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samples/texts/102047/page_1.md

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+ON TWO POLYNOMIAL SPACES
+ASSOCIATED WITH A BOX SPLINE
+
+CARL DE BOOR, NIRA DYN, AND AMOS RON
+
+The polynomial space $\mathcal{H}$ spanned by the integer translates of a box spline $M$ admits a well-known characterization as the joint kernel of a set of homogeneous differential operators with constant coefficients. The dual space $\mathcal{H}^*$ has a convenient representation by a polynomial space $\mathcal{P}$, explicitly known, which plays an important role in box spline theory as well as in multivariate polynomial interpolation.
+
+In this paper we characterize the dual space $\mathcal{P}$ as the joint kernel of simple differential operators, each one a power of a directional derivative. Various applications of this result to multivariate polynomial interpolation, multivariate splines and duality between polynomial and exponential spaces are discussed.
+
+**1. Introduction.** The space $H(\phi)$ of all exponentials in the linear span $S(\phi)$ of the integer translates of a compactly supported distribution $\phi$ is of basic importance in multivariate spline theory since, in principle, it allows the construction of good approximation maps to $S(\phi)$ from spaces containing $S(\phi)$. Generically, $H(\phi)$ is $D$-invariant (i.e., closed under differentiation), hence is the joint kernel for a set $\{p(D): p \in I_{H(\phi)}\}$ of differential operators with constant coefficients, with $I_{H(\phi)}$ a polynomial ideal of finite codimension (in the space $\Pi$ of all multivariate polynomials, i.e., an ideal of transcendental dimension 0, hence with finite variety). An understanding of the interplay between the space $H(\phi)$ and its associated ideal $I_{H(\phi)}$ is useful in the determination of the basic properties of $H(\phi)$ such as its spectrum, its dimension, and its local approximation order.
+
+For the important special case when $\phi$ is a polynomial box spline (and $\mathcal{H} := H(\phi)$ is thus a polynomial space), an explicit set of generators for the ideal $I_\mathcal{H}$ is known [BH1], but nevertheless, the construction of their joint kernel was found to be very difficult. At the same time, a polynomial space $\mathcal{P}$ (of very simple structure) is known which serves as a natural dual for $\mathcal{H}$ and is of substantial use in the analysis of $\mathcal{H}$. Specifically, the duality between $\mathcal{H}$ and $\mathcal{P}$ has been used in [DR1] in the investigation of the local approximation order of some exponential spaces, in [DR1,2] in the solution of an

samples/texts/102047/page_10.md

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+On the other hand, by (4.4) and (4.6),
+
+$$ \dim(\Pi_j \cap \mathcal{P}(X)) = \dim(\Pi_j \cap \mathcal{H}(X)), \quad \forall j, $$
+
+and (5.3) now follows from the fact that $\Pi(M_X) = \mathcal{H}(X)$. $\square$
+
+We note that the result is no longer valid if we drop the unimodularity assumption (cf. [R; Ex. 4.1]).
+
+**6. A remark on (2.11) Lemma.** A careful examination of the proof of (2.11) Lemma shows that the details of the connection between the ideal $I^X$ and its kernel $I^X \perp$ enter into the argument in only a minor way. The only facts used are: (i) the map $\Pi \to L(\Pi): p \mapsto p(D)$ is a ring homomorphism; and (ii) for any basis $B$ in $X$, $\dim I^B \perp = 1$.
+This suggests the following result.
+
+(6.1) PROPOSITION. Let $M: \Pi \to L(V)$ be a ring-homomorphism into the ring of linear maps on the linear space $V$. For a given multiset $X$ of directions, define
+
+$$ I_M^X \perp := \bigcap_{p \in I^X} \ker M(p). $$
+
+Then
+
+$$ \dim I_M^X \perp \leq \sum_{B \in \mathfrak{B}(X)} \dim I_B^X \perp. $$
+
+The proof of the proposition follows entirely that of (2.11) Lemma, with the obvious modifications whenever the induction hypothesis is applied.
+
+It would be nice to identify other settings rather than the one utilized in this paper, where the above proposition is of use.
+
+REFERENCES
+
+[BH1] C. de Boor and K. Höllig, *B-splines from parallelepipeds*, J. d'Anal. Math., **42** (1982/3), 99-115.
+
+[BH2] __________, *Bivariate box splines and smooth pp functions on a three-direction mesh*, J. Comput. Applied Math., **9** (1983), 13-28.
+
+[BR1] C. de Boor and A. Ron, *On multivariate polynomial interpolation, Constructive Approx.*, **6** (1990), 287-302.
+
+[BR2] __________, *On polynomial ideals of finite codimension with application to box spline theory*, J. Math. Anal. Appl., to appear.
+
+[DM1] W. Dahmen and C. A. Micchelli, *Translates of multivariate splines*, Linear Algebra and Appl., **52/3** (1983), 217-234.

samples/texts/102047/page_12.md

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+interpolation problem induced by $\mathcal{H}$, in [J] in the construction of
+linear projectors onto a box spline space, and in [DR1] in the com-
+putation of the homogeneous degrees of $\mathcal{H}$ (which is equivalent to
+computing the Hilbert function of $I_{\mathcal{H}}$). See also [DM3, § 3].
+
+It is the purpose of this paper to establish the surprising result that
+$\mathcal{P}$, too, is the joint kernel of a rather simple set of constant coefficient
+differential operators, each being just a power of a directional deriva-
+tive. This allows us to characterize $\mathcal{P}$ in terms of the degrees of its
+polynomials when restricted to certain linear manifolds. Various ap-
+plications of this result to multivariate polynomial interpolation, box
+spline theory, and duality between polynomial and exponential spaces
+are discussed as well.
+
+In §2, after defining the space $\mathcal{P}$ and its associated differential op-
+erators, we prove that $\mathcal{P}$ is indeed the joint kernel of these operators.
+In §3, we identify $\mathcal{P}$ as a space of least degree among all polyno-
+mial spaces that interpolate correctly on certain subsets of the integer
+lattice. As a matter of fact, the discussion in that section may have
+an independent value: this discussion illustrates how the interpolating
+space of least degree from [BR1] may be computed using the tech-
+nique from [BR2] of “perturbing the generators of a homogeneous
+ideal”, hence in a computationally painless way.
+
+Section 4 is devoted to the more general discussion of duality be-
+tween finite-dimensional polynomial and exponential spaces, a dis-
+cussion which improves proofs and results from [DM2] and [DR1].
+Finally, we discuss in §5 the construction of piecewise polynomials on
+the support of a box spline and improve thereby an observation in
+[R].
+
+**2. The main result.** Let $X$ be a multiset of vectors in $\mathbb{R}^s \setminus \{0\}$. We will at times think of $X$, equivalently, as a real matrix, of size $(s \times |X|)$. Let $\mathbb{H}(X)$ denote the collection of all hyperplanes (i.e., linear subspaces of codimension 1) which are spanned by some columns of $X$. We associate with each $h \in \mathbb{H}(X)$ a nontrivial linear polynomial which vanishes on $h$, and write this polynomial
+
+$$ (h^\perp, \cdot), $$
+
+thus using $h^\perp$ to stand for any particular nonzero vector normal to $h$. We are interested in the ideal $I^X$ generated by all polynomials of the form
+
+$$ (2.1) \qquad p_h := p_{h,X} := \langle h^\perp, \cdot \rangle^{(X\setminus h)}, \quad h \in \mathbb{H}(X), $$
+
+where $X\setminus h := \{x \in X \mid x \notin h\}$.

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+(2.9) LEMMA.
+
+$$
+b(X) \leq \dim \mathcal{P}(X).
+$$
+
+(2.10) LEMMA.
+
+$$
+\mathcal{P}(X) \subset I^{\times}_{\perp}.
+$$
+
+(2.11) LEMMA.
+
+$$
+\dim I^{\mathcal{X}} \perp \leq b(X).
+$$
+
+Proof [DR1] of (2.9) Lemma. Every polynomial
+
+$$
+q_B := \prod_{x \in X \setminus B} (\langle x, \cdot \rangle - \lambda_x),
+$$
+
+with $B \in \mathbb{B}(X)$ and $(\lambda_x)$ arbitrary constants, is in $\mathcal{P}(X)$, as follows
+readily by multiplying out. For each $B \in \mathbb{B}(X)$, there is a unique
+common zero of the $s$ linear factors $\langle x, \cdot \rangle - \lambda_x$, $x \in B$, which do
+not occur in the associated $q_B$; call that point $\theta_B$. Choose now, as
+we may, the constants $(\lambda_x)$ in such a way that $\theta_B \neq \theta_{B'}$ whenever
+$B \neq B'$. (In fact, almost every choice of the $\lambda_x$ would satisfy this
+condition.) It then follows that
+
+$$
+q_B(\theta_{B'}) = 0 \Leftrightarrow B \neq B',
+$$
+
+proving the linear independence of the collection $(q_B)_{B \in \mathbb{B}(X)}$ in
+$\mathcal{P}(X)$. $\square$
+
+*Proof of (2.10) Lemma.* We have to prove that, for each $h \in \mathbb{H}(X)$,
+$p_h(D) = (D_{h^\perp})^{#(X\setminus h)}$ annihilates $\mathcal{P}(X)$, i.e., that $p_h(D)p_V = 0$ for
+every $V \subset X$ for which $X\setminus V$ contains a basis. For this, we note that
+$D_{h^\perp}p_{V \cap h} = 0$; hence
+
+$$
+p_h(D)p_V = (p_{V \cap h}) p_h(D) p_{V \setminus h}.
+$$
+
+On the other hand, since $X\setminus V$ contains a basis, $V\setminus h$ cannot co-
+incide with $X\setminus h$, hence $\deg p_{V\setminus h} < \#X\setminus h = \deg p_h$ and therefore
+$p_h(D)p_{V\setminus h} = 0$. $\square$
+
+*Proof of (2.11) Lemma.* We prove the lemma by induction on #X.
+For the case #X = s we observe that I^X is generated by s linearly in-
+dependent linear homogeneous polynomials, consequently I^X ⊥ con-
+tains only constants, and so dim I^X ⊥ = 1 = b(X). Assume now that
+#X > s. We follow the argument used in the proof of [DM3; Thm.
+3.1], decompose X as
+
+$$
+X := X' \cup X,
+$$

