plate, A =
\(\begin{array}{l}A = \frac{d}{\epsilon_{0}\epsilon _{r}C}\end{array} \)
= (50 × 10^-12 × 10^-3)/(3 × 8.85 × 10^-12)
= 1.9 × 10^-3 m²
= 19 cm²
Therefore, the minimum area of the plates, A = 19 cm²
Q 2.34) Describe schematically the equipotential surfaces corresponding to
(a) a constant electric field in the z-direction
(b) a field that uniformly increases in magnitude but remains in a constant (say, z) direction
(c) a single positive charge at the origin
(d) a uniform grid consisting of long, equally spaced parallel charged wires in a plane
(a) Planes parallel to the x-y plane.
(b) Same as in (a), except that planes differing by a fixed potential get closer as the field increases.
(c) Concentric spheres centred at the origin.
(d) A periodically varying shape near the grid, which gradually reaches the shape of planes parallel to the grid at far distances.
Q 2.35) A small sphere of radius r[1] and charge q[1] is enclosed by a spherical shell of radius r[2] and charge q[2]. Show that if q[1] is positive, the charge will necessarily flow from the sphere
to the shell (when the two are connected by a wire) no matter what the charge q[2] on the shell is
By Gauss’s law, the electric field between the sphere and the shell is determined only by the charge q[1]. Hence, the potential difference between the sphere and the shell is independent of q[2]. If
q[1] is positive, then the potential difference is also positive.
Q 2.36) Answer the following:
(a) The top of the atmosphere is at about 400 kV with respect to the surface of the earth, corresponding to an electric field that decreases with altitude. Near the surface of the earth, the field is
about 100 Vm^–1. Why then do we not get an electric shock as we step out of our house into the open? (Assume the house to be a steel cage, so there is no field inside!)
(b) A man fixes outside his house one evening a two-metre high insulating slab carrying on its top a large aluminium sheet of area 1m^2. Will he get an electric shock if he touches the metal sheet
the next morning?
(c) The discharging current in the atmosphere due to the small conductivity of air is known to be 1800 A on average over the globe. Why then does the atmosphere not discharge itself completely in due
course and become electrically neutral? In other words, what keeps the atmosphere charged?
(d) What are the forms of energy into which the electrical energy of the atmosphere is dissipated during lightning?
(Hint: The earth has an electric field of about 100 Vm^–1 at its surface in the downward direction, corresponding to a surface charge density = –10^–9 C m^–2. Due to the slight conductivity of the
atmosphere up to about 50 km (beyond which it is a good conductor), about + 1800 C is pumped every second into the earth as a whole. The earth, however, does not get discharged since thunderstorms
and lightning occurring continually all over the globe pump an equal amount of negative charge on the earth.
(a) Our body and the ground form an equipotential surface. As we step out into the open, the original equipotential surfaces of open-air change, keeping our heads and the ground at the same
(b) Yes, the steady discharging current in the atmosphere charges up the aluminium sheet gradually and raises its voltage to an extent depending on the capacitance of the capacitor (formed by the
sheet, slab and the ground).
(c) The atmosphere is continually being charged by thunderstorms and lightning all over the globe and discharged through regions of ordinary weather. The two opposing currents are, on average, in
(d) Light energy involved in lightning; heat and sound energy in the accompanying thunder.
NCERT Class 12 Physics Solutions for Electrostatic Potential and Capacitance
Chapter 2, Electrostatic Potential and Capacitance of Class 12 Physics, is prepared as per the current CBSE Syllabus 2023-24. Also, this chapter provides good marks weightage to derivations and
numerical problems for the board exam. The derivation of topics like potential energy of the system of charges, potential due to electric dipole and energy stored in the capacitor is frequently asked
in exams. Numerical problems based on the effective capacitance of a complex combination of capacitors are asked in exams regularly. All the topics in the chapter are covered in the NCERT Solutions
for Class 12 Physics Chapter 2.
Subtopics of Class 12 NCERT Physics Chapter 2 – Electrostatic Potential and Capacitance
Section Number Topic
2.1 Introduction
2.2 Electrostatic Potential
2.3 Potential Due to a Point Charge
2.4 Potential Due to an Electric Dipole
2.5 Potential Due to a System of Charges
2.6 Equipotential Surfaces
2.6.1 Relation Between Field and Potential
2.7 Potential Energy of a System of Charges
2.8 Potential Energy in an External Field
2.8.1 Potential Energy of a Single Charge
2.8.2 Potential Energy of a System of Two Charges in an External Field
2.8.3 Potential Energy of a Dipole in an External Field
2.9 Electrostatics of Conductors
2.10 Dielectrics and Polarization
2.11 Capacitors and Capacitance
2.12 The Parallel Plate Capacitor
2.13 Effect of Dielectric on Capacitance
2.14 Combination of Capacitors
2.14.1 Capacitors in Series
2.14.2 Capacitors in Parallel
2.15 Energy Stored in a Capacitor
2.16 Van De Graaff Generator
Students can utilise the NCERT Solutions for Class 12 Physics Chapter 2 for any quick references to comprehend these and other complex topics easily. Referring to these solutions provides clarity on
key concepts and problem-solving skills that are essential for the board examination.
Why select BYJU’S?
BYJU’S is India’s largest Learning App, with subject experts from across the nation. Finding notes, question papers, NCERT Solutions, and solutions for various other subjects like Mathematics,
Biology, Chemistry, and many competitive exams are very easy as these materials are available in the form of free PDF downloads. Enjoy learning with a great experience. Stay tuned to know more about
NCERT Solutions and preparation tips.
Disclaimer –
Dropped Topics –
2.15 Energy Stored in a Capacitor (delete only derivation)
Exercises 2.12 to 2.36
Frequently Asked Questions on NCERT Solutions for Class 12 Physics Chapter 2
What are the main topics and subtopics covered in Chapter 2 of NCERT Solutions for Class 12 Physics?
The main topics and sub-topics covered in Chapter 2 of NCERT Solutions for Class 12 Physics are listed below:
2.1 Introduction
2.2 Electrostatic Potential
2.3 Potential Due to a Point Charge
2.4 Potential Due to an Electric Dipole
2.5 Potential Due to a System of Charges
2.6 Equipotential Surfaces
2.6.1 Relation between Field and Potential
2.7 Potential Energy of a System Of Charges
2.8 Potential Energy in an External Field
2.8.1 Potential Energy of a Single Charge
2.8.2 Potential Energy of a System of Two Charges in an External Field
2.8.3 Potential Energy of a Dipole in an External Field
2.9 Electrostatics of Conductors
2.10 Dielectrics and Polarization
2.11 Capacitors and Capacitance
2.12 The Parallel Plate Capacitor
2.13 Effect of Dielectric on Capacitance
2.14 Combination of Capacitors
2.14.1 Capacitors in Series
2.14.2 Capacitors in Parallel
2.15 Energy Stored in a Capacitor
2.16 Van De Graaff Generator
Are the NCERT Solutions for Class 12 Physics Chapter 2 the best reference guide for the students?
To prepare well for the board exams, students should choose the perfect reference guide which helps them to grasp the concepts effortlessly. For this purpose, the expert teachers at BYJU’S have
prepared chapter-wise solutions to guide the students, and these are the best reference guide for the students. The solutions can be referred to while answering the textbook questions to get a clear
idea about the concepts, which are important from the exam perspective.
Why should I refer to BYJU’S NCERT Solutions for Class 12 Physics Chapter 2?
The subject experts at BYJU’S have prepared the NCERT Solutions for Class 12 Physics to help students understand the concepts efficiently. Simple and interactive language is used in the solutions,
which makes it interesting for the students to learn. Shortcut tips and techniques to remember important topics are highlighted with the aim of helping the students to score more marks in the board
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Core of Data Analysis Archives - Deep Mukhopadhyay, Ph.D.
The phrases “Science” and “Management” of data analysis were introduced by Manny Parzen (2001) while discussing Leo Breiman’s Paper on “Statistical Modeling: The Two Cultures,” where he pointed out:
“Management seeks profit, practical answers (predictions) useful for decision making in the short run. Science seeks truth, fundamental knowledge about nature which provides understanding and control
in the long run.” Management = Algorithm, prediction and inference is undoubtedly the most useful and “sexy” part of Statistics. Over the past two decades, there have been tremendous advancements
made in this front, leading to a growing number of literature and excellent textbooks like Hastie, Tibshirani, and Friedman (2009) and more recently Efron and Hastie (2016). Nevertheless, we surely
all agree that algorithms do not arise in a vacuum and our job as a Statistical scientist should be better than just finding another “gut” algorithm. It has long been observed that elegant
statistical learning methods can be often derived from something more fundamental. This forces us to think about the guiding principles for designing (wholesale) algorithms. The “Science” of data
analysis = Algorithm discovery engine (Algorithm of Algorithms). Finding such a consistent framework of Statistical Science (from which one might be able to systematically derive a wide range of
working algorithms) promises to not be trivial. Above all, I strongly believe the time has come to switch our focus from “management” to the heart of the matter: how can we create an inclusive and
coherent framework of data analysis (to accelerate the innovation of new versatile algorithms)–“A place for everything, and everything in its place”– encoding the fundamental laws of numbers. In this
(difficult yet rewarding) journey, we have to remind ourselves constantly the enlightening piece of advice from Murray Gell-Mann (2005): “We have to get rid of the idea that careful study of a
problem in some NARROW range of issues is the only kind of work to be taken seriously, while INTEGRATIVE thinking is relegated to cocktail party conversation”
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Citation metrics
Publish or Perish calculates the following citation metrics:
• Hirsch's h-index and related parameters, shown as h-index and Hirsch a=y.yy, m=z.zz in the output. Also Zhang's e-index.
• Egghe's g-index, shown as g-index in the output
• The contemporary h-index, shown as hc-index and ac=y.yy in the output
• Three variations of the individual h-index, shown as hI-index, hI,norm, and hm-index in the output
• The average annual increase in the individual h-index, shown as hI,annual
• An analysis of the number of authors per paper.
Please note that these metrics are only as good as their input. We recommend that you consult the following topics for information about the limitations of the citation metrics and the underlying
sources that Publish or Perish uses:
Basic metrics
The basic metrics are quite straightforward and are calculated as follows in Publish or Perish.
Total number of papers
This is simply the number of papers returned by Google Scholar or Microsoft Academic Search in reply to a query.
Total number of citations
The sum of the citation counts across all papers.
Average number of citations per paper
The sum of the citation counts across all papers, divided by the total number of papers. The median and mode are also calculated.
Number of citations per author
For each paper, its citation count is divided by the number of authors for that paper to give the normalized per-author citation count for the paper. The normalized citation counts are then
summed across all papers to give the number of citations per author over the result set.
Number of citations per author per year
This is the number of citations per author as above, divided by the number of years covered by the result set.
Number of papers per author
For each paper, 1/author_count is calculated to give the normalized author count for the paper. The normalized author counts are then summed across all papers to give the number of papers per
Average number of authors per paper
The sum of the author counts across all papers, divided by the total number of papers. The median and mode are also calculated.