samples/texts/102047/page_2.md

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+this notion of the least term $f_{\perp}$ of $f$, we then define
+
+$$ (3.2) \qquad H_{\perp} := \operatorname{span}\{f_{\perp} : f \in H\}, $$
+
+and have [BR1]
+
+$$ (3.3) \qquad \dim H_{\perp} = \dim H. $$
+
+We say that a pointset $\Theta \subset \mathbb{R}^s$ is correct for the polynomial space $P$
+if the restriction map $P \to C^\Theta: p \mapsto p|_\Theta$ is invertible. Equivalently, $\Theta$
+is correct for $P$ if, for every data $(d_0)_{\theta \in \Theta}$, there is exactly one $p \in P$
+for which $p(\theta) = d_0$ for all $\theta \in \Theta$. In other words, interpolation
+from $P$ at the points of $\Theta$ is correct.
+
+**RESULT [BR1].** $\Theta$ is correct for $(\exp_{\Theta})_{\downarrow}$. Moreover, among all polynomial spaces $P$ for which $\Theta$ is correct, $(\exp_{\Theta})_{\downarrow}$ is "of least degree" in the sense that
+
+$$ (3.4) \qquad \dim(P \cap \Pi_j) \le \dim((\exp_{\Theta})_{\downarrow} \cap \Pi_j), \quad \forall j. $$
+
+Here,
+
+$$ (3.5) \qquad \exp_{\Theta} := \operatorname{span}\{e_{\theta}\}_{\theta \in \Theta}, $$
+
+where $e_{\theta}: x \mapsto e^{<\theta, x>}$ is the exponential function (with frequency $\theta$).
+
+In view of this result, it is useful to be able to identify the “least”
+space $(\exp_{\Theta})_{\downarrow}$ for given $\Theta$. This we now do for certain pointsets
+$\Theta = \nu_X$ associated with the box splines $M_X$. Our tool is the following
+result
+
+(3.6) **RESULT [BR2].**
+
+$$ H_{\perp} = (I_{H_{\perp}})^{\perp}, $$
+
+which obtains $H_{\perp}$ as the kernel of the ideal generated by the leading
+terms of the annihilator of $H$. Precisely, [BR2; (4.3) Theorem (b)]
+provides the statement that, for any polynomial ideal $I$ of finite codi-
+mension,
+
+$$ (I^{\perp})_{\downarrow} = (I_{\uparrow})^{\perp}, $$
+
+and (3.6) Result is obtained by applying this to
+
+$$ (3.7) \qquad I = I_H := \{p \in \Pi : p(D)H = 0\}, $$
+
+which is an ideal since $H$ is closed under differentiation. In fact, with
+$H = \exp_{\Theta}$, $I_H$ consists of all polynomials which vanish at $\Theta$; hence
+$I_H$ has finite codimension and
+
+$$ H = I_H^{\perp}, $$

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+(cf. [BR2, §3]). From $I_H$, we obtain its homogeneous counterpart $I_{H\uparrow}$ as
+
+$$ (3.8) \qquad I_{H\uparrow} := \operatorname{span}\{p_\uparrow : p \in I_H\}, $$
+
+where $p_\uparrow$ is the leading term of the polynomial $p$, namely the homogeneous polynomial satisfying
+
+$$ \deg(p - p_\uparrow) < \deg p. $$
+
+The result (3.6) is of interest here since it is easy to identify elements of $I_{H\uparrow}$ in case $H = \exp_\Theta$: If $(p_j)$ are linear homogeneous polynomials for which the union of the corresponding hyperplanes $\{x \in \mathbb{R}^s : p_j(x) = c_j\}$ (for suitable choices of the constants $c_j$) contains $\Theta$, then $p := \prod_j (p_j - c_j) \in I_H$; hence $\prod_j p_j \in (I_{H\uparrow})$. If we obtain enough of these $p$ to generate all of $I_{H\uparrow}$, then we know by (3.6) Result that $H_↓$ is the joint kernel of all the corresponding differential operators $p(D)$. In fact, since we know from (3.3) that $\dim H_↓ = \dim H$, we can already reach this conclusion when we only know that the $p$ so constructed generate an ideal $J$ of codimension $\le \dim H$.
+
+(3.9) RESULT [BR2]. If the ideal $J$ generated from the leading terms of some polynomials in $I_{H\uparrow}$ has codimension $\le \dim H$, then $J = I_{H\uparrow}$; therefore
+
+$$ H_↓ = J \perp. $$
+
+In our case, we have identified (in (2.7) Theorem) $\mathcal{P}(X)$ as the joint kernel of the differential operators $(D_h)^{#\{X\setminus h\}}$ (with $h$ running over $\mathbb{H}(X)$), hence are entitled to conclude that $(\exp_\Theta)_↓ = \mathcal{P}(X)$ whenever we can find, for each such $h$, constants $c_{j,h}$ so that
+
+$$ \prod_{j=1}^{#\{X\setminus h\}} ((h^\perp, \cdot) - c_{j,h}) $$
+
+vanishes on $\Theta$, and know additionally that $\# \Theta \ge \dim \mathcal{P}(X)$.
+
+Just such a pointset is, under certain assumptions on $X$, provided by
+
+$$ (3.10) \quad \nu_X := \nu_X(z) := \left\{ \alpha \in \mathbb{Z}^s : z - \alpha = \sum_{x \in X} t_x x, 0 < t_x < 1, \forall x \right\}, $$
+
+with $z \in \mathbb{R}^s \setminus \bigcup_{h \in \mathbb{H}(X)} (h + \mathbb{Z}^s)$. The set $-\nu_X$ comprises the integer points in the support of the shifted box spline $M_X(\cdot + z)$ (cf. [DM2]).

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+Let $I_X$ be the ideal generated by all polynomials of the form $p_V = \prod_{v \in V} (v, \cdot)$ with $V \subset X$ and $\text{span}(X \setminus V) \neq \mathbb{R}^s$. This means that
+
+$$ (4.1) \qquad I_X := \text{ideal}\{p_{X\setminus h} : h \in \mathbb{H}(X)\}. $$
+
+Let $\mathcal{H}(X)$ denote the kernel of $I_X$, i.e.,
+
+$$ \mathcal{H}(X) := I_X \perp . $$
+
+It is known [BH1], [DM1] that $\mathcal{H}(X)$ is a finite-dimensional polynomial space and [DM2] that
+
+$$ (4.2) \qquad \dim \mathcal{H}(X) = b(X). $$
+
+The spaces $\mathcal{P}(X)$ and $\mathcal{H}(X)$ are dual to each other in the following sense. Each $p \in \Pi$ gives rise to a linear functional $p^*$ on $\Pi$ (and even on a larger space of smooth functions), viz. the linear functional
+
+$$ p^*: q \mapsto p(D)q(0). $$
+
+This allows us to consider, for any two finite-dimensional linear polynomial spaces $Q$ and $R$, the map
+
+$$ M: Q \to R^* : p \mapsto p^*|_R. $$
+
+If $M$ is invertible, we say that $Q$ is *dual to* $R$ (in the sense that we can then use the elements of $Q$ in this fashion to represent uniquely the dual of $R$). Note that the dual to $M$ carries $R^{**} = R$ in the same way to $Q^*$; hence $Q$ is dual to $R$ iff $R$ is dual to $Q$.
+
+A necessary and sufficient condition for such duality is that $M$ be 1-1 and $\dim R \le \dim Q$ (since then $M$ is necessarily onto). In particular, if $\dim Q = \dim R$, then such duality is assured as soon as we know that, for every $q \in Q \setminus 0$, there is an $r \in R$ for which $q^*(r) \ne 0$. By the duality already mentioned, this is equivalent to having, for every $r \in R \setminus 0$, a $q \in Q$ for which $q^*(r) \ne 0$.
+
+For $q \in Q$, the linear functional $q^*$ cannot tell the difference between $f$ and $T_k f := f_0 + \cdots + f_{k-1} := \text{the power expansion of } f$ up to order $k$, if $k$ is sufficiently large. This allows us to extend this notion of duality to pairs $Q, R$ in which $R$ is a finite-dimensional space of smooth functions.
+
+We will eventually make use of the following observation:
+
+(4.3) **THEOREM.** Let $P$ be an $n$-dimensional homogeneous polynomial space. Let $H$ be an $n$-dimensional space of entire functions. If $P$ is dual to $H_\downarrow$, then $P$ is dual to $H$.
+
+*Proof.* For any $f \in H\setminus 0$, $f_\downarrow \in H_\downarrow\setminus 0$; hence, by assumption, $p^*(f_\downarrow) \ne 0$ for some $p \in P$. Further, since $P$ is spanned by homogeneous