The h-index was proposed by J.E. Hirsch in his paper An index to quantify an individual's scientific research output, arXiv:physics/0508025 v5 29 Sep 2005. It is defined as follows:
A scientist has index h if h of his/her N[p] papers have at least h citations each, and the other (N[p]-h) papers have no more than h citations each.
It aims to measure the cumulative impact of a researcher's output by looking at the amount of citation his/her work has received. Publish or Perish calculates and displays the h index proper, its
associated proportionality constant a (from N[c,tot] = ah^2), and the rate parameter m (from h ~ mn, where n is the number of years since the first publication).
The properties of the h-index have been analyzed in various papers; see for example Leo Egghe and Ronald Rousseau: An informetric model for the Hirsch-index, Scientometrics, Vol. 69, No. 1 (2006),
pp. 121-129.
Publish or Perish also calculates the e-index as proposed by Chun-Ting Zhang in his paper The e-index, complementing the h-index for excess citations, PLoS ONE, Vol 5, Issue 5 (May 2009), e5429. The
e-index is the (square root) of the surplus of citations in the h-set beyond h^2, i.e., beyond the theoretical minimum required to obtain a h-index of 'h'. The aim of the e-index is to differentiate
between scientists with similar h-indices but different citation patterns.
h-index, Hirsch a=y.yy, m=z.zz, and e-index in the output.
The g-index was proposed by Leo Egghe in his paper Theory and practice of the g-index, Scientometrics, Vol. 69, No 1 (2006), pp. 131-152. It is defined as follows:
[Given a set of articles] ranked in decreasing order of the number of citations that they received, the g-index is the (unique) largest number such that the top g articles received (together) at
least g^2 citations.
It aims to improve on the h-index by giving more weight to highly-cited articles.
g-index in the output.
Contemporary h-index
The Contemporary h-index was proposed by Antonis Sidiropoulos, Dimitrios Katsaros, and Yannis Manolopoulos in their paper Generalized h-index for disclosing latent facts in citation networks,
arXiv:cs.DL/0607066 v1 13 Jul 2006.
It adds an age-related weighting to each cited article, giving (by default; this depends on the parametrization) less weight to older articles. The weighting is parametrized; the Publish or Perish
implementation uses gamma=4 and delta=1, like the authors did for their experiments. This means that for an article published during the current year, its citations count four times. For an article
published 4 years ago, its citations count only once (4/4). For an article published 6 years ago, its citations count 4/6 times, and so on.
hc-index and ac=y.yy in the output.
Individual h-index (3 variations)
The Individual h-index was proposed by Pablo D. Batista, Monica G. Campiteli, Osame Kinouchi, and Alexandre S. Martinez in their paper Is it possible to compare researchers with different scientific
interests?, Scientometrics, Vol 68, No. 1 (2006), pp. 179-189.
It divides the standard h-index by the average number of authors in the articles that contribute to the h-index, in order to reduce the effects of co-authorship; the resulting index is called h[I].
Publish or Perish also implements an alternative individual h-index, h[I,norm], that takes a different approach: instead of dividing the total h-index, it first normalizes the number of citations for
each paper by dividing the number of citations by the number of authors for that paper, then calculates h[I,norm] as the h-index of the normalized citation counts. This approach is much more
fine-grained than Batista et al.'s; we believe that it more accurately accounts for any co-authorship effects that might be present and that it is a better approximation of the per-author impact,
which is what the original h-index set out to provide.
The third variation is due to Michael Schreiber and first described in his paper To share the fame in a fair way, h[m] modifies h for multi-authored manuscripts, New Journal of Physics, Vol 10
(2008), 040201-1-8. Schreiber's method uses fractional paper counts instead of reduced citation counts to account for shared authorship of papers, and then determines the multi-authored h[m] index
based on the resulting effective rank of the papers using undiluted citation counts.
hI-index (Batista et al.'s), hI,norm (PoP's), and hm-index (Schreiber's) in the output.
Average annual increase in individual h-index
The individual, average annual increase of the h-index called hI,annual was proposed by Anne-Wil Harzing, Satu Alakangas and David Adams in their paper hIa: An individual annual h-index to
accommodate disciplinary and career length differences, Scientometrics, vol. 99, no. 3, pp. 811-821, which is available online on the Harzing.com web site.
As of release 4.3 Publish or Perish calculates and displays this new index. The average annual increase in the individual h-index is useful for the following reasons:
• In common with the hI,norm index, it removes to a considerable extent any discipline-specific publication and citation patterns that otherwise distort the h-index.
• It also reduces the effect of career length and provides a fairer comparison between junior and senior researchers.
The hI,annual is meant as an indicator of an individual's average annual research impact, as opposed to the lifetime score that is given by the h-index or hI,norm.
hI,annual in the output.
Age-weighted citation rate (AWCR, AWCRpA) and AW-index
The age-weighted citation rate was inspired by Bihui Jin's note The AR-index: complementing the h-index, ISSI Newsletter, 2007, 3(1), p. 6.
The AWCR measures the number of citations to an entire body of work, adjusted for the age of each individual paper. It is an age-weighted citation rate, where the number of citations to a given paper
is divided by the age of that paper. Jin defines the AR-index as the square root of the sum of all age-weighted citation counts over all papers that contribute to the h-index.
However, in the Publish or Perish implementation we sum over all papers instead, because we feel that this represents the impact of the total body of work more accurately. (In particular, it allows
younger and as yet less cited papers to contribute to the AWCR, even though they may not yet contribute to the h-index.)
The AW-index is defined as the square root of the AWCR to allow comparison with the h-index; it approximates the h-index if the (average) citation rate remains more or less constant over the years.
The per-author age-weighted citation rate is similar to the plain AWCR, but is normalized to the number of authors for each paper.
AWCR, AWCRpA and AW-index in the output.
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Python Built-in Constants (With Examples) - MachineLearningTutorials.org
Python Built-in Constants (With Examples)
Python, as a versatile and powerful programming language, comes with a wide range of built-in constants that provide essential information or fixed values for various purposes. Constants are
predefined values that remain unchanged during the course of a program’s execution. They are used to enhance code readability, improve maintainability, and reduce the likelihood of errors caused by
hardcoding values.
In this tutorial, we will explore the most commonly used built-in constants in Python, along with examples demonstrating their usage and significance.
Table of Contents
1. Introduction to Built-in Constants
2. Numeric Constants
• True, False, and None
• NotImplemented and Ellipsis
• Inf and NaN
1. String Constants
• __doc__ and __file__
• __name__ and __package__
1. Examples
• Mathematical Calculations
• Boolean Logic
1. Conclusion
1. Introduction to Built-in Constants
Built-in constants in Python are predefined values that are accessible without the need for explicit definition. They offer a convenient way to use essential values throughout your codebase. Python
provides various types of constants, including numeric constants, string constants, and special-purpose constants.
In this tutorial, we will focus on numeric constants, which include boolean values, special numeric values like infinity and NaN (Not-a-Number), and string constants related to module and package
2. Numeric Constants
True, False, and None
Python’s boolean constants are True and False, representing the truth values in logical expressions. They are often used in conditions and control flow statements to make decisions based on certain
is_valid = True
if is_valid:
print("This is valid.")
is_checked = False
if not is_checked:
print("Not checked yet.")
The constant None is used to represent the absence of a value or a null value. It is often used to initialize variables or indicate that a function has no return value.
result = None
def do_something():
# Do something here
return None
NotImplemented and Ellipsis
The constant NotImplemented is used to indicate that a certain functionality or method is not implemented yet. It’s often returned by methods that subclasses need to override.
class MyCustomClass:
def my_method(self):
raise NotImplementedError("This method needs to be implemented in subclasses.")
The constant Ellipsis represents an ellipsis and is used to denote incomplete code, placeholders, or slicing in certain contexts, especially in NumPy arrays.
Inf and NaN
Python provides constants for representing infinity and NaN (Not-a-Number) in floating-point arithmetic.
positive_infinity = float('inf')
negative_infinity = float('-inf')
not_a_number = float('nan')
3. String Constants
__doc__ and __file__
The __doc__ constant provides access to the docstring of a module, class, function, or method. It contains the documentation and description of the object.
def my_function():
"""This function does something."""
The __file__ constant holds the path of the current module’s source file. It’s particularly useful when you want to determine the location of the file in which the code is executing.
__name__ and __package__
The __name__ constant holds the name of the current module. When a module is run as the main program, its __name__ attribute is set to '__main__'.
if __name__ == '__main__':
print("This module is the main program.")
The __package__ constant holds the name of the package to which the current module belongs. It is mainly used in modules that are part of a package.
4. Examples
Mathematical Calculations
Constants play a significant role in mathematical calculations. Let’s consider an example of calculating the area of a circle using the constant pi from the math module.
import math
radius = 5
area = math.pi * (radius ** 2)
print(f"The area of the circle with radius {radius} is {area:.2f}")
Boolean Logic
Boolean constants True and False are essential for implementing conditional statements and boolean logic. Here’s an example of using boolean constants to check if a number is positive.
def is_positive(number):
return number > 0
num = 10
if is_positive(num):
print(f"{num} is a positive number.")
print(f"{num} is not a positive number.")
5. Conclusion
Built-in constants are powerful tools that enhance code readability, improve maintainability, and reduce the risk of errors by avoiding hardcoded values. In this tutorial, we explored some of the
most commonly used built-in constants in Python, including boolean values, special numeric values like infinity and NaN, and string constants related to module and package attributes.
By leveraging these constants in your Python code, you can write more clear and expressive programs that are easier to understand and maintain. Whether you’re performing mathematical calculations,
implementing boolean logic, or documenting your code, built-in constants are valuable resources that contribute to the elegance and effectiveness of your codebase.
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The STANDARD-DEVIATION Function
The STANDARD-DEVIATION function returns a numeric value that approximates the standard deviation of its arguments. The type of this function is numeric.
General Format
1. Argument-1 must be class numeric.
Returned Values
1. The returned value is the approximation of the standard deviation of the argument-1 series.
2. The returned value is calculated as follows:
a. The difference between each argument-1 value and the arithmetic mean of the argument-1 series is calculated and squared.
b. The values obtained are added. This quantity is divided by the number of values in the argument-1 series.
c. The square root of the quotient obtained is calculated. The returned value is the absolute value of this square root.
3. If the argument-1 series consists of only one value, or if the argument-1 series consists of all variable occurrence data items and the total number of occurrences for all of them is one, the
returned value is zero.
4. Floating-point format is used for numeric non-integer results.
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Consequences of Lorentz Transformation
Introduction to Lorentz Transformation
Welcome, fellow enthusiasts of the cosmos, to an exciting exploration of the consequences of the Lorentz Transformation. In the world of physics, few concepts are as mind-bending and transformative
as those proposed by Albert Einstein in his Special Theory of Relativity. The Lorentz Transformation is a mathematical framework that has profound implications for our understanding of space, time,
and reality itself. This blog post will delve into the intriguing consequences of the Lorentz Transformation, focusing on length contraction, time dilation, and the simultaneity of events. Prepare to
have your perceptions of reality stretched and reshaped!