samples/texts/102047/page_7.md

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+ideal (viz. $I_X$) whose kernel is $\mathcal{H}(X)$. Specifically, given any map
+$\Gamma: X \to \mathbb{C}: x \mapsto \Gamma_x$, we consider the ideal
+
+$$
+(4.7) \qquad I_{\Gamma} := \operatorname{ideal}\{q_h : h \in \mathbb{H}(X)\},
+$$
+
+with
+
+$$
+q_h := \prod_{x \in X \setminus h} (\langle x, \cdot \rangle - \Gamma_x).
+$$
+
+Then $\mathcal{H}(\Gamma) := I_{\Gamma} \perp$ is an exponential space (i.e., a space which is spanned by certain products of exponentials with polynomials), and [BR2]
+
+$$
+(4.8) \qquad \mathcal{H}(\Gamma)_\downarrow = \mathcal{H}(X).
+$$
+
+The diagram below illustrates for a unimodular $X$ the various connections established so far between the ideals $I_X^x$, $I_X$, their kernels and the associated exponential spaces.
+
+$$
+\begin{tikzcd}[column sep=2.5em, row sep=2.5em]
+(\exp_{\nu_X})_\downarrow \arrow[r] & \mathcal{P}(X) \arrow[r] & I_X^\perp \\
+& \arrow[uur]^{dual} & \\
+\mathcal{H}(\Gamma)_\downarrow & \mathcal{H}(X) & I_X^\perp
+\end{tikzcd}
+$$
+
+Our first corollary improves [DM2; Thm. 4.1]:
+
+(4.9) COROLLARY. Let $X$ be unimodular. Then $\nu_X$ is correct for $\mathcal{H}(X)$, and $\mathcal{H}'(X)$ is of least degree among all polynomial spaces for which $\nu_X$ is correct.
+
+*Proof*. We apply (4.3) Theorem with $H = \exp_{\nu_X}$ and $P = \mathcal{H}(X)$.
+By (3.12) Theorem, $H_\downarrow = \mathcal{P}(X)$, while by (4.6) Result, $\mathcal{P}(X)$ is
+dual to $\mathcal{H}(X)$. Since also $\mathcal{H}(X)$ is homogeneous (as the kernel of
+a homogeneous ideal), (4.3) Theorem implies that $\mathcal{H}(X)$ is dual to
+$\exp_{\nu_X}$, which is equivalent (cf. [BR1; §4]) to the correctness of $\nu_X$ for
+$\mathcal{H}(X')$. Moreover, since $\mathcal{P}(X)$ and $\mathcal{H}(X)$ are both homogeneous
+and, by (4.6) Result, dual to each other, we must have
+
+$$
+(4.10) \qquad \dim(\Pi_j \cap \mathcal{H}(X)) = \dim(\Pi_j \cap \mathcal{P}(X))
+$$
+
+for every $j$, by (4.4) Proposition. Since we already know by (3.12)
+Theorem that $\mathcal{P}(X)$ is of least degree among all polynomial spaces
+for which $\nu_X$ is correct, it follows from (4.10) that $\mathcal{H}(X)$ has the
+same property.
+$\square$

samples/texts/102047/page_8.md

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+(4.11) COROLLARY [DR1]. The spaces $\mathcal{P}(X)$ and $\mathcal{H}(\Gamma)$ are dual to each other.
+
+*Proof*. Take $P = \mathcal{P}(X)$ and $H = \mathcal{H}(\Gamma)$ in (4.3) Theorem. Since, by (4.8), $\mathcal{H}(\Gamma)_\downarrow = \mathcal{H}(X)$, and, by (4.6) Result, $\mathcal{H}(X)$ is dual to $\mathcal{P}(X)$, (4.3) Theorem provides the desired result. $\square$
+
+We refer to [DR1; §7] for a discussion of the interpolation conditions induced by $\mathcal{H}(\Gamma)$.
+
+5. **Application to box splines.** In this section we point out some connections between the results of the previous sections and the theory of multivariate splines. In the discussion here the (polynomial) box spline $M_X$ associated with a set of directions $X$ plays a central role. For our purposes, it is sufficient to note that $M_X$ is a piecewise-polynomial function supported on
+
+$$ \Omega_X := \{Xt : t \in [0, 1]^{\#X}\} $$
+
+and satisfies
+
+$$ \Pi(M_X) = \mathcal{H}(X), $$
+
+where, for a general compactly supported $\phi$, the notation $\Pi(\phi)$ stands for the space of polynomials spanned by the integer translates of $\phi$.
+
+We make use of the following result, which is a special case of [R; Thm.1.1]:
+
+(5.1) **RESULT.** Let $P$ be a translation-invariant space of polynomials, and $\Omega$ a compact subset of $\mathbb{R}^s$ with boundary $\partial\Omega$. Then the following conditions are equivalent:
+
+(a) There exists a function $\phi$ supported in $\Omega$ and satisfying $\Pi(\phi) = P$, $\tilde{\phi}(0) \neq 0$.
+
+(b) For every $z \in \mathbb{R}^s \setminus \bigcup_{\alpha \in Z^s} \alpha + \partial\Omega$, the set
+
+$$ v_{\Omega} := v_{\Omega}(z) := \{\alpha \in \mathbb{Z}^s : z - \alpha \in \Omega\} $$
+
+is total for $P$, i.e., no element of $P\backslash 0$ vanishes on this set.
+
+Note that when we take $\Omega$ to be $\Omega_X = \text{supp } M_X$, the sets $v_\Omega$ are identical with the sets $v_X$ from (3.10). Thus, by appealing to (3.12) Theorem, we deduce from the implication (b) ⇒ (a) of (5.1) Result

samples/texts/1072043/page_13.md

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+Criterion (AIC), as shown in equation (16), is used to calculate a comparison of models with different structures.
+
+$$
+AIC = \log V + \frac{2d}{N} \tag{16}
+$$
+
+Waveforms of input and output from the experimental setup consist of DC voltage, DC output current, AC voltage and AC output current. Model properties, estimators, percentage of accuracy, Final Prediction Error - FPE and Akaikae Information Criterion - AIC of the model are shown in Table 3. Examples of voltage and current output waveforms of multi input-multi output (MIMO) model in steady state condition (FVMC) having accuracy 97.03% and 91.7 % are shown in Fig 21.
+
+<table>
+    <thead>
+        <tr>
+            <th>Type</th>
+            <th>I/P</th>
+            <th>O/P</th>
+            <th>Linear model parameters<br/>[nb<sub>1</sub> nb<sub>2</sub> nb<sub>3</sub> nb<sub>4</sub>] poles<br/>[nf<sub>1</sub> nf<sub>2</sub> nf<sub>3</sub> nf<sub>4</sub>] zeros<br/>[nk<sub>1</sub> nk<sub>2</sub> nk<sub>3</sub> nk<sub>4</sub>] delays</th>
+            <th>% fit Voltage Current</th>
+            <th>FPE</th>
+            <th>AIC</th>
+        </tr>
+    </thead>
+    <tbody>
+        <tr>
+            <td colspan="7">Steady state conditions</td>
+        </tr>
+        <tr>
+            <td rowspan="3">FCLV</td>
+            <td rowspan="3">DZ</td>
+            <td rowspan="3">DZ</td>
+            <td>[4 4 3 5];</td>
+            <td>87.3</td>
+            <td rowspan="3">3,080.90</td>
+            <td rowspan="3">10.9</td>
+        </tr>
+        <tr>
+            <td>[5 5 3 6];</td>
+            <td>85.7</td>
+        </tr>
+        <tr>
+            <td>[3 4 4 2]</td>
+            <td></td>
+        </tr>
+        <tr>
+            <td rowspan="3">FCMV</td>
+            <td rowspan="3">PW</td>
+            <td rowspan="3">PW</td>
+            <td>[5 2 4 4];</td>
+            <td>84.5</td>
+            <td rowspan="3">729.03</td>
+            <td rowspan="3">6.59</td>
+        </tr>
+        <tr>
+            <td>[4 2 3 4];</td>
+            <td>86.4</td>
+        </tr>
+        <tr>
+            <td>[2 2 4 3];</td>
+            <td></td>
+        </tr>
+        <tr>
+            <td rowspan="3">FCHV</td>
+            <td rowspan="3">ST</td>
+            <td rowspan="3">ST</td>
+            <td>[2 2 3 4];</td>
+            <td>89.5</td>
+            <td rowspan="3">26.27</td>
+            <td rowspan="3">3.26</td>
+        </tr>
+        <tr>
+            <td>[1 2 1 2];</td>
+            <td>88.7</td>
+        </tr>
+        <tr>
+            <td>[2 1 3 2];</td>
+            <td></td>
+        </tr>
+        <tr>
+            <td rowspan="3">FVLC</td>
+            <td rowspan="3">SN</td>
+            <td rowspan="3">SN</td>
+            <td>[3 6 3 2];</td>
+            <td>56.8</td>
+            <td rowspan="3">0.07</td>
+            <td rowspan="3">2.57</td>
+        </tr>
+        <tr>
+            <td>[8 5 4 3];</td>
+            <td>60.5</td>
+        </tr>
+        <tr>
+            <td>[2 4 3 5];</td>
+            <td></td>
+        </tr>
+        <tr>
+            <td rowspan="3">FVMC</td>
+            <td rowspan="3">WN</td>
+            <td rowspan="3">WN</td>
+            <td>[3 4 2 5];</td>
+            <td>97.03</td>
+            <td rowspan="3">254.45</td>
+            <td rowspan="3">7.89</td>
+        </tr>
+        <tr>
+            <td>[4 2 3 4];</td>
+            <td>91.7</td>
+        </tr>
+        <tr>
+            <td>[2 3 2 4];</td>
+            <td></td>
+        </tr>
+        <tr>
+            <td rowspan="3">FVHC</td>
+            <td rowspan="3">WN</td>
+            <td rowspan="3">WN</td>
+            <td>[1 4 3 5];</td>
+            <td>88</td>
+            <td rowspan="3">3,079.8</td>
+            <td rowspan="3">10.33</td>
+        </tr>
+        <tr>
+            <td>[5 2 3 5];</td>
+            <td>94</td>
+        </tr>
+        <tr>
+            <td>[1 3 2 4];</td>
+            <td></td>
+        </tr>
+        <tr>
+            <td colspan="7">Transient conditions</td>
+        </tr>
+        <tr>
+            <td rowspan="4">Step Up</td>
+            <td rowspan="4">DZ</td>
+            <td rowspan="4">DZ</td>
+            <td>[3 4 2 4];</td>
+            <td>91.75</td>
+            <td rowspan="4">3,230</td>
+            <td rowspan="4">7.40</td>
+        </tr>
+        <tr>
+            <td>[4 5 4 3];</td>
+            <td>87.20</td>
+        </tr>
+        <tr>
+            <td>[2 3 5 5];</td>
+            <td></td>
+        </tr>
+        <tr>
+            <td>[4 5 2 2];</td>
+            <td></td>
+        </tr>
+        <tr>
+            <td rowspan="4">Step Down</td>
+            <td rowspan="4">PW</td>
+            <td rowspan="4">PW</td>
+            <td>[3 5 5 3];</td>
+            <td>85.99</td>
+            <td rowspan="4">3,233</td>
+            <td rowspan="4">10.0</td>
+        </tr>
+        <tr>
+            <td>[3 5 4 3];</td>
+            <td>85.12</td>
+        </tr>
+        <tr>
+            <td>[3 5 5 4];</td>
+            <td></td>
+        </tr>
+        <tr>
+            <td>[4 4 4 1];</td>
+            <td></td>
+        </tr>
+    </tbody>
+</table>
+
+Table 3. Results of a PV inverter modeling using a Hammerstein-Wiener model

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+## 6. Applications: Power quality problem analysis
+
+A power quality analysis from the model follows the Standard IEEE 1159 Recommend Practice for Monitoring Electric Power Quality [59]. In this Standard, the definition of power quality problem is defined. In summary, a procedure of this Standard when applied to operating systems can be divided into 3 stages (i) Measurement Transducer, (ii) Measurement Unit and (iii) Evaluation Unit. In comparing operating systems and modeling, modeling is more advantageous because of its predictive power, requiring no actual monitoring. Based on proposed modeling, the measurement part is replaced by model prediction outputs, electrical values such as RMS and peak values, frequency and power are calculated, rather than measured. The actual evaluation is replaced by power quality analysis. The concept representation is shown in Fig.22.
+
+Fig. 22. Diagram of power quality analysis from IEEE 1159 and application to modeling
+
+### 6.1 Model output prediction
+
+In this stage, the model output prediction is demonstrated. From the 8 operation conditions selected in experimental, we choose two representative case. One is the steady state Fix Voltage High Current (FVHC) condition, the other the transient step down condition. To illustrate model predictive power, Fig.23 shows an actual and predictive output current waveforms of the transient step down condition. We see good agreement between experimental results and modeling results.
+
+### 6.2 Electrical parameter calculation
+
+In this stage, output waveforms are used to calculate RMS, peak and per unit (p.u.) values, period, frequency, phase angle, power factor, complex power (real, reactive and apparent power), Total Harmonic Distortion - THD.
+
+#### 6.2.1 Root mean square
+
+RMS values of voltage and current can be calculated from the following equations:

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+$$V_{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} v^2(t) dt} \qquad (17)$$
+
+$$I_{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} v^2(t) dt} \qquad (18)$$
+
+Fig. 23. Prediction and experiment results of AC output current under a transient step down condition
+
+### 6.2.2 Period, frequency and phase angle
+
+We calculate a phase shift between voltage and current from the equation (19), and the frequency (f) from equation (20).
+
+$$\phi = \frac{\Delta t(\text{ms}) \cdot 360^\circ}{T \text{ ms}} \qquad (19)$$
+
+$$f = \frac{1}{T} \qquad (20)$$
+
+$\Delta t$ is time lagging or leading between voltage and current (ms), T is the waveform period.

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+## 3.2 Hammerstein-Wiener (HW) nonlinear model
+
+In this section, a combination of the Wiener model and the Hammerstein model called the Hammerstein-Wiener model is introduced, shown in Fig. 9. In the Wiener model, the front part being a dynamic linear block, representing the system, is cascaded with a static nonlinear block, being a sensor. In the Hammerstein model, the front block is a static nonlinear actuator, in cascading with a dynamic linear block, being the system. This model enables combination of a system, sensors and actuators in one model. The described dynamic system incorporates a static nonlinear input block, a linear output-error model and an output static nonlinear block.
+
+Fig. 9. Structure of Hammerstein-Weiner Model
+
+General equations describing the Hammerstein-Wiener structure are written as the Equation (1)
+
+$$ 
+\left. 
+\begin{aligned}
+w(t) &= f(u(t)) \\
+x(t) &= \sum_{i=1}^{n_u} \frac{B_i(q)}{F_i(q)} w(t-n_k) \\
+y(t) &= h(x(t))
+\end{aligned}
+\right\} \qquad (1)
+$$
+
+Which $u(t)$ and $y(t)$ are the inputs and outputs for the system. Where $w(t)$ and $x(t)$ are internal variables that define the input and output of the linear block.
+
+### 3.2.1 Linear subsystem
+
+The linear block is similar to an output error polynomial model, whose structure is shown in the Equation (2). The number of coefficients in the numerator polynomials $B(q)$ is equal to the number of zeros plus 1, $b_n$ is the number of zeros. The number of coefficients in denominator polynomials $F(q)$ is equal to the number of poles, $f_n$ is the number of poles. The polynomials B and F contain the time-shift operator $q$, essentially the z-transform which can be expanded as in the Equation (3). $n_u$ is the total number of inputs. $n_k$ is the delay from an input to an output in terms of the number of samples. The order of the model is the sum of $b_n$ and $f_n$. This should be minimum for the best model.
+
+$$ x(t) = \sum_{i=1}^{n_u} \frac{B_i(q)}{F_i(q)} w(t-n_k) \qquad (2) $$
+
+$$ 
+\begin{aligned}
+B(q) &= b_1 + b_2 q^{-1} + \dots + b_n q^{-b_n+1} \\
+F(q) &= 1 + f_1 q^{-1} + \dots + f_n q^{-f_n}
+\end{aligned}
+\qquad (3)
+$$

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+### 3.2.2 Nonlinear subsystem
+
+The Hammerstein-Wiener Model composes of input and output nonlinear blocks which contain nonlinear functions $f(\bullet)$ and $h(\bullet)$ that corresponding to the input and output nonlinearities. The both nonlinear blocks are implemented using nonlinearity estimators. Inside nonlinear blocks, simple nonlinear estimators such deadzone or saturation functions are contained.
+
+i. The dead zone (DZ) function generates zero output within a specified region, called its dead zone or zero interval which shown in Fig. 10. The lower and upper limits of the dead zone are specified as the start of dead zone and the end of dead zone parameters. Deadzone can define a nonlinear function $y = f(x)$, where $f$ is a function of $x$. It composes of three intervals as following in the equation (4)
+
+$$ \left. \begin{array}{ll} x \le a & f(x) = x - a \\ a < x < b & f(x) = 0 \\ x \ge b & f(x) = x - b \end{array} \right\} \qquad (4) $$
+
+when $x$ has a value between $a$ and $b$, when an output of the function equal to $F(x)=0$, this zone is called as a "zero interval".
+
+Fig. 10. Deadzone function
+
+ii. **Saturation (ST) function** can define a nonlinear function $y = f(x)$, where $f$ is a function of $x$. It composes of three interval as the following characteristics in the equation (5) and Fig. 11. The function is determined between $a$ and $b$ values. This interval is known as a "linear interval"
+
+$$ \left. \begin{array}{ll} x > a & f(x) = a \\ a < x < b & f(x) = x \\ x \le b & f(x) = b \end{array} \right\} \qquad (5) $$
+
+Fig. 11. Saturation function