The Roots of Lorentz Transformation
To appreciate the consequences of the Lorentz Transformation, we must first understand its origins. The Lorentz Transformation equations were derived by Hendrik Lorentz, but it was Albert Einstein
who fully realized their implications in his 1905 paper on Special Relativity. These equations describe how space and time coordinates change between two observers moving relative to each other at a
constant velocity. The transformation preserves the speed of light, which remains constant regardless of the motion of the observer or the source of light. This seemingly simple idea leads to a
series of astonishing consequences that defy our everyday experiences.
Setting the Stage: Inertial Frames of Reference
Before we dive into the specific consequences of Lorentz Transformation, it’s crucial to understand the concept of inertial frames of reference. An inertial frame is one in which an object remains at
rest or in uniform motion unless acted upon by a force. In the context of Special Relativity, the Lorentz Transformation connects the space and time coordinates of events as measured in different
inertial frames. This connection is the key to unlocking the mysteries of length contraction, time dilation, and the relativity of simultaneity.
Length Contraction: Shrinking Distances
Understanding Length Contraction
One of the most fascinating consequences of the Lorentz Transformation is length contraction. Imagine you’re on a spaceship traveling at a significant fraction of the speed of light. According to the
principles of Special Relativity, objects in the direction of your travel will appear shorter than when they are at rest relative to you. This phenomenon is known as length contraction.
The Lorentz Factor and Length Contraction
The degree of length contraction depends on the Lorentz factor, denoted by the Greek letter gamma (γ). The Lorentz factor is defined as:
γ=11−v2c2\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}γ=1−c2v21
where vvv is the relative velocity and ccc is the speed of light. As your velocity vvv approaches ccc, γ\gammaγ increases, leading to more pronounced length contraction. For example, if you’re
traveling at 80% of the speed of light, the length of objects in your direction of travel will appear reduced by a factor of about 0.6.
Time Dilation: Slowing Down the Clock
The Concept of Time Dilation
Another mind-blowing consequence of Lorentz Transformation is time dilation. If you’re moving at a high velocity relative to a stationary observer, time will appear to pass more slowly for you than
for the observer. This effect becomes significant as you approach the speed of light. Time dilation means that if you embark on a journey at near-light speeds, you would age more slowly compared to
your friends and family back on Earth.
Mathematical Representation of Time Dilation
Time dilation can be quantified using the Lorentz factor. If Δt\Delta tΔt represents the time interval for an observer at rest and Δt′\Delta t’Δt′ represents the time interval for a moving observer,
the relationship between them is:
Δt′=γΔt\Delta t’ = \gamma \Delta tΔt′=γΔt
This equation shows that the time interval for the moving observer (Δt′\Delta t’Δt′) is longer than that for the stationary observer (Δt\Delta tΔt), indicating that time is dilated for the moving
The Relativity of Simultaneity
What Is Simultaneity?
In our everyday lives, we take simultaneity for granted. If two events happen at the same time, we assume they are simultaneous for everyone, everywhere. However, the Lorentz Transformation reveals
that simultaneity is relative. Events that appear simultaneous to one observer may not be simultaneous to another observer moving at a different velocity.
Illustrating Relativity of Simultaneity
Consider a train moving at a constant speed along a track. Two lightning bolts strike the ends of the train simultaneously, according to an observer standing on the ground. However, an observer on
the moving train might perceive the lightning bolt at the front of the train to strike first. This difference in perception arises because the train is moving relative to the ground, and the Lorentz
Transformation alters the time coordinates of the events.
Implications for Space Travel
Traveling to Distant Stars
The consequences of Lorentz Transformation have profound implications for space travel. If humanity ever develops the technology to travel at a significant fraction of the speed of light, astronauts
could experience dramatic time dilation. For them, a journey to a distant star system might take only a few years, while many decades pass on Earth. This concept, known as the “twin paradox,”
highlights the difference in aging between a space-traveling twin and their Earth-bound sibling.
Navigating Length Contraction
Length contraction also affects space travel. As a spacecraft approaches the speed of light, the distances to distant stars and galaxies will appear shorter to the travelers. This contraction of
space could make interstellar travel more feasible, at least from the perspective of the astronauts. However, these effects come with significant engineering and energy challenges, as reaching such
high velocities requires enormous amounts of energy.
Consequences in Everyday Technology
GPS and Relativity
The consequences of Lorentz Transformation are not limited to theoretical physics and space travel; they also impact everyday technology. The Global Positioning System (GPS) relies on a network of
satellites orbiting Earth. These satellites move at high speeds relative to the ground, and their onboard clocks experience time dilation. Engineers must account for these relativistic effects to
ensure the accuracy of GPS signals. Without these adjustments, GPS would quickly become inaccurate, leading to navigation errors.
Particle Accelerators
Particle accelerators, such as the Large Hadron Collider (LHC), accelerate particles to speeds close to the speed of light. At these velocities, both time dilation and length contraction come into
play. Physicists use the principles of Lorentz Transformation to predict and control the behavior of particles in these high-energy environments, allowing them to probe the fundamental properties of
matter and the forces that govern the universe.
Theoretical Implications
Unifying Quantum Mechanics and Relativity
The consequences of Lorentz Transformation extend into the realm of theoretical physics, particularly in the quest to unify quantum mechanics and relativity. Quantum electrodynamics (QED) and other
quantum field theories incorporate the principles of Lorentz Transformation to describe how particles interact with electromagnetic fields. The challenge of reconciling general relativity with
quantum mechanics remains one of the most significant puzzles in modern physics, and understanding the consequences of Lorentz Transformation is crucial to this endeavor.
Exploring the Nature of Time
The relativity of simultaneity and time dilation force us to rethink our understanding of time. In the classical Newtonian view, time is absolute and flows uniformly for all observers. However, the
Lorentz Transformation shows that time is flexible and relative, depending on the observer’s motion. This new perspective has profound implications for our philosophical and scientific understanding
of time’s nature.
Experimental Evidence
Testing Time Dilation with Atomic Clocks
One of the most compelling pieces of evidence for time dilation comes from experiments with atomic clocks. In the 1970s, scientists placed highly accurate atomic clocks on airplanes and flew them
around the world. Upon their return, the airborne clocks showed a slight difference in elapsed time compared to identical clocks left on the ground. This difference matched the predictions of time
dilation according to the Lorentz Transformation, providing direct experimental confirmation of this consequence.
Muons and Relativistic Lifetimes
Another striking example comes from the observation of muons, subatomic particles created in the upper atmosphere. Muons travel towards the Earth’s surface at speeds close to the speed of light. Due
to time dilation, muons have a longer lifespan from our perspective on the ground than they would if they were at rest. This extended lifetime allows more muons to reach the Earth’s surface than
would be possible without relativistic effects, offering further evidence of time dilation.
Misconceptions and Clarifications
Faster-than-Light Travel
A common misconception is that the Lorentz Transformation allows for faster-than-light travel. However, according to Special Relativity, nothing with mass can reach or exceed the speed of light. As
an object approaches light speed, its energy requirement grows infinitely, making such travel impossible with our current understanding of physics. The Lorentz Transformation ensures that the speed
of light remains the ultimate speed limit in the universe.
Time Travel
While time dilation allows for “time travel” to the future—by moving at high velocities—traveling to the past remains firmly in the realm of science fiction. The Lorentz Transformation does not
provide a mechanism for backward time travel, as it would violate causality and lead to paradoxes that our current physical theories cannot resolve. The idea of traveling back in time, despite its
allure, is not supported by the consequences of Lorentz Transformation.
Lorentz Transformation in Popular Culture
Science Fiction Inspirations
The consequences of Lorentz Transformation have inspired countless works of science fiction. Concepts like time dilation and the relativity of simultaneity are central themes in many books, movies,
and TV shows. While these stories often take creative liberties, they spark curiosity and provide a gateway for the public to engage with complex scientific ideas. Classics like H.G. Wells’ “The Time
Machine” and more recent works like the film “Interstellar” explore these themes, blending scientific theory with imaginative storytelling.
Educational Outreach
Educational programs and public lectures frequently use the consequences of Lorentz Transformation to illustrate the counterintuitive nature of modern physics. By presenting thought experiments and
real-world analogies, educators can make these abstract concepts more accessible and engaging. Programs like NOVA, Cosmos, and popular science books by authors like Stephen Hawking and Brian Greene
have brought the wonders of relativity to a broad audience, fostering a deeper appreciation for the beauty and complexity of the universe.
Philosophical Reflections
Rethinking Reality
The consequences of Lorentz Transformation challenge our conventional notions of reality. The relativity of simultaneity, in particular, forces us to reconsider the nature of events and their
temporal order. In a universe where time and space are not absolute, our understanding of existence itself becomes more fluid and dynamic. Philosophers and scientists alike grapple with these
implications, exploring the profound questions raised by the flexible fabric of spacetime.
The Human Experience
On a more personal level, contemplating the consequences of Lorentz Transformation can alter our perception of the human experience. The idea that time can stretch and contract, that distances can
shrink, and that simultaneity is relative invites us to see our place in the universe in a new light. It underscores the interconnectedness of all things and the profound ways in which our reality is
shaped by the motion and energy of the cosmos.
Conclusion: Embracing the Consequences
Continuing the Journey
The consequences of Lorentz Transformation—length contraction, time dilation, and the relativity of simultaneity—are more than just theoretical curiosities. They are fundamental aspects of our
universe that have been confirmed through rigorous experimentation and observation. As we continue to explore the frontiers of physics, these concepts will remain central to our understanding of
space, time, and the nature of reality itself.
Inspiring Future Generations
By embracing the consequences of Lorentz Transformation, we inspire future generations to question, explore, and dream. The story of relativity is a testament to human curiosity and the relentless
pursuit of knowledge. It encourages us to push the boundaries of our understanding, to seek out the unknown, and to marvel at the intricate dance of space and time that shapes our existence.
So, as you ponder the mysteries of the universe, remember that the consequences of Lorentz Transformation are more than just equations and concepts. They are a gateway to a deeper understanding of
the cosmos—a journey that continues to captivate and inspire us all.
Add a Comment
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1. the work-in-process inventory account of a manufacturing company
Read the scenario and answer the questions in no less than 200 words each. Include APA-formatted citations and references.
William is a 44-year old project manager for a large commercial construction firm. He started out as a gifted carpenter who greatly enjoyed designing and building custom furniture. However, after
several promotions, he focuses on bringing in new business. He spends many work hours at his computer or on the telephone. He is divorced and rarely sees his two daughters. In recent years, he has
gained weight and is displeased with his appearance, but has no interest in or energy for exercise. He does not sleep well because he worries about business problems at night. He was recently
diagnosed with high blood pressure. Although he is financially secure, he rarely takes vacations or socializes outside of the office. He has begun to feel that his life is pointless.
1. Discuss William’s situation from the perspective of traditional psychology. What information would be most important? What conclusions and recommendations might be made by a psychologist working
from the disease model?
2. Discuss William’s situation from the perspective of positive psychology. What information would be most important? What conclusions and recommendations might be made by a psychologist working from
the positive psychology model?
Conduct a poll of at least five people by asking the following questions. Evaluate each answer and decide whether you would consider it as hedonic or eudaimonic, and record it in the table. Finally,
answer the question below the table in 200-350 words.
· What makes you happy?
· Would you say you are living “the good life?” Why or why not?
· If you could make any changes you wished that would make you happier, what would those be?
Write a summary of your results. What common beliefs about happiness were evident in your results?
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Welcome to the Golf Club Atlas Discussion Group!Each user is approved by the Golf Club Atlas editorial staff. For any new inquiries, please contact us.
I had to look up the word and couldn't even understand the 15 sentences I read...
No matter how non-determinant a hole may be when comparing various strategies, the purpose of playing a golf hole is to get the ball in the hole. That means that inherently, all golf holes are
determinant in nature, and one strategy will always be favored over another given the current variables (weather, conditioning, pin position, tee position, etc.) to get the ball down in the fewest
strokes.For example, think of #10 at Riviera. Is it always 50/50 on whether to lay up or go for the green no matter what the conditions are? I don't think it is. And I don't think any golf hole ever
is completely stochastic in nature, though I think designing to that ideal can make a great hole. But some great holes are completely determinant in strategy. So there you go.
No matter how non-determinant a hole may be when comparing various strategies, the purpose of playing a golf hole is to get the ball in the hole. That means that inherently, all golf holes are
determinant in nature, and one strategy will always be favored over another given the current variables (weather, conditioning, pin position, tee position, etc.) to get the ball down in the fewest
strokes.For example, think of #10 at Riviera. Is it always 50/50 on whether to lay up or go for the green no matter what the conditions are? I don't think it is. And I don't think any golf hole ever
is completely stochastic in nature, though I think designing to that ideal can make a great hole. But some great holes are completely determinant in strategy. So there you go. I don't think it's
50-50 for hole #10..do you? If one believes that strategy begins at the green and goes backwards then stochastic process has to be there.Here's what I'm doing. Bragging a little here.... My nephew
just graduated a Valedictorian at NC State in the Engineering School. Friday he called to say he has been offered a full ride with monthly stipend to study for his Masters leading to his Doctorate at
MIT. But he can't decide because Stanford offered the same thing Thursday. As I was congratulating him we started a nerd discussion on golf design strategies and he told me what he wanted to do....so
he is working this thing out for me....
Mike,Congrats to your nephew, that's an impressive resume and set of choices to make. I don't think #10 is 50/50 at all. But only based on changing variables. If strategy begins at the green and goes
back, then I think it's even more determinant what may be the best strategy on any given day. I just don't believe there is ever a situation in golf where there isn't a best way and slightly worse
way, and so on and so forth.
Mike,Congrats to your nephew, that's an impressive resume and set of choices to make. I don't think #10 is 50/50 at all. But only based on changing variables. If strategy begins at the green and goes
back, then I think it's even more determinant what may be the best strategy on any given day. I just don't believe there is ever a situation in golf where there isn't a best way and slightly worse
way, and so on and so forth. Let's go nerd for a minute....( his words)In general one calculates the number of possible combinations by using the binomial coefficient. (http://en.wikipedia.org/wiki/
Binomial_coefficient). Here n represents the number of "objects" to choose from, and k represents the number of "objects" chosen.To make par on a par 4 hole, you have to use 4 strokes. Therefore
there should be only 1 choice (using 4 strokes).If one means: How many possibilities of making par or under when starting at the tee, then there are 4 possible outcomes (at least theoretically). You
can make the 1st, 2nd, 3rd, or 4th shot. Once you have hit the ball once and do not make it in the hole, you now have 3 possible outcomes: Sinking the ball on the 2nd, 3rd, or 4th shots. This should
be pretty intuitive though.In general it would be interesting to model golf play as a stochastic process. This could be done by sectioning the course into finite elements, each of which has a
probability of being reached on the 1st, 2nd, 3rd, etc. shots. Of course the model can become quite complex because the location of the ball after the nth shot is dependent on the previous location
of the ball, and so on. Additional complexity can be added if hazards, such as lakes and trees are added. Of course all of this would require observed data to calculate the mean and variance of the
shot distance and direction.
In general it would be interesting to model golf play as a stochastic process.
Ben,That's my point...you can't predetermine strategy...so why does it matter? And don't you think the young tour pros keep proving this to us more and more...they don't even think about it...
I vote Stanford too!! I did follow and understand the flat screen over the urinal line much better.
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Daily Themed Crossword September 24 2024 Crossword Clue
Daily Themed Crossword September 24 2024
The Daily Themed Crossword Sep 24 2024 Answers were just published after we played around with it and solved today’s puzzle in a timely matter. This puzzle special in the sense that everyday it
allows you to play with a different theme hence the name Daily Themed Crossword. The clues are grouped by their orientation on the puzzle grid and by their number.
For another Daily Themed Crossword Answers go in the homepage.
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The graded module category of a generalized Weyl algebra
Printable PDF
Department of Mathematics,
University of California San Diego
Final Defense
Robert Won
The graded module category of a generalized Weyl algebra
The first Weyl algebra $A = k\langle x,y \rangle/(xy - yx - 1)$ is a well-studied noncommutative $\mathbb {Z}$-graded ring. Generalized Weyl algebras, introduced by Bavula, are a class of
noncommutative $\mathbb {Z}$-graded rings which generalize the Weyl algebra. In this talk, we investigate the category of graded modules over certain generalized Weyl algebras and construct
commutative rings with equivalent graded module categories. Along the way, we will learn about graded rings, noncommutative projective schemes, and how to do geometry without a geometric space.
Advisor: Daniel Rogalski
May 2, 2016
4:00 PM
AP&M 6402
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Filters, part III: correcting a mistake, and pretopologies
It turns out I made a mistake in my last post on filters.
We had ended up in the situation where I claimed that the topological filter spaces, that is, the filters spaces that actually arise from a topological space, are those that satisfy the extra axiom:
• if (F[i])[i][ in ][I] is any family of filters on X such that F[i] → x for every i in I, then F → x where F is the intersection of the F[i]s.
Certainly, any topological space satisfies this extra axiom. The problem is in the converse implication.
The filter spaces that satisfy the above extra axiom are the pretopological filter spaces. Every topological filter space is pretopological, but the converse is wrong. Here is a counterexample,
given to me by Frédéric Mynard yesterday. Let X=R^2. Given a point x=(s,t) in X, draw a small cross centered at (s,t), with arms of length 2. Formally, for a>0, let C[x] be the set of points of
the form (s+δ,t) or (s,t+δ) where -1 < δ < 1. Say that a filter F converges to x=(s,t) if and only if C[x] is an element of F. One checks easily that this is a pretopological notion of convergence.
If it were topological, then it would be the notion of convergence for Top(X). Let us elucidate the topology of the latter. The opens of Top(X) are those subsets U such that for every element x of
U, every filter that converges to x contains U, i.e., every filter that contains C[x] also contains U. Using the filter of all supersets of C[x], one sees that the opens are those subsets that are
“neighborhoods” of all of their elements, where U is a “neighborhood” of x if and only if U contains C[x]. So, if U is a non-empty open subset, let x be one if its elements. The whole of C[x] is
included in U. In particular, every point y obtained by translating x horizontally along a distance < 1 is in U. Applying the same argument with y, and moving vertically, now, we can reach any
point z in the open square centered at x with side length 2, while staying in U. Repeating the argument, we can actually reach any point in X while staying in U. So U=X, meaning that the topology
of Top(X) is indiscrete. In particular, every filter converges to any point in Top(X). This is really far from our original notion of convergence!
And therefore, certainly, X is pretopological, but not topological.
Neighborhood systems.
What happens in this counterexample can be generalized to the following. Call a neighborhood system on a set X the data of a family N[x] of subsets of X containing x, for each point x in X. (The
notion is due to Felix Hausdorff [1].) Given any neighborhood system, one can define a notion on convergence in the usual way: a filter F converges to x if and only if N[x] is included in F. It is
an easy exercise to show that this is a notion of convergence, and that it is always pretopological.
Conversely, given a pretopological notion of convergence →, one obtains a neighborhood system in the exact same way we (erroneously) tried to prove that pretopological notions of convergence were
topological at the end of part II. Consider the family of all filters that converge to x, and take their intersection. Call this intersection filter N[x]. The extra axiom defining pretopologies
implies that N[x] → x, and then that a filter converges to x iff it contains N[x]. So neighborhood systems and pretopologies are essentially the same thing. (To make this precise, you will have to
strengthen the definition of a neighborhood system so that N[x] is a filter, not just a family of subsets. That is no essential difference, since every family of sets gives rise to a unique smallest
filter containing it.)
What went wrong last time? Let me cite: “From N[x], we retrieve a topology by declaring open any set U that is a neighborhood of each of its points, i.e., such that U is in N[x] for every x in U: in
particular, N[x] is really the filter of neighborhoods of x: our filter space is indeed topological.” Of course, this gives you a topology, but N[x] will in general be a superset of the set of all
neighborhoods of x, and not necessarily equal to it. In Frédéric’s example, N[x] would be the filter generated by C[x], that is, the set of all supersets of C[x]. In Top(X), there is only one
neighborhood of x: the whole space X itself. This is a much smaller set of neighborhoods!
And topologies?
So how can we characterize those filter spaces that are topological?
You need yet one more axiom, which essentially says that limits of limits are limits. This is probably a bit too complex to be put here, and is best defined using convergence of ultrafilters instead
of filters. In short, there is a space UX of all ultrafilters on a given filter space X. UX can be given a natural notion of convergence by saying that an ultrafilter of ultrafilters A (in UUX!)
converges to an ultrafilter a (in UX) if and only if for every element u of A, for every element u of A, there is an element a of u and an element x of u such that a converges to x. One can also
flatten out an ultrafilter of ultrafilters by the so-called Kowalsky sum operation µ[X]: for A in UUX, µ[X](A) is the collection of subsets u of X such that u^# is in A… where u^# is the set of all
ultrafilters a (in UX ) such that u is in a. (This is the so-called multiplication operation of the U monad.)
Oops… don’t worry if you don’t understand, that is precisely why I don’t want to explain. I’m not sure I understand too much of it either.
Anyway, to finish the story, a filter space is topological if and only if for every ultrafilter of ultrafilters A that converges (in UX) to some ultrafilter a that itself converges (in X) to x, then
µ[X](A) converges to x. This was proved by Dirk Hofmann and Walter Tholen [2].
As Walter Tholen mentioned last Saturday at the categorical topology session I was at, the novelty here is that this new axiom, together with the axiom that the ultrafilter at x converges to x (the
first axiom of filter spaces), are enough to define exactly the topological filter spaces. All the other axioms, including the fact that if a filter contains a filter that converges to x, then it
already converges to x (the second axiom of filter spaces), and mainly that intersections of filters that converge to a point again converge to this point (pretopologies) are all redundant.
— Jean Goubault-Larrecq (January 21st, 2014)
1. Felix Hausdorff, Grundzüge der Mengenlehre. Teubner, Leipzig, 1914.
2. Dirk Hofmann, Walter Tholen. Kleisli compositions for topological spaces.
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Associative algebra
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by
mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an associative algebra A is an
associative ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition
and a semigroup under multiplication such that multiplication distributes over addition...
that has a compatible structure of a vector space over a certain
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction,
multiplication, and division, satisfying certain axioms...
K or, more generally, of a
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over a
commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in
which the operations are performed does not matter as long as the sequence of the operands is not...
of multiplication and
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
, as well as compatible multiplication by the elements of the field K or the ring R.