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+wavelet dilation coefficient, cs and cw are scaling translation and wavelet translation coefficients. The scaling function f(.) and the wavelet function g(.) are both radial functions, and can be written as the equation (9)
+
+$$ 
+\begin{aligned}
+f(u) &= \exp(-0.5 * u' * u) \\
+g(u) &= (\dim(u) - u'*u) * \exp(-0.5 * u'*u)
+\end{aligned} 
+\tag{9} $$
+
+In a system identification process, the wavelet coefficient (a), the dilation coefficient (b) and the translation coefficient (c) are optimized during model learning steps to obtain the best performance model.
+
+### 3.3 MIMO Hammerstein-Wiener system identification
+
+The voltage and current are two basic signals considered as input/output of PV grid connected systems. The measured electrical input and output waveforms of a system are collected and transmitted to the system identification process. In Fig. 13 show a PV based inverter system which are considered as SISO (single input-single output) or MIMO (multi input-multi output), depending on the relation of input-output under study [57]. In this paper, the MIMO nonlinear model of power inverters of PV systems is emphasized because this model gives us both voltage and current output prediction simultaneously.
+
+Fig. 13. Block diagram of nonlinear SISO and MIMO inverter model
+
+For one SISO model, there is only one corresponding set of nonlinear estimators for input and output, and one set of linear parameters, i.e. pole $b_n$, zero $f_n$ and delay $n_k$, as written in the equation (9). For SIMO, MISO and MIMO models, there would be more than one set of

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+nonlinear estimators and linear parameters. The relationships between input-output of the MIMO model have been written in the equation (10) whereas vdc is DC voltage, idc DC current, vac AC voltage, iac AC current. q is shift operator as equivalent to z transform. f(•) and h(•) are input and output nonlinear estimators. In this case a deadzone and saturation are selected into the model. In the MIMO model the relation between output and input has four relations as follows (i) DC voltage (vdc) - AC voltage (vac), (ii) DC voltage (vdc) - AC current (iac), (iii) DC current (idc) - AC voltage (vac) and (iv) DC current(vdc)-AC voltage (vac).
+
+$$
+\left.
+\begin{aligned}
+v_{ac}(t) &= \frac{B(q)}{F(q)} f(v_{dc}(t-n_k)) + e(t) \\
+i_{ac}(t) &= \frac{B(q)}{F(q)} f(i_{dc}(t-n_k)) + e(t)
+\end{aligned}
+\right\} \tag{10}
+$$
+
+$$
+\begin{equation}
+\begin{split}
+V_{ac}(t) &= h\left(\frac{B_1(q)}{F_1(q)}f(v_{dc}(t-n_{k1})) + e(t)\right) \otimes h\left(\frac{B_2(q)}{F_2(q)}f(i_{dc}(t-n_{k2})) + e(t)\right) \\
+I_{ac}(t) &= h\left(\frac{B_3(q)}{F_3(q)}f(v_{dc}(t-n_{k3})) + e(t)\right) \otimes h\left(\frac{B_4(q)}{F_4(q)}f(i_{dc}(t-n_{k4})) + e(t)\right)
+\end{split}
+\tag{11}
+\end{equation}
+$$
+
+$$
+B_i(q) = b_1 + b_2 + \dots + b_{n_{bi}} q^{-n_{bi}+1}
+$$
+
+$$
+F_i(q) = f_1 + f_2 + \dots + f_{n_{fi}} q^{-n_{fi} + 1} \quad (12)
+$$
+
+Where $n_{bi}$, $n_{fi}$ and $n_{ki}$ are pole, zero and delay of linear model. Where as number of subscript i are 1,2,3 and 4 which stand for relation between DC voltage-AC voltage, DC current-AC voltage, DC voltage-AC current and DC current-AC current respectively. The output voltage and output current are key components for expanding to the other electrical values of a system such power, harmonic, power factor, etc. The linear parameters, zeros, poles and delays are used to represent properties and relation between the system input and output. There are two important steps to identify a MIMO system. The first step is to obtain experimental data from the MIMO system. According to different types of experimental data, the second step is to select corresponding identification methods and mathematical models to estimate model coefficients from the experimental data. The model is validated until obtaining a suitable model to represent the system. The obtained model provides properties of systems. State-space equations, polynomial equations as well as transfer functions are used to describe linear systems. Nonlinear systems can be described by the above linear equations, but linearization of the nonlinear systems has to be carried out. Nonlinear estimators explain nonlinear behaviors of nonlinear system. Linear and nonlinear graphical tools are used to describe behaviors of systems regarding controllability, stability and so on.
+
+**4. Experimental**
+
+In this work, we model one type of a commercial grid connected single phase inverters,
+rating at 5,000 W. The experimental system setup composes of the inverter, a variable DC
+power supply (representing DC output from a PV array), real and complex loads, a digital
+power meter, a digital oscilloscope, , a AC power system and a computer, shown

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+The system identification scheme is shown in Fig.15. Good accuracy of models are achieved by selecting model structures and adjusting the model order of linear terms and nonlinear estimators of nonlinear systems. Finally, output voltage and current waveforms for any type of loads and operating conditions are then constructed from the models. This allows us to study power quality as required.
+
+## 4.1 Steady state conditions
+
+To emulate working conditions of PVGCS systems under environment changes (irradiance and temperature) affecting voltage and current inputs of inverters, six conditions of DC voltage variations and DC current variations. The six conditions are listed as Table 1. They are 3 conditions of a fixed DC current with DC low, medium and high voltage, i.e., FCLV (Fixed Current Low Voltage), FCMV (Fixed Current Medium Voltage) and FCHV (Fixed Current High Voltage) which shown in Fig. 16. The other three corresponding conditions are a DC fixed voltage with DC low, medium and high current, i.e., FVLC (Fixed Voltage Low Current), FVMC (Fixed Voltage Medium Current), and FVHC (Fixed Voltage High Current) as shown in Fig.17.
+
+Fig. 16. AC voltage and current waveforms corresponding to FCLV, FCMV and FCHV conditions
+
+Fig. 17. AC voltage and current waveforms corresponding to FVLC, FVMC and FVHC conditions

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+<table><thead><tr><th>No.</th><th>Case</th><th>Idc (A)</th><th>Vdc (V)</th><th>Pdc (W)</th><th>Iac (A)</th><th>Vac (A)</th><th>Pac (VA)</th></tr></thead><tbody><tr><td>1</td><td>FCLV</td><td>12</td><td>210</td><td>2520</td><td>11</td><td>220</td><td>2420</td></tr><tr><td>2</td><td>FCMV</td><td>12</td><td>240</td><td>2880</td><td>13</td><td>220</td><td>2860</td></tr><tr><td>3</td><td>FCHV</td><td>12</td><td>280</td><td>3360</td><td>15</td><td>220</td><td>3300</td></tr><tr><td>4</td><td>FVLC</td><td>2</td><td>235</td><td>470</td><td>2</td><td>220</td><td>440</td></tr><tr><td>5</td><td>FVMC</td><td>10</td><td>240</td><td>2,400</td><td>10</td><td>220</td><td>2,200</td></tr><tr><td>6</td><td>FVHC</td><td>21</td><td>245</td><td>5,145</td><td>23</td><td>220</td><td>5,060</td></tr></tbody></table>
+
+Table 1. DC and AC parameters of an inverter under changing operating conditions
+
+## 4.2 Transient conditions
+
+Transient conditions are studied under two cases which composed of step up power transient and step down power transient. The step up condition is done by increasing power output from 440 to 1,540 W, and the step down condition from 1,540 to 440 W, shown in Table 2. Power waveform data of the two conditions are divided in two groups, the first group is used to estimate model, the second group to validate model. Examples of captured voltage and current waveforms under the step-up power transient condition (440 W or 2 A) to 1540 W or 7 A) and the step-down power transient condition (1540 W or 7 A) to 440 W (2 A) are shown in Fig. 18 and 19, respectively.
+
+Fig. 18. AC voltage and current waveforms under the step up transient condition