In some areas of mathematics, associative algebras are typically assumed to have a multiplicative unit, denoted 1. To make this extra assumption clear, these associative algebras are called unital
Formal definition
Let R be a fixed
commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
. An associative R-algebra is an additive
abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian
groups generalize the arithmetic of addition of integers...
A which has the structure of both a
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition
and a semigroup under multiplication such that multiplication distributes over addition...
and an
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
in such a way that ring multiplication is R-bilinear:
for all r ∈ R and x, y ∈ A.
We say A is unital if it contains an element 1 such that
for all x ∈ A.
If A itself is commutative (as a ring) then it is called a commutative R-algebra.
From R-modules
Starting with an R-module A, we get an associative R-algebra by equipping A with an R-bilinear mapping A × A → A such that
for all x, y, and z in A. This R-bilinear mapping then gives A the structure of a ring and an associative R-algebra. Every associative R-algebra arises this way.
Moreover, the algebra A built this way will be unital if and only if
This definition is equivalent to the statement that a unital associative R-algebra is a
Monoid (category theory)
In category theory, a monoid in a monoidal category is an object M together with two morphisms* \mu : M\otimes M\to M called multiplication,* and \eta : I\to M called unit,...
in R-Mod (the
monoidal category
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up
to a natural isomorphism...
of R-modules).
From rings
Starting with a ring A, we get a unital associative R-algebra by providing a
ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
Center (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum,
meaning "center". More specifically:...
of A. The algebra A can then be thought of as an R-module by defining
for all r ∈ R and x ∈ A.
If A is commutative then the center of A is equal to A, so that a commutative R-algebra can be defined simply as a homomorphism
Algebra homomorphisms
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning
"shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
between two associative R-algebras is an R-linear
ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
. Explicitly,
For a homomorphism of unital associative R-algebras, we also demand that
The class of all unital associative R-algebras together with algebra homomorphisms between them form a
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the
existence of an identity arrow for each object. A simple example is the category of sets, whose...
, sometimes denoted R-Alg.
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively,
a subcategory of C is a category obtained from C by "removing" some of its objects and...
of commutative R-algebras can be characterized as the coslice category R/CRing where CRing is the category of commutative rings.
• The square n-by-n matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements
isMatrices of the same size can be added or subtracted element by element...
with entries from the field K form a unitary associative algebra over K.
• The complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the
number line for the real part and adding a vertical axis to plot the imaginary part...
s form a 2-dimensional unitary associative algebra over the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
• The quaternions form a 4-dimensional unitary associative algebra over the reals (but not an algebra over the complex numbers, since if complex numbers are treated as a subset of the quaternions,
complex numbers and quaternions do not commute).
• The 2 × 2 real matrices form an associative algebra useful in plane mapping.
• The polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative
integer exponents...
s with real coefficients form a unitary associative algebra over the reals.
• Given any Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a
norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X, the continuous linear operators A : X → X form a unitary associative algebra (using composition of operators as multiplication); this is a Banach algebra
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach
• Given any topological space
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or
X, the continuous real- or complex-valued functions on X form a real or complex unitary associative algebra; here the functions are added and multiplied pointwise.
• An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
as x nears infinity is zero.
• The Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.
The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
s, which are useful in geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of
nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
• Incidence algebra
Incidence algebra
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for any locally finite partially ordered setand commutative ring with unity.-Definition:...
s of locally finite partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset
consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
s are unitary associative algebras considered in combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size ,
deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
• Any ring A can be considered as a Z-algebra in a unique way. The unique ring homomorphism from Z to A is determined by the fact that it must send 1 to the identity in A. Therefore rings and
Z-algebras are equivalent concepts, in the same way that abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order .
Abelian groups generalize the arithmetic of addition of integers...
s and Z-modules are equivalent.
• Any ring of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the
additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
n is a (Z/nZ)-algebra in the same way.
• Any ring A is an algebra over its center
Center (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum,
meaning "center". More specifically:...
Z(A), or over any subring of its center.
• Any commutative ring R is an algebra over itself, or any subring of R.
• Given an R-module M, the endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
of M, denoted End[R](M) is an R-algebra by defining (r·φ)(x) = r·φ(x).
• Any ring of matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements
isMatrices of the same size can be added or subtracted element by element...
with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication. This coincides with the previous example when M is a finitely-generated, free
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a
set is a special case of a free module on a set.-Definition:...
• Every polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial
rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
R[x[1], ..., x[n]] is a commutative R-algebra. In fact, this is the free commutative R-algebra on the set {x[1], ..., x[n]}.
• The free R-algebra
Free algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring .-Definition:...
on a set E is an algebra of polynomials with coefficients in R and noncommuting indeterminates taken from the set E.
• The tensor algebra
Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
of an R-module is naturally an R-algebra. The same is true for quotients such as the exterior
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
and symmetric algebra
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
s. Categorically speaking, the functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the
category of small categories....
which maps an R-module to its tensor algebra is left adjoint to the functor which sends an R-algebra to its underlying R-module (forgetting the ring structure).
• Given a commutative ring R and any ring A the tensor product R⊗[Z]A can be given the structure of an R-algebra by defining r·(s⊗a) = (rs⊗a). The functor which sends A to R⊗[Z]A is left adjoint to
the functor which sends an R-algebra to its underlying ring (forgetting the module structure).
Subalgebras: A subalgebra of an R-algebra A is a subset of A which is both a
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative
identity of R...
and a submodule of A. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of A.
Quotient algebras: Let A be an R-algebra. Any ring-theoretic
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like
"even number" or "multiple of 3"....
I in A is automatically an R-module since r·x = (r1
)x. This gives the
quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient
spaces of linear algebra...
A/I the structure of an R-module and, in fact, an R-algebra. It follows that any ring homomorphic image of A is also an R-algebra.
Direct products: The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication.
Free products: One can form a
free product
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated
by the elements of these subgroups, and is the “most general” group having these properties...
of R-algebras in a manner similar to the free product of groups. The free product is the
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the
direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
in the category of R-algebras.
Tensor products: The tensor product of two R-algebras is also an R-algebra in a natural way. See
tensor product of algebras
Tensor product of algebras
In mathematics, the tensor product of two R-algebras is also an R-algebra. This gives us a tensor product of algebras. The special case R = Z gives us a tensor product of rings, since rings may be
regarded as Z-algebras....
for more details.
Associativity and the multiplication mapping
Associativity was defined above quantifying over all elements of A. It is possible to define associativity in a way that does not explicitly refer to elements. An algebra is defined as a map M
(multiplication) on a vector space A:
An associative algebra is an algebra where the map M has the property
Here, the symbol
function composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument
of f instead of x...
, and Id : A → A is the
identity map
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
on A.
To see the equivalence of the definitions, we need only understand that each side of the above equation is a function that takes three arguments. For example, the left-hand side acts as
Similarly, a unital associative algebra can be defined in terms of a unit map
which has the property
Here, the unit map η takes an element k in K to the element k1 in A, where 1 is the unit element of A. The map s is just plain-old scalar multiplication:
An associative unitary algebra over K is based on a
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
A×A→A having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K→A identifying the scalar multiples of the multiplicative identity. These two morphisms can be
dualized using categorial duality by reversing all arrows in the
commutative diagram
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to
the same result by composition...
s which describe the algebra
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other
words, an axiom is a logical statement that is assumed to be true...
s; this defines the structure of a
In mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
There is also an abstract notion of
In mathematics, specifically in category theory, an F-coalgebra is a structure defined according to a functor F. For both algebra and coalgebra, a functor is a convenient and general way of
organizing a signature...
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these
abstract algebraic structures...
of an algebra is a linear map ρ: A → gl(V) from A to the general linear algebra of some vector space (or module) V that preserves the multiplicative operation: that is, ρ(xy)=ρ(x)ρ(y).
Note, however, that there is no natural way of defining a
tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other
structures or objects. In each case the significance of the symbol is the same: the most general...
of representations of associative algebras, without somehow imposing additional conditions. Here, by tensor product of representations, the usual meaning is intended: the result should be a linear
representation on the product vector space. Imposing such additional structure typically leads to the idea of a
Hopf algebra
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover
is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
or a
Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the
concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
, as demonstrated below.
Motivation for a Hopf algebra
Consider, for example, two representations
However, such a map would not be linear, since one would have
for k ∈ K. One can rescue this attempt and restore linearity by imposing additional structure, by defining a map Δ: A → A × A, and defining the tensor product representation as
Here, Δ is a comultiplication. The resulting structure is called a
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra, such that these structures are compatible....
. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be unital as well. Note that
bialgebras leave multiplication and co-multiplication unrelated; thus it is common to relate the two (by defining an antipode), thus creating a
Hopf algebra
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover
is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
Motivation for a Lie algebra
One can try to be more clever in defining a tensor product. Consider, for example,
so that the action on the tensor product space is given by
This map is clearly linear in x, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:
But, in general, this does not equal
Equality would hold if the product xy were antisymmetric (if the product were the
Lie bracket
Lie bracket
Lie bracket can refer to:*A bilinear binary operation defined on elements of a Lie algebra*Lie bracket of vector fields...
, that is,
Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the
concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
The source of this article is
, the free encyclopedia. The text of this article is licensed under the
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• SBI PO Exam Quantitative Aptitude Study Material
Digitization help student to explore and study their academic courses online, as this gives them flexibility and scheduling their learning at their convenience. Kidsfront has prepared unique course
material of Quantitative Aptitude Area and Volume for SBI PO Exam student. This free online Quantitative Aptitude study material for SBI PO Exam will help students in learning and doing practice on
Area and Volume topic of SBI PO Exam Quantitative Aptitude. The study material on Area and Volume, help SBI PO Exam Quantitative Aptitude students to learn every aspect of Area and Volume and prepare
themselves for exams by doing online test exercise for Area and Volume, as their study progresses in class. Kidsfront provide unique pattern of learning Quantitative Aptitude with free online
comprehensive study material and loads of SBI PO Exam Quantitative Aptitude Area and Volume exercise prepared by the highly professionals team. Students can understand Area and Volume concept easily
and consolidate their learning by doing practice test on Area and Volume regularly till they excel in Quantitative Aptitude Area and Volume.
Area and Volume
a) 4cm
b) 2cm
c) 3cm
d) 6cm
Solution Is : According to question
Volume of sphere= surfae area of sphere
(4/3)πr^3 = 4πr^2
⇒ r = 3cm diameter = 6 cm
^2 of floor area and 100m^3 space for air then the height of the cone of smallest size to accommodate these persons would be?
a) 18.75m
b) 16m
c) 10.25m
d) 20m
Solution Is :
a) 12cm^2
b) 72cm^2
c) 36√3cm^2
d) 144√3cm^2
Solution Is :
Let ABC is equilateral triangle.
Where sides is a cm.
⇒ Area of equilateral triangle =(?3/4)a^2)
⇒ (1/2)*9*AD = (?3/4)a^2
⇒ (1/2)*12?3 =(?3/4)a
⇒ a=24cm.
Now, Area of triangle =((?3/4)a^2)
=(?3/4)*24*24 = 148?3cm
a) 3sq.unit
b) 1 1/2sq.unit
c) 1sq.unit
d) 4 1/2sq.unit
Solution Is : X=0. 2X+3Y=6. 2X+Y=3.
2x + 3y = 6 ...(ii)
x = 0, y = 2
x = 3, y = 0
From eqn.
(iii) x+ y = 3
x = 0, y = 3
x = 3, y = 0
Area made by these three lines
= Area of triangle OBC – Area of OAC
=(1/2)*3*3-(1/2)*2*3 =(9/2)-3 =3/2 = 1 1/2 sq.
a) 1:3
b) 1:2
c) 3:4
d) 2:3
Solution Is :
a) 80
b) 70
c) 72
d) 75
Solution Is : (4) Let A,B,C,Dand E are in kg.represent their weights.Then Therefore,A+B+C=84*3=252 kg A+B+C+D=80*4=320 kg Therefore,D=(320-252)kg=68 kg E=68+3=71kg. B+C+D+E=79*4=316kg. Now, (A+B+C+D)
-(B+C+D+E)=320-316 A-E=4kg. Therefore, A=4+E=4+71=75kg.
a) 4 sq. unit
b) 3 sq. unit
c) 6 sq. unit
d) 12 sq. unit
Solution Is : X=4
❑(⇒┴ equation of a line parallel to y-axis. y=3)
❑(⇒┴ equation of a line parallel to x-axis)
Putting x=0 in the equation
3×0+4y=12 ❑(⇒┴ y=12/4=3)
?co-ordinates of the point of
intersection on y-axis=(0,3)
Again putting y=0 in the
equation 3x+4y=12,
3x+4×0=12 ❑(⇒┴ )x=12/3=4
?co-ordinates of the point of
intersection on x-axis=(4,0) AC = 3units,BC= 4 units Therefore Area of ΔABC =1/2 ×BC×AC =1/2 ×4×3 = 6 sq.units
^2 cm^2,100√3 cm and 7200 cm’ ‘respectively, then the value of P will be
a) √3
b) 3/2
c) 2/√3
d) 4
Solution Is :
a) 13
b) 9
c) 18
d) 15
Solution Is :
a) 13 cm
b) 12 cm
c) 10 cm
d) 15, cm
Solution Is :
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3.9 — Logarithmic Regression — R Practice | ECON 480: Econometrics
3.9 — Logarithmic Regression — R Practice
Set Up
To minimize confusion, I suggest creating a new R Project (e.g. regression_practice) and storing any data in that folder on your computer.
Alternatively, I have made a project in R Studio Cloud that you can use (and not worry about trading room computer limitations), with the data already inside (you will still need to assign it to an
Question 1
We are returning to the speeding tickets data that we began to explore in R Practice 3.4 on Multivariate Regression and R Practice 3.7 on Dummy Variables & Interaction Effects. Download and read in
(read_csv) the data below.
This data again comes from a paper by Makowsky and Strattman (2009) that we will examine later. Even though state law sets a formula for tickets based on how fast a person was driving, police
officers in practice often deviate from that formula. This dataset includes information on all traffic stops. An amount for the fine is given only for observations in which the police officer decided
to assess a fine. There are a number of variables in this dataset, but the one’s we’ll look at are:
Variable Description
Amount Amount of fine (in dollars) assessed for speeding
Age Age of speeding driver (in years)
MPHover Miles per hour over the speed limit
Black Dummy \(=1\) if driver was black, \(=0\) if not
Hispanic Dummy \(=1\) if driver was Hispanic, \(=0\) if not
Female Dummy \(=1\) if driver was female, \(=0\) if not
OutTown Dummy \(=1\) if driver was not from local town, \(=0\) if not
OutState Dummy \(=1\) if driver was not from local state, \(=0\) if not
StatePol Dummy \(=1\) if driver was stopped by State Police, \(=0\) if stopped by other (local)
We again want to explore who gets fines, and how much.
Question 2
Run a regression of Amount on Age. Write out the estimated regression equation, and interpret the coefficient on Age.
Question 3
Is the effect of Age on Amount nonlinear? Let’s run a quadratic regression.
Part A
Create a new variable for \(Age^2\). Then run a quadratic regression.
Part B
Try running the same regression using the alternate notation: lm(Y~X+I(X^2)). This method allows you to not have to create a new variable first. Do you get the same results?
Part C
Write out the estimated regression equation.
Part D
Is this model an improvement from the linear model?Check \(R^2\).
Part E
Write an equation for the marginal effect of Age on Amount.
Part F
Predict the marginal effect on Amount of being one year older when you are 18. How about when you are 40?
Part G
Our quadratic function is a \(U\)-shape. According to the model, at what age is the amount of the fine minimized?
Part H
Create a scatterplot between Amount and Age and add a a layer of a linear regression (as always), and an additional layer of your predicted quadratic regression curve. The regression curve, just like
any regression line, is a geom_smooth() layer on top of the geom_point() layer. We will need to customize geom_smooth() to geom_smooth(method="lm", formula="y~x+I(x^2) (copy/paste this verbatim)!
This is the same as a regression line (method="lm"), but we are modifying the formula to a polynomial of degree 2 (quadratic): \(y=a+bx+cx^2\).
Part I
It’s quite hard to see the quadratic curve with all those data points. Redo another plot and this time, only keep the quadratic stat_smooth() layer and leave out the geom_point() layer. This will
only plot the regression curve.
Question 4
Should we use a higher-order polynomial equation? Run a cubic regression, and determine whether it is necessary.
Question 5
Run an \(F\)-test to check if a nonlinear model is appropriate. Your null hypothesis is \(H_0: \beta_2=\beta_3=0\) from the regression in pert (h). The command is linearHypothesis(reg_name, c("var1",
"var2")) where reg_name is the name of the lm object you saved your regression in, and var1 and var2 (or more) in quotes are the names of the variables you are testing. This function requires
(installing and) loading the “car” package (additional regression tools).
Question 6
Now let’s take a look at speed (MPHover the speed limit).
Part A
Creating new variables as necessary, run a linear-log model of Amount on MPHover. Write down the estimated regression equation, and interpret the coefficient on MPHover \((\hat{\beta_1})\). Make a
scatterplot with the regression line.Hint: The simple geom_smooth(method="lm") is sufficient, so long as you use the right variables on the plot!
Part B
Creating new variables as necessary, run a log-linear model of Amount on MPHover. Write down the estimated regression equation, and interpret the coefficient on MPHover \((\hat{\beta_1})\). Make a
scatterplot with the regression line.Hint: The simple geom_smooth(method="lm") is sufficient, so long as you use the right variables on the plot!
Part C
Creating new variables as necessary, run a log-log model of Amount on MPHover. Write down the estimated regression equation, and interpret the coefficient on MPHover \((\hat{\beta_1})\). Make a
scatterplot with the regression line.Hint: The simple geom_smooth(method="lm") is sufficient, so long as you use the right variables on the plot!
Part D
Which of the three log models has the best fit?Hint: Check \(R^2\)
Question 7
Return to the quadratic model. Run a quadratic regression of Amount on Age, Age\(^2\), MPHover, and all of the race dummy variables. Test the null hypothesis: “the race of the driver has no effect on
Question 8
Now let’s try standardizing variables. Let’s try running a regression of Amount on Age and MPHover, but standardizing each variable.
Part A
Create new standardized variables for Amount, Age, and MPHover.Hint: use the scale() function inside of mutate()
Part B
Run a regression of standardized Amount on standardized Age and MPHover. Interpret \(\hat{\beta_1}\) and \(\hat{\beta_2}\) Which variable has a bigger effect on Amount?
Question 9
Make a regression output table with huxtable of your regressions in Questions 2, 3, 4, 6a, 6b, 6c, 7 and 8.
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Possible speed increase for parts of the Expressway to Wellington
According to Waka Kotahi NZ Transport Agency the speed limit on the Mackays to Peka Peka and Peka Peka to Ōtaki expressways could be lifted to 110 Kph in 2024.
Waka Kotahi is working to see if speed limits can be increased to 110 km/h on the Mackays to Peka Peka and Peka Peka to Ōtaki expressways next year.
Emma Speight, Director, Regional Relationships, says a speed management technical review is being done and the intention is to make a decision on speed limits for the expressways in early 2024.
“The review, along with public consultation, will see if it is appropriate to increase the speed limit on these expressways without compromising driver safety. We can consider increasing the
posted speed limit when a road is designed and constructed to modern safety standards.”
Sounds great, but before you make too many plans for how to spend all the extra time this will save you, consider that it’ll shave as much as 2 minutes off your journey. The two sections are 18 Km
and 13 Km long. In the Travel Time Calculation, “How long does it take to travel 31 Km if travelling at 100 Kph compared with travelling at 110 Kph”:
To compare the time it takes to travel a certain distance at different speeds, you can use the formula:
Time = Distance / Speed
Let’s calculate the time it takes to travel 31 kilometers at both 100 kilometers per hour (Kph) and 110 kilometers per hour (Kph):
For 100 Kph:
Time = 31 km / 100 km/h = 0.31 hours
For 110 Kph:
Time = 31 km / 110 km/h ≈ 0.2818 hours
Now, let’s convert both times from hours to minutes:
For 100 Kph:
0.31 hours * 60 minutes/hour = 18.6 minutes
For 110 Kph:
0.2818 hours * 60 minutes/hour ≈ 16.908 minutes
Comparing the times:
At 100 Kph, it takes approximately 18.6 minutes to travel 31 kilometers.
At 110 Kph, it takes approximately 16.908 minutes to travel 31 kilometers.
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Kondo spectral functions at low-tempera
SciPost Submission Page
Kondo spectral functions at low-temperatures: A dynamical-exchange-correlation-field perspective.
by Zhen Zhao
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users): Zhen Zhao
Submission information
Preprint Link: scipost_202407_00039v1 (pdf)
Date submitted: 2024-07-21 13:35
Submitted by: Zhao, Zhen
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties: • Condensed Matter Physics - Theory
Approach: Theoretical
We calculate the low-temperature spectral function of the symmetric single impurity Anderson model using a recently proposed dynamical exchange-correlation (xc) field formalism. The xc field, coupled
to the one-particle Green's function, is obtained through analytic analysis and numerical extrapolation based on finite clusters. In the Kondo regime, the xc field consists of a complex constant term
and a main quasiparticle-like oscillation term. The constant term represents the Hubbard side-band contribution, containing a bath-induced broadening effect, while the quasiparticle-like term is
related to the Kondo resonance peak at low-temperature. We illustrate these features in terms of analytical and numerical calculations for small and medium-size finite clusters, and in the
thermodynamic limit. The results indicate that the xc field formalism provides a good trade-off between accuracy and complexity in solving impurity problems. Consequently, it can significantly reduce
the complexity of the many-body problem faced by first-principles approaches to strongly correlated materials.
Author indications on fulfilling journal expectations
• Provide a novel and synergetic link between different research areas.
• Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
• Detail a groundbreaking theoretical/experimental/computational discovery
• Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Has been resubmitted
Reports on this Submission
This paper applies a novel method, the so called "dynamical exchange-correlation field" (Ref. 37), to the Anderson impurity model in order to compute spectral functions in the Kondo regime.