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+# OPEN CHALLENGE '06
+
+## 1. MAGICAL MAZE
+
+$$ \begin{array}{ccc} \boxed{19} & \times & \boxed{91} & \times & \boxed{95} & \times & \boxed{38} & \times & \boxed{179} = & \boxed{1117262510} \\ 1419 = \text{IN} & \rightarrow & I = 14 & N = 19 & & & & & \\ 202625 = \text{OUT} & \rightarrow & O = 20 & U = 26 & T = 25 & & & & \end{array} $$
+
+Code number is alphabet position + 5
+$1117262510$ = FLUTE
+Thus the password is FLUTE.
+
+## 2. A LITTLE LIGHT MOVEMENT
+
+The least possible labour involves 18 moves as follows:
+
+Move to empty room Wardrobe; Bookcase; Piano; Wardrobe; Bookcase; Chest of Drawers; Cabinet;
+Bookcase; Wardrobe; Piano; Chest of Drawers; Wardrobe; Bookcase; Cabinet; Wardrobe; Chest of
+Drawers; Piano; Bookcase.
+
+## 3. MIDDLE C
+
+The clarinettist is at the centre of a circle of radius r.
+
+$$ \text{Let } \angle OFP = \theta $$
+
+$$ \begin{aligned} \cos\theta &= \frac{14^2 + 15^2 - 13^2}{2 \times 14 \times 15} \\ &= \frac{3}{5} \end{aligned} $$
+
+$$ \text{Thus } \sin\theta = \frac{4}{5} $$
+
+$$ \text{Now } \angle OCP = 2\theta $$
+
+$$ \begin{aligned} \text{Thus } r &= \frac{13}{2\sin\theta} = \frac{65}{8} \\ &= 8\frac{1}{8} \text{ feet.} \end{aligned} $$
+
+Thus the clarinettist is $8 \text{ feet} 1\frac{1}{2} \text{ inches}$ from the other three.

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+### 4. THE TRIANO
+
+The arrangement of notes
+within the basic pattern is:
+
+### 5. CANDLIGHT SONATA
+
+The lengths of candles A and B are both functions of time such that:
+
+$$A(t) = \alpha - \frac{\alpha t}{4} \qquad B(t) = \alpha - \frac{\alpha t}{5}, \text{ where } \alpha \text{ is the original length of both candles.}$$
+
+At time T
+
+$$A(T) = \frac{1}{4} B(T)$$
+
+$$\alpha - \frac{\alpha T}{4} = \frac{1}{4} \left( \alpha - \frac{\alpha T}{5} \right)$$
+
+$$4\alpha - \alpha T = \alpha - \frac{\alpha T}{5}$$
+
+Thus
+
+$$\frac{4}{5}T = 3$$
+
+$$T = 3\frac{3}{4} \text{ hours}$$
+
+Thus Wolfgang practised for 3hours 45 minutes.
+
+### 6. SALIERI'S CONFESSION
+
+NOW AT MY PASSING I ADMIT A GRAVE SIN. CONSUMED BY JEALOUSY I
+ADMINISTERED THE POISON ARSENIC TO WOLFGANG AMADEUS MOZART AND
+THUS PROCURED HIS END AND MY ADVANCEMENT.
+ANTONIO SALIERI
+
+The poison was arsenic.
+
+The substitution code gives:
+
+A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
+T C H A I K O V S Y Z X W U R Q P N M L J G F E D B
+
+Thus the code word used was TCHAIKOVSKY but as he was not born until 1840 his name would have been unknown to Salieri who died in 1825. Hence researchers are convinced that this is not genuine.

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+and rational functions $f_i$ on $U_i$ such that $f_{ij} = f_i - f_j$ is a regular function on $U_i \cap U_j$. We can define, as for (multiplicative) divisors, the notions of definition functions of an additive divisor, equivalence between two additive divisors, etc. We find, for example, that $H^1(X, \mathcal{O})$ is equal to the group of classes of additive divisors on $X$.
+
+Let $D$ be a (multiplicative) divisor on $X$. We define $\mathrm{supp}D$ to be the set of points $x \in X$ such that $D(x)$ is not the identity element in $\mathcal{R}^{\times}/\mathcal{O}_x^{\times}$, i.e. such that every definition function of $D$ at $x$ is either not defined at $x$ or takes the value 0 at $x$.
+
+**Proposition 7.** *The support of a divisor *D* on a variety *X* is a closed subset ≠ *X* of *X*, and *D* = 0 if and only if the support is empty.*
+
+*Proof.* The latter claim is trivial. For the former, we prove that the set $E$ of points $x \in X$
+such that every definition function of $D$ at $x$ belongs to $\mathcal{O}_x^\times$ is a non-empty open subset;
+indeed, if we take a definition function $g$ of $D$ at $x$, then it is also a definition function of
+$D$ in an open subset $U$ that contains $x$. By hypothesis, if $x \in E$, then $g$ is regular at $x$ and
+$g(x) \neq 0$, and we can choose $U$ such that $g$ is regular and invertible on $U$, which proves
+that $E$ is open.
+$\square$
+
+**Proposition 8.** If *D* is a divisor on a normal variety *X*, then *supp*(*D* is a union of hyper-surfaces (i.e. of closed subvarieties of codimension 1).
+
+*Proof.* If $f$ is a function on a normal variety $Y$, we know that, if $f$ is not defined at $x \in Y$, then $x$ belongs to a variety of poles or zeros of $f$ (i.e. to an irreducible component of the closure of the set of points $x \in Y$ such that $f(x) \in \{0, \infty\}$). So, if we take $f$ to be a definition function for $D$ in some open subset $U \subset X$, then $\text{supp} D \cap U$ is the union of the pole and zero varieties of $f$ in $U$, and we know that these varieties are of codimension 1 ([2, chapitre III]).
+
+**Remark.** If *X* is not normal, then the support of a divisor *D* on *X* is not necessarily of codimension 1. It is easy to define an affine variety *X* of dimension > 1 that is normal everywhere except at a single point *x*₀ (for example, the point (*a*,*ab*,*b*², *b*³) in the four-dimensional space *K*⁴). There exists a function *u* that is everywhere defined on *X*, and which is entire on the local ring of *x*₀, but which is not contained in this ring; by adding, if necessary, a constant, we can assume that *x*₀ is not a zero of *u*. There is then an open neighbourhood *X*' of *x*₀ such that the divisor of the function induced by *u* on *X*' has support equal to the single point *x*₀.
+
+Suppose that X is a normal variety, and D is a divisor on X. Let S be a hypersurface of X. If f is a definition function of D at x ∈ S, then the order of f on S ([2]) does not depend on the choice of f, nor on x ∈ S. We denote this integer by ord_S D. It is easy to see that ord_S D = 0 if and only if S ⊂ supp D. If we now take the formal combination C = ∑_S (ord_S D) ⋅ S, where S runs over the set of all hypersurfaces of X, then C is a cycle of codimension 1, and we call it the associated cycle of the divisor D.
+
+**Proposition 9.** Let X be a normal variety. The map that sends a divisor D to its associated cycle of codimension 1 is an injective homomorphism from the group of divisors on X to the group of cycles of codimension 1.

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+*Proof.* The proof is trivial.
+
+**Proposition 10.** If X is further a non-singular variety, then the homomorphism that sends a divisor to its associated cycle is bijective.
+
+*Proof.* It suffices to show that, for every hypersurface $S$, there exists a divisor $D$ such that the cycle $1 \cdot S$ is the cycle associated to $D$. Since $X$ is non-singular, for every $x \in X$, the local ring $\mathcal{O}_x$ is factorial [3]; thus, for every $x \in S$, $S$ is defined by one single equation in a neighbourhood of $x$. So there exists a cover $\{U_i\}_{i=1,\dots,p}$ of $S$ by open subsets $U_i$ of $X$, and, for each $i$, a regular function $f_i$ on $U_i$ that is non-zero outside of $U_i \cap S$ in $U_i$ with $\text{ord}_S f_i = 1$. It then follows that $f_i/f_j$ is an invertible regular function in $U_i \cap U_j$. Now take the cover $\{U_i\}_{i=0,\dots,p}$, where $U_0 = CS$, and take $f_0 = 1$. IT is easy to see that the divisor $D$ for which $f_i$ is a definition function of $D$ on $U_i$ is such that the cycle associated to $D$ is $1 \cdot S$. So the proposition is proven. $\square$
+
+p. 4-09
+
+**Remark.** The proposition is not necessarily true if $X$ is non-singular. For example, for the cone $xy-zw = 0$ in $K^4$, the cycle defined by $x = z = 0$ is not a cycle associated to any divisor.
+
+## References
+
+[1] Cartier, P. Diviseurs et dérivations en géométrie algébrique. (Thèse Sc. math. Paris. 1958) (To appear in *Bull. Soc. math. France*).
+
+[2] Chevalley, C. Fondements de la Géométrie algébrique Paris, Secrétariat mathématique, 1958, multigraphed. (Class taught at the Sorbonne in 1957–58).
+
+[3] Godement, R. Propriétés analytiques des localités In *Séminarie Cartan-Chevalley*, vol. **8**, 1955–56. (Talk number 19).
+
+[4] Grothendieck, A. Sur les faisceaux algébriques et les faisceaux analytiques cohérents. In *Séminaire H. Cartan*, vol. **9**. (Talk number 2).
+
+[5] Serre, J.-P. Faisceaux algébriques cohérents. Ann. Math. **61** (1955), 197–279.
+
+[6] Serre, J.-P. Sur la cohomologie des variétés algébriques. *J. Math. pures et appl.* **36** (1957), 1–16.
+
+[7] Weil, A. Fibre spaces in algebraic geometry (Notes taken by A. Wallace, 1952) Chicago, University of Chicago, 1955.