I find the paper quite interesting. But I think the paper lacks clarity in a couple of places. The paper would also benefit from better explanations in some parts, since the dynamical
exchange-correlation field is a novel approach. I have a couple of comments and questions that the author should address before I can agree to publication:
(1) I was at first confused by Eq. (11): How could the dynamic exchange correlation hole for $r^{\prime\prime}=r$ become time-independent and just equal to the negative density? But this follows from
Eq.(8) and the fact that the second order Green's function $G^{(2)}(r,r^\prime,r^{\prime\prime};t)$ (using the notation of Ref. 37) vanishes for $r^{\prime\prime}=r$, and thus also the correlation
function $g(r,r^\prime,r;t)=0$. I think the author should give this explanation after Eq. (11) to help the reader.
(2) Eqs. (13) and (14) are the Lehmann representations of the Green's function (which should be mentioned), and the denominators are just the partition functions $Z$. I think the equations would
become clearer if $Z$ was introduced and used. In the following equation (20), the denominators cancel anyway.
(3) To help the reader, it should be explicitly stated that Eq. (20) follows from applying the equation of motion (15) to the Lehmann representation and solving for Vxc.
(4) Sec. 3.1, after Eq. (22): I am not sure whether it is appropriate to speak of "Kondo regime" in the context of the Anderson dimer. The Kondo effect is usually associated with an impurity coupled
to a continuous band of conduction electrons.
(5) The last two sentences of Sec. 3.1, p. 8: I think this explanation for the temperature induced broadening follows simply from the Lehmann representation of the GF (13,14) which the author used to
obtain the approximation for the dynamic Vxc.
(6) Is the Vxc given by Eq. (30) valid only for $t>0$? If so, what is the corresponding equation for $t<0$? I think it would also be interesting to see Vxc in the frequency domain, i.e. the Fourier
transform of Eq. (30), which could then be compared to the self-energy for the SIAM. I suspect they must be very similar in the case of the SIAM.
(7) How did the author arrive at the hyperbolic-tangent form for $R(L)$ fitted to the data in Fig. 3b? Is that based on some theoretical background? Otherwise I think the actual functional form
cannot be extrapolated from the calculated data, since the data is still largely in the linear regime. Very different functional forms leading to very different limits $R(L=\infty)$could be
compatible with the data.
(8) It would be nice if in Figs. 4 and 5 the calculated spectra would be directly compared to the NRG spectra of Refs. [28] and [47].
I would like to thank the referee for the report. Please find the "ResponseSciPost.pdf" document below that answers directly to both referees' questions and suggestions. A revised manuscript will be
submitted with a formal list of changes.
The present manuscript applies the formalism of the so-called
"dynamical exchange-correlation (xc) field" of Ref.[37] to the single-impurity Anderson model. The authors proposes a rather simple ansatz (Eq.(30)) for this dynamical xc field to obtain the spectral
function of the Anderson model in the Kondo regime where the parameters are fixed by using the known peak positions and widths of the spectral peaks. This ansatz seems to work surprisingly well given
its simplicity.
While I find the paper interesting in general, I still have a number of
points which I would like the author to address:
1. The coefficient a_{n+,m} is defined just after Eq.(14). Shouldn't this also be sigma-dependent?
2. On Eq.(20): first, I suppose it is only meant to be valid for t>0, no?
Second, I am a bit confused about its form: why is there no explicit
dependence on the interaction U? Shouldn't it (loosely speaking) be
something like U G^(2)(t)/G(t) where G^(2) is the two-particle Green
function? Also, I don't understand the factor \aN{n+,m} \omega_{n^+,m} in the denominator. I would have expected this to be
<m| \hat{n}_{-\sigma} f_{\sigma} |n+><n+|f^{\dagger}_{\sigma} |m>.
Please clarify!
3.Please give more details on what is actually done in Sec. 3.2 and how, such that interested readers could repeat the calculations. The time-dependent variational principle is used to obtain which
quantity, the one-particle Green function of the cluster?
4.In Fig.4: could the author plot the NRG results on top of the present results for better comparison? The same applies for Fig.5.
5.In Eq.(37): I assume that the parameter \Omega_T is temperature dependent? How is this parameter determined in practice? Is it used as a fit parameter to reproduce known spectral functions? Please
show its evolution as function of temperature!
6.Finally, I noticed a typo in line 141: it should be "emphasize" instead of
To summarize, before I can recommend this manuscript for publication in SciPost Physics I would like to see the issues raised above being addressed.
I would like to thank the referee for the report. Please find the "ResponseSciPost.pdf" document that answers directly to both referees' questions and suggestions. A revised manuscript will be
submitted with a formal list of changes.
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Computational Category Theory
by D.E. Rydeheard, R.M. Burstall
Number of pages: 263
This book is an account of a project in which basic constructions of category theory are expressed as computer programs. The programs are written in a functional programming language, called ML, and
have been executed on examples. The authors have used these programs to develop algorithms for the unification of terms and to implement a categorical semantics. In general, this book is a
bridge-building exercise between category theory and computer programming. These efforts are a first attempt at connecting the abstract mathematics with concrete programs, whereas others have applied
categorical ideas to the theory of computation.
Download or read it online for free here:
Download link
(0.9MB, PDF)
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Fluid Mechanics, 6th edition
Fluid Mechanics, 6th edition
By John F. Douglas, et al.
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Pearson Education
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ISBN: 9780273717720
1012 pages
$127.50 Paper Original
The sixth edition of this established, popular textbook provides an excellent and comprehensive treatment of fluid mechanics that is concisely written and supported by numerous worked examples.
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Appendix 1 Some Properties of Common Fluids
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Return to main page of Trans-Atlantic Publications
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We’ve looked at what it means to multiply fractions, including whole and mixed numbers; now it’s time for division of fractions. We’ll start here with pictures, similar to what we did for
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Simplifying Sums and Quotients of Radicals
(A new question of the week) A recent question asked about the reasons for differences in the work of simplifying different kinds of radical expressions. We’ll look at that general question, with two
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Multiplying Fractions by Whole or Mixed Numbers
Last week we looked at how to multiply fractions, and why we do it that way. But what do we do when one of the numbers is a whole number, or when one or both are mixed numbers? And do we have to do
it the way we are taught?
Multiplying Fractions
Last week we looked at some questions about multiplication that arise once students learn to multiply fractions or decimals. Let’s turn to the underlying question: How do you multiply fractions, and
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How Can Multiplication Make It Smaller?
A fairly common question arises when students learn to multiply or divide fractions and decimals: They discover that multiplication, which always used to make numbers larger (2, multiplied by 3,
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3.7 Device Simulation
Next: 3.8 Electrical Key-Parameter Extraction Up: 3. The TCAD Concept Previous: 3.6 Contact Definition
Finally, the data structure is ready for processing with device simulation. The main inputs for device simulation are:
1. Doping concentration of different doping species (e.g. Arsen, Phosphorus, Antimony, Boron etc.) on a mesh.
2. Structural information about the shape of the region which has to be evaluated with device simulation. This information includes material types of layers, topological variation of layers and the
detailed surface and interface shapes of the materials present in the structure.
3. Named contacts as source for adjustable boundary conditions of the device simulation.
An example for a typical input structure for device simulation is shown in Figure 3.12. The three main input classes (doping, structural and contact information) can be clearly seen.
Figure 3.12: Example for a typical input structure for device simulation comprising of doping concentration (a) including the different doping species e.g. Boron (b) on a mesh and the topological
structure and contact definition (d).
The semiconductor device simulators are fairly similar in their solution approach. They all solve a system of partial differential equations describing the potential distribution and carrier
transport in a doped semiconducting material. The standard semi-classical transport theory is based on the BOLTZMANN equation [131],[132]
where 3.1) is substituted with a phenomenological term
where AXWELL-BOLTZMANN distribution function
where 3.3) for semiconductors is justified in equilibrium as long as degeneracy is not present. the carrier density
which is of general applicability. The significance of the momentum relaxation time can be understood if the electric field is switched off instantaneously and a space-independent distribution is
considered. The resulting BOLTZMANN equation is then
which shows that the ralaxation time is a characteristic decay constant for the return to the equilibrium state.
The often used drift-diffusion current equations
can be easily derived directly from the BOLTZMANN equation as outlined in Appendix D. All device simulators use the drift-diffusion approach as the simplest model to cover the transport effects
inside the semiconductor material.
Next: 3.8 Electrical Key-Parameter Extraction Up: 3. The TCAD Concept Previous: 3.6 Contact Definition R. Minixhofer: Integrating Technology Simulation into the Semiconductor Manufacturing
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Volume of
Volume of cylinders
Cylinders are ubiquitous in everyday life. For example tinned food normally comes in a can whose shape is a cylinder.
If we slice a cylinder parallel to its base, then each cross-section is a circle of the same size as the base.
Thus a cylinder has the same basic property as a prism, and we will take the formula for the volume of a cylinder to be the area of the circular base times the height. We cannot prove this formula
rigorously at this stage, because the proof involves constructing the cylinder as a limit of prisms.
If the base circle of the cylinder has radius r, then we know that the area of the circle is A = πr². If the height of the cylinder is h, then its volume is:
Volume of a cylinder = Ah = πr²h.
Example 5
A thermos flask of height 30 cm is in the shape of two cylinders, one inside the other. It has an inner radius of 8 cm and an outer radius of 10 cm. What is the volume between the two cylinders?
Solution method 1
The volume between the cylinders will be the difference in volume.
\text{Volume of the outer cylinder} &= \text{area of the base}\ ×\ \text{height}\\ &= π × 10^2 × 30\\ &= 3000 π\ \text{cm³}.\\ \text{Volume of the inner cylinder}\ &= π × 8^2 × 30\\ &= 1920π\ \text
{cm³}.\\ \text{Hence the volume between the cylinders}\ &= 3000π\ – 1920π\\ &= 1080π\ \text{cm³}\\ &\approx 3392.92\ \text{cm³ (correct to two decimal places)}.
Solution method 2
\text{Volume}\ &= \text{area of the base × height}\\ \text{Area of the base}\ &= \text{area of the annulus pictured} \text{Area}\ &= π × 10^2\ –\ π × 8^2\\ &= π × (10^2\ –\ 8^2 )\\ &= π × (100\ –\
64)\\ &=36π\ \text{cm²}.\\ \text{Hence volume}\ &= 36π × 30\\ &=1080π\ \text{cm³}\\ &\approx 3392.92\ \text{cm³ (correct to two decimal places)}.
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What Is Time Value of Money: Does It Apply to Crypto? - Phemex Academy
A dollar today is worth more than a dollar tomorrow. This well-known rule among investors is what underpins the concept of the Time Value of Money (TVM). TVM is a crucial financial principle, and
many financial products are based on utilizing it.
What Is the Time Value of Money (TVM)?
The time value of money (TVM) implies that a certain amount of money today is worth more than the same nominal amount at some point in the future. TVM is one of the most fundamental concepts in
finance and is used extensively in asset valuation, budgeting, and forecasting.
There are two principal reasons why money today is worth more than in the future:
Inflation and Returns potential
How does inflation affect the time value of money?