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+# An Exponential Time Parameterized Algorithm for Planar Disjoint Paths*
+
+Daniel Lokshtanov†
+
+Pranabendu Misra‡
+
+Michal Pilipczuk§
+
+Saket Saurabh¶
+
+Meirav Zehavi||
+
+August 30, 2021
+
+## Abstract
+
+In the Disjoint Paths problem, the input is an undirected graph $G$ on $n$ vertices and a set of $k$ vertex pairs, $\{s_i, t_i\}_{i=1}^k$, and the task is to find $k$ pairwise vertex-disjoint paths such that the $i$'th path connects $s_i$ to $t_i$. In this paper, we give a parameterized algorithm with running time $2^{\mathcal{O}(k^2)} n^{\mathcal{O}(1)}$ for Planar Disjoint Paths, the variant of the problem where the input graph is required to be planar. Our algorithm is based on the unique linkage/treewidth reduction theorem for planar graphs by Adler et al. [JCTB 2017], the algebraic cohomology based technique of Schrijver [SICOMP 1994] and one of the key combinatorial insights developed by Cygan et al. [FOCS 2013] in their algorithm for Disjoint Paths on directed planar graphs. To the best of our knowledge our algorithm is the first parameterized algorithm to exploit that the treewidth of the input graph is small in a way completely different from the use of dynamic programming.
+
+* A preliminary version of this paper appeared in the proceedings of STOC 2020.
+† University of California, Santa Barbara, USA. daniello@ucsb.edu
+‡ Max Planck Institute for Informatics, Saarbrucken, Germany. pmisra@mpi-inf.mpg.de
+§ Institute of Informatics, University of Warsaw, Poland. michal.pilipczuk@mimuw.edu.pl
+¶ The Institute of Mathematical Sciences, HBNI, Chennai, India. saket@imsc.res.in
+|| Ben-Gurion University, Beersheba, Israel. meiravze@bgu.ac.il

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+Figure 12: Crossing (left) and non-crossing (right) walks.
+
+# 5 Discrete Homotopy
+
+The purpose of this section is to assert that rather than working with homology (Definition 3.2) or the standard notion of homotopy, to obtain our algorithm it will suffice to work with a notion called *discrete homotopy*. Working with discrete homotopy will substantially shorten and simplify our proof, particularly Section 8. Translation from discrete homotopy to homology is straightforward, thus readers are invited to skip the proofs in this section when reading the paper for the first time. We begin by defining the notion of a weak linkage. This notion is a generalization of a linkage (see Section 3) that concerns walks rather than paths, and which permits the walks to intersect one another in vertices. Here, we only deal with walks that may repeat vertices but which do not repeat edges. Moreover, weak linkages concern walks that are *non-crossing*, a property defined as follows (see Fig. 12).
+
+**Definition 5.1 (Non-Crossing Walks).** Let $G$ be a plane graph, and let $W$ and $W'$ be two edge-disjoint walks in $G$. A crossing of $W$ and $W'$ is a tuple $(v, e, \hat{e}, e', \tilde{e}')$ where $e, \hat{e}$ are consecutive in $W$, $e', \tilde{e}'$ are consecutive in $W'$, $v \in V(G)$ is an endpoint of $e, \hat{e}, e'$ and $\tilde{e}',$ and when the edges incident to $v$ are enumerated in clockwise order, then exactly one edge in $\{e', \tilde{e}'\}$ occurs between $e$ and $\hat{e}$. We say that $W$ is self-crossing if, either it has a repeated edge, or it has two edge-disjoint subwalks that are crossing.
+
+We remark that when we say that a collection of edge-disjoint walks is non-crossing, we mean that none of its walks is self-crossing and no pair of its walks has a crossing.
+
+**Definition 5.2 (Weak Linkage).** Let $G$ be a plane graph. A weak linkage in $G$ of order $k$ is an ordered family of $k$ edge-disjoint non-crossing walks in $G$. Two weak linkages $W = (W_1, \dots, W_k)$ and $Q = (Q_1, \dots, Q_k)$ are aligned if for all $i \in \{1, \dots, k\}$, $W_i$ and $Q_i$ have the same endpoints. Given an instance $(G, S, T, g, k)$ of Planar Disjoint Paths, a weak linkage $W$ in $G$ (or $H_G$) is sensible if its order is $k$ and for every terminal $s \in S$, $W$ has a walk with endpoints $s$ and $g(s)$.
+
+The following observation is clear from Definitions 5.1 and 5.2.
+
+**Observation 5.1.** Let $G$ be a plane graph, and let $W$ be a weak linkage in $G$. Let $e_1, e_2$ and $e_3, e_4$ be two pairs of edges in $E(W)$ that tare all distinct and incident on a vertex $v$, and there is some walk in $W$ where $e_1, e_2$ are consecutive, and likewise for $e_3, e_4$. Then, in a clockwise enumeration of edges incident to $v$, the pairs $e_1, e_2$ and $e_3, e_4$ do not cross, that is, they do not occur as $e_1, e_3, e_2, e_4$ in clockwise order (including cyclic shifts).
+
+We now define the collection of operations applied to transform one weak linkage into another weak linkage aligned with it (see Fig. 13). We remark that the face push operation is not required for our arguments, but we present it here to ensure that discrete homotopy defines an equivalence relation (in case it will find other applications that need this property).
+
+**Definition 5.3 (Operations in Discrete Homotopy).** Let $G$ be a triangulated plane graph with a weak linkage $W$, and a face $f$ that is not the outer face with boundary cycle $C$. Let $W \in W$.
+
+* **Face Move.** Applicable to $(W, f)$ if there exists a subpath $P$ of $C$ such that (i) $P$ is a subwalk of $W$, (ii) $1 \le |E(P)| \le |E(f)| - 1$, and (iii) no edge in $E(C) \setminus E(P)$ belongs to

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+Figure 13: Face operations.
+
+any walk in W. Then, the face move operation replaces P in W by the unique subpath of C between the endpoints of P that is edge-disjoint from P.
+
+* **Face Pull.** Applicable to (W, f) if C is a subwalk Q of W. Then, the face pull operation replaces Q in W by a single occurrence of the first vertex in Q.
+
+* **Face Push.** Applicable to (W, f) if (i) no edge in E(C) belongs to any walk in W, and (ii) there exist two consecutive edges e, e' in W with common vertex v ∈ V(C) (where W visits e first, and v is visited between e and e') and an order (clockwise or counter-clockwise) to enumerate the edges incident to v starting at e such that the two edges of E(C) incident to v are enumerate between e and e', and for any pair of consecutive edges of W' for all W' ∈ W incident to v, it does not hold that one is enumerated between e and the two edges of E(C) while the other is enumerated between e' and the two edges of E(C). Let $\tilde{e}$ be the first among the two edges of E(C) that is enumerated. Then, the face push operation replaces the occurrence of v between e and e' in W by the traversal of C starting at $\tilde{e}$.
+
+We verify that the application of a single operation results in a weak linkage.
+
+**Observation 5.2.** Let $G$ be a triangulated plane graph with a weak linkage $W$, and a face $f$ that is not the outer face. Let $W \in W$ with a discrete homotopy operation applicable to $(W, f)$. Then, the result of the application is another weak linkage aligned to $W$.
+
+Then, discrete homotopy is defined as follows.
+
+**Definition 5.4 (Discrete Homotopy).** Let $G$ be a triangulated plane graph with weak linkages $W$ and $W'$. Then, $W$ is discretely homotopic to $W'$ if there exists a finite sequence of discrete homotopy operations such that when we start with $W$ and apply the operations in the sequence one after another, every operation is applicable, and the final result is $W'$.
+
+We verify that discrete homotopy gives rise to an equivalence relation.
+
+**Lemma 5.1.** Let $G$ be a triangulated plane graph with weak linkages $W, W'$ and $W''$. Then, (i) $W$ is discretely homotopic to itself, (ii) if $W$ is discretely homotopic to $W'$, then $W'$ is discretely homotopic to $W$, and (iii) if $W$ is discretely homotopic to $W'$ and $W'$ is discretely homotopic to $W'',$ then $W$ is discretely homotopic to $W''$.
+
+*Proof.* Statement (i) is trivially true. The proof of statement (ii) is immediate from the observation that each discrete homotopy operation has a distinct inverse. Indeed, every face move operation is invertible by a face move operation (applied to the same walk and cycle). Additionally, every face pull operation is invertible by a face push operation (applied to the same walk and cycle), and vice versa. Hence, given the sequence of operations to transform $W$ to $W'$, say $\phi$, the sequence of operations to transform $W'$ to $W$ is obtained by first writing the