Inflation is a familiar companion of the economy. As inflation occurs, money loses its real value over time. For example, in the US today, you would need to pay around $123 for goods that cost you
only $100 ten years ago. In other words, that $100 has lost a certain amount of its real value over these ten years due to inflation.
It should be noted that inflation is not a guaranteed economic event, at least not in the shorter-term. There have been some periods in the history of various countries’ economies when deflation, and
overall reduction in prices, also known as negative inflation, has occurred for extended periods of time.
Examples of Time Value of Money
For example, the US economy experienced deflation during the subprime mortgage crisis in 2007/2008. Some other OECD countries, e.g., Switzerland and Greece, had deflation in 2020.
When deflation occurs, the same amount of money normally appreciates in value over time. This contradicts with the concept of the TVM. However, deflationary periods have been few and far between over
the last many decades compared to the times of inflation. As such, the overall concept of the TVM still applies in most cases. Additionally, inflation is not the only reason why money today is worth
more than in the future. The second reason is the returns potential.
Returns potential
One hundred dollars you have access to now can be invested to generate returns over a period of time, e.g., one year. On the other hand, if you received the same $100 in a year’s time, you will have
missed out on the returns potential. This potential to generate returns is the second concept underpinning the TVM.
If you were given $100 five years ago, in November 2016, and invested it in the S&P 500-based stock market index, that investment would be worth $215 now.
The S&P 500 chart from November 2016 to November 2021 (Source: Yahoo Finance)
However, if you were simply promised to be given that $100 five years later, in November 2021, you would be handed that same $100 bill now, a loss of a $115 opportunity.
How to Calculate TVM?
The TVM formula helps you estimate the future value of a certain amount you have on your hands now. Here is the base formula for TVM:
FV = CV * (1 + r) ⁿ
FV = Future Value
CV = Current Value
r = interest rate
n = number of years (of investment)
Let’s have a look at an example with a $100 amount and estimate its future value two years from now, assuming that we invest it with a yearly return of 30%.
FV = $100 * (1 + 30%) ² = $169
Thus, our $100 has a future value of $169 two years from today if invested with an expected annual return of 30%.
The base TVM formula above is often adjusted when calculating future values for a variety of financial products. That is because many investments have periodic yield payouts multiple times a year,
e.g., on a monthly or quarterly basis.
The base TVM formula does not account for such periodicity and simply assumes that the return is paid out on a yearly basis.
The Adjusted TVM Formula
However, if periodic returns are paid out and added to the principal amount, the investment can benefit from the effect of compounding. In this case, the future value will be somewhat different.
Below is the adjusted TVM formula that takes into account periodic payouts that compound the initial investment:
FV = CV * (1 + (r / t)) ^(ⁿ * ^t)
t = number of compounding periods per year
If we assume that there are quarterly compounding periods, i.e., four per year, in our earlier example, then the adjusted future value of that $100 will now be equal to $178:
FV = $100 * (1 + (30% / 4)) ^(2 * 4) = $178
How to Adjust the TVM Calculation for Inflation?
Both of the formulas you saw in the previous section account for the expected return on investment when calculating the future value of money. However, as noted earlier, the TVM is affected both by
our expected returns as well as by the inflation rate.
Thus, to make our estimates as precise as possible, it is recommended that we adjust the investment return figures for inflation. This is known as the inflation-adjusted return:
IAR = (1 + return rate) / (1 + inflation rate) – 1
In our example, if we assume that over the time of our investment, the annual inflation is going to be 3%, then instead of the 30% unadjusted return figure, we would use 26.2% since:
IAR = (1 + 30%) / (1 + 3%) – 1 = 26.2%
Does Time Value of Money (TVM) Apply to Cryptocurrency?
The TVM concept is applicable to any asset, either crypto or from the world of traditional finance, that may be subject to inflationary effects and/or could earn you positive returns over a period of
As such, it is applicable to the majority of cryptocurrencies. Only highly deflationary crypto assets with virtually no returns potential might be worth more today than in the future, i.e., they
might not align with the concept of TVM.
The Time Value of Bitcoin
Let’s consider the example of Bitcoin (BTC), the world’s largest cryptocurrency. Bitcoin is often believed to be a deflationary cryptocurrency. However, Bitcoin has only limited deflationary
potential. When people talk about Bitcoin’s deflation, they presume the halving of mining rewards every four years, which is in-built into Bitcoin’s operational mechanism.
However, the halving only reduces the rate at which Bitcoin’s supply grows, it does not reduce the overall supply over time. In fact, Bitcoin’s supply has been growing, albeit at an increasingly
slower rate, since its launch in 2009.
Thus, Bitcoin does have a positive inflation rate, which currently stands at around 1.8%. The maximum supply of Bitcoin is capped at 21 million, and only when this limit is reached, the
cryptocurrency will stop being inflationary. However, this is projected to happen only by February 2140. Unless you plan to board a time machine, you can safely assume that Bitcoin will have at least
some degree of inflation in your lifetime.
In addition to inflation, the second condition that an asset must satisfy to provide a positive future value is an expectation of positive returns.
Similar to fiat money, Bitcoin could be invested for yield, as it is a relatively liquid asset that is easily exchanged for fiat currencies or other cryptos. Bitcoin itself, within its own blockchain
ecosystem, cannot be invested to earn yield. The BTC network has little in-built capacity for such decentralized finance (DeFi) functionality.
However, Bitcoin could be easily swapped for other crypto assets, including the so-called wrapped Bitcoin tokens, and invested for yield on blockchain platforms that run these crypto deposit apps.
The majority of these apps reside on Ethereum (ETH).
Alternatively, Bitcoin funds may be exchanged for fiat currency, such as USD, on crypto exchanges and then invested in any traditional financial product, e.g., stocks, index funds, mutual funds,
exchange-traded funds (ETFs), derivatives, commodities, and many more.
Increasingly, some financial brokerages and funds accept payments in BTC, so the intermediary step of exchanging BTC to fiat currency may not be needed.
The only issue with BTC investing, whether it is on DeFi platforms or in the traditional financial markets, is the transaction fees incurred due to all these transfers away from or back to the BTC
network. These fees may add up easily and erode potential investment earnings. Thus, it is important to account for these expenses when calculating any potential returns on an investment that takes
BTC funds off the platform.
Why is Time Value of Money Important?
TVM is a fundamental concept in finance, implying that a sum of money in the present is worth more than the same amount in the future. Money is said to have this characteristic for two key reasons –
inflation, which reduces the real value of money over time, and earnings potential, the opportunity to invest the money and earn interest/yield from it.
TVM is applicable to any asset with at least some inflationary properties and/or potential for positive returns in the future. This applies to the majority of cryptocurrencies, including Bitcoin. The
Bitcoin you own in the present can be invested for yield, either by transferring it to platforms that support yield-earning DeFi apps or by transferring the funds to yield-earning traditional
financial products, such as stocks, index funds, derivatives, and others.
Bitcoin also has some inflationary characteristics, despite many people’s assumption that it is a “deflationary coin.” The deflationary property of Bitcoin refers to the halving of mining rewards
every four years, which decelerates but does not eliminate Bitcoin’s inflation, as measured by the increase in its supply. Bitcoin will stop being truly inflationary only by around early 2140.
Until that time, and while Bitcoin can be legally invested for a positive yield, the TVM concept will apply to the world’s leading cryptocurrency.
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Understanding the True Costs and Benefits of Refinancing a Property: A Closer Look at Mortgage Calculations
When it comes to property investment, understanding the financials behind your mortgage is essential to ensuring long-term profitability. Let’s delve into the specifics of refinancing a property,
particularly focusing on how much you can borrow, what that borrowing will cost you, and what your true net cash flow looks like after all expenses are considered.
Step 1: Refinancing Your Property
So, you’ve purchased a property below market value, invested in renovations, and now it’s time to refinance. Imagine your property is now valued at £100,000. The key question is: how much will the
bank lend you against this newly appraised value?
In most cases, banks will lend up to 75% of the property’s value. This percentage is known as the loan-to-value (LTV) ratio. For a property valued at £100,000, the bank would offer you a mortgage of
£75,000 (calculated as £100,000 x 0.75). This is the amount you can leverage to either recoup your initial investment or reinvest in other opportunities.
Step 2: Calculating the Mortgage Cost
Next, you need to consider the cost of borrowing this money. Mortgage interest rates fluctuate, but let’s work with a conservative estimate of 6%, which represents a worst-case scenario in the
current market. To calculate your monthly mortgage payment, you’ll apply the following steps:
1. Determine Annual Interest: Multiply the mortgage amount by the interest rate.£75,000×0.06=£4,500 per year£75,000 \times 0.06 = £4,500 \text{ per year}£75,000×0.06=£4,500 per year
2. Convert to Monthly Payment: Divide the annual interest by 12 to get the monthly cost.£4,500/12=£375 per month£4,500 / 12 = £375 \text{ per month}£4,500/12=£375 per month
So, your mortgage payment to the bank is £375 per month. This figure represents the cost of borrowing the £75,000.
Step 3: Understanding Rental Income and Expenses
Now that we’ve established the cost of the mortgage, let’s compare it to the income generated by renting out the property. Suppose you can rent the property for £750 per calendar month (PCM). This
rental income is the first step in calculating your net cash flow, but it’s important to remember that it isn’t the entire picture.
Deducting Management and Maintenance Costs
Managing a rental property comes with additional costs, primarily letting and management fees and a reserve for maintenance expenses:
1. Letting and Management Fees: Typically, you’ll pay your letting agent around 10% of the rental income for their services. In this case, that’s £75 per month.
2. Maintenance Reserve: It’s wise to set aside another 10% of the rental income (£75 per month) for ongoing maintenance and unexpected repairs. This reserve ensures you’re covered for minor issues
like broken door handles or more significant repairs.
Step 4: Calculating True Net Cash Flow
To determine your true net cash flow, you subtract all these expenses from your rental income:
1. Total Expenses: Add the mortgage payment, letting and management fees, and the maintenance reserve:£375+£75+£75=£525 per month£375 + £75 + £75 = £525 \text{ per month}£375+£75+£75=£525 per month
2. Net Cash Flow: Subtract the total expenses from the rental income:£750−£525=£225 per month£750 – £525 = £225 \text{ per month}£750−£525=£225 per month
This £225 is your true net cash flow, representing the money that remains in your pocket after all costs are accounted for.
Step 5: The Importance of Understanding Net Cash Flow
Understanding your true net cash flow is crucial because it gives you a realistic view of the profitability of your investment. It’s easy to overlook costs like management fees and maintenance, but
failing to account for them can lead to overestimating your profits.
In this example, while the gross rental income is £750, the true profit, after accounting for all expenses, is £225 per month. This figure assumes your tenant pays rent consistently each month
without any issues.
Final Thoughts
The process of refinancing a property and calculating the associated costs and benefits can seem complex, but it’s essential for any property investor to master. By understanding how much you can
borrow against your property’s value, the cost of that borrowing, and your true net cash flow after all expenses, you can make informed decisions about your investments.
Remember, while the allure of a high rental income might be tempting, it’s the net cash flow that ultimately determines the success of your property investment. Always factor in all possible
expenses, including management fees and maintenance reserves, to get an accurate picture of your investment’s profitability. This careful calculation is the key to sustainable and profitable property
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