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+# 1 Introduction
+
+In the Disjoint Paths problem, the input is an undirected graph $G$ on $n$ vertices and a set of $k$ pairwise disjoint vertex pairs, $\{s_i, t_i\}_{i=1}^k$, and the task is to find $k$ pairwise vertex-disjoint paths connecting $s_i$ to $t_i$ for each $i \in \{1, \dots, k\}$. The Disjoint Paths problem is a fundamental routing problem that finds applications in VLSI layout and virtual circuit routing, and has a central role in Robertson and Seymour's Graph Minors series. We refer to surveys such as [21, 43] for a detailed overview. The Disjoint Paths problem was shown to be NP-complete by Karp (who attributed it to Knuth) in a followup paper [25] to his initial list of 21 NP-complete problems [24]. It remains NP-complete even if $G$ is restricted to be a grid [33, 30]. On directed graphs, the problem remains NP-hard even for $k = 2$ [20]. For undirected graphs, Perl and Shiloach [35] designed a polynomial time algorithm for the case where $k = 2$. Then, the seminal work of Robertson and Seymour [37] showed that the problem is polynomial time solvable for every fixed $k$. In fact, they showed that it is fixed parameter tractable (FPT) by designing an algorithm with running time $f(k)n^3$. The currently fastest parameterized algorithm for Disjoint Paths has running time $h(k)n^2$ [26]. However, all we know about $h$ and $f$ is that they are computable functions. That is, we still have no idea about what the running time dependence on $k$ really is. Similarly, the problem appears difficult in the realm of approximation, where one considers the optimization variant of the problem where the aim is to find disjoint paths connecting as many of the $\{s_i, t_i\}$ pairs as possible. Despite substantial efforts, the currently best known approximation algorithm remains a simple greedy algorithm that achieves approximation ratio $O(\sqrt{n})$.
+
+The Disjoint Paths problem has received particular attention when the input graph is re-
+stricted to be planar [2, 17, 42, 14]. Adler et al. [2] gave an algorithm for Disjoint Paths on
+planar graphs (Planar Disjoint Paths) with running time $2^{2^{\mathcal{O}(k)}} n^2$, giving at least a concrete form
+for the dependence of the running time on $k$ for planar graphs. Schrijver [42] gave an algo-
+rithm for Disjoint Paths on directed planar graphs with running time $n^{\mathcal{O}(k)}$, in contrast to the
+NP-hardness for $k=2$ on general directed graphs. Almost 20 years later, Cygan et al. [14] im-
+proved over the algorithm of Schrijver and showed that Disjoint Paths on directed planar graphs
+is FPT by giving an algorithm with running time $2^{2^{\mathcal{O}(k^2)}} n^{\mathcal{O}(1)}$. The Planar Disjoint Paths prob-
+lem is well-studied also from the perspective of approximation algorithms, with a recent burst
+of activity [7, 8, 9, 10, 11]. Highlights of this work include an approximation algorithm with
+factor $\mathcal{O}(n^{9/19} \log^{\mathcal{O}(1)} n)$ [8] and, under reasonable complexity-theoretic assumptions, hardness
+of approximating the problem within a factor of $2^{\Omega(\frac{1}{(\log \log n)^2})}$ [10].
+
+In this paper, we consider the parameterized complexity of **Planar Disjoint Paths**. Prior to our work, the fastest known algorithm was the $2^{2^{\mathcal{O}(k)}} n^2$ time algorithm of Adler et al. [2]. Double exponential dependence on *k* for a natural problem on planar graphs is something of an outlier–the majority of problems that are FPT on planar graphs enjoy running times of the form $2^{\mathcal{O}(\sqrt{k} \text{ polylog } k)} n^{\mathcal{O}(1)}$ (see, e.g., [15, 18, 19, 29, 36]). This, among other reasons (discussed below), led Adler [1] to pose as an open problem in GROW 2013¹ whether **Planar Disjoint Paths** admits an algorithm with running time $2^{k^{\mathcal{O}(1)}} n^{\mathcal{O}(1)}$. By integrating tools with origins in algebra and topology, we resolve this problem in the affirmative. In particular, we prove the following.
+
+**Theorem 1.1.** *The Planar Disjoint Paths problem is solvable in time 2*$^{{\mathcal{O}(k^2)}}$*n*^{{\mathcal{O}}(1)}.²*
+
+In addition to its value as a stand-alone result, our algorithm should be viewed as a piece of an
+on-going effort of many researchers to make the Graph Minor Theory of Robertson and Seymour
+algorithmically efficient. The graph minors project is abound with powerful algorithmic and
+
+¹The conference version of [2] appeared in 2011, before [1]. The document [1] erroneously states the open problem for Disjoint Paths instead of for Planar Disjoint Paths—that Planar Disjoint Paths is meant is evident from the statement that a $2^{2^{\mathcal{O}(k)}} n^{\mathcal{O}(1)}$ time algorithm is known.
+
+²In fact, towards this we implicitly design a w*$^{\mathcal{O}(k)}$-time algorithm, where w is the treewidth of the input graph.

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+operations of $\phi$ in reverse order and then inverting each of them. Finally, Statement (iii) follows by considering the sequence of discrete homotopy operations obtained by concatenating the sequence of operations to transform $W$ to $W'$ and the sequence of operations to transform $W'$ to $W''$. $\square$
+
+Towards the translation of discrete homotopy to homology, we need to associate a flow with every weak linkage and thereby extend Definition 3.4.
+
+**Definition 5.5.** Let $(D, S, T, g, k)$ be an instance of Directed Planar Disjoint Paths. Let $W$ be a sensible weak linkage in $D$. The flow $\phi: A(D) \to \text{RW}(T)$ associated with $W$ is defined as follows. For every arc $e \in A(D)$, define $\phi(e) = 1$ if there is no walk in $W$ that traverses $e$, and $\phi(e) = t$ otherwise where $t \in T$ is the end-vertex of the (unique) walk in $W$ that traverses $e$.
+
+Additionally, because homology concerns directed graphs, we need the following notation. Given a graph $G$, we let $\vec{G}$ denote the directed graph obtained by replacing every edge $e \in E(G)$ by two arcs of opposite orientations with the same endpoints as $e$. Notice that $G$ and the underlying graph of $\vec{G}$ are not equal (in particular, the latter graph contains twice as many edges as the first one). Given a weak linkage $W$ in $G$, the weak linkage in $\vec{G}$ that corresponds to $W$ is the weak linkage obtained by replacing each edge $e$ in each walk in $W$, traversed from $u$ to $v$, by the copy of $e$ in $\vec{G}$ oriented from $u$ to $v$.
+
+Now, we are ready to translate discrete homotopy to homology.
+
+**Lemma 5.2.** Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths where $G$ is triangulated. Let $W$ be a sensible weak linkage in $G$. Let $W'$ be a weak linkage discretely homotopic to $W'$. Let $\tilde{W}$ and $\tilde{W}'$ be the weak linkages corresponding to $W$ and $W'$ in $\vec{G}$, respectively. Then, the flow associated with $\tilde{W}$ is homologous to the flow associated with $\tilde{W}'$.
+
+*Proof.* Let $\phi$ and $\psi$ be the flows associated with $\tilde{W}$ and $\tilde{W}'$, respectively, in $\vec{G}$. Consider a sequence $O_1, O_2, \dots, O_\ell$ of discrete homotopy operations that, starting from $W$, result in $W'$. We prove the lemma by induction on $\ell$. Consider the case when $\ell = 1$. Then, the sequence contains only one discrete homotopy operation, which a face move, face pull or face push operation. Let this operation be applied to a face $f$ and a walk $W \in W$, where the walk $W$ goes from $s \in S$ to $g(s) \in T$. Let $C$ be the boundary cycle of $f$ in $G$, and let $\vec{C}$ denote the collection of arcs in $G$ obtained from the edges of $C$. After this discrete homotopy operation, we obtain a walk $W' \in W'$, which differs from $W$ only in the subset of edges $C$. All other walks are identical between $W$ and $W'$. Hence, $\tilde{W}$ and $\tilde{W}'$ differ in $\vec{G}$ only in a subset of $\vec{C}$. Then observe that the flows $\phi$ and $\psi$ are identical everywhere in $A(\vec{G})$ except for a subset of $\vec{C}$. More precisely, Let $P = \vec{C} \cap \tilde{W}$ and $P' = \vec{C} \cap \tilde{W}'$. Then $\phi(e) = g(s)$ if $e \in P$ and $\phi(e) = 1$ if $e \in \vec{C} - P$; a similar statement holds for $\psi$ and $P'$. Furthermore, it is clear from the description of each of the discrete homotopy operations that $P$ and $P'$ have no common edges and $P \cup P'$ is the (undirected)¹¹ cycle $\vec{C}$ in $\vec{G}$.
+
+It only remains to describe the homology between the flows $\phi$ and $\psi$, which is exhibited by a function $h$ on the faces of $\vec{G}$. Then $h$ assigns 1 to all faces of $\vec{G}$ that lie in the exterior of $\vec{C}$, and $g(s)$ to all the faces that lie in the interior of $\vec{C}$. Note that $h$ assigns 1 to the outer face of $\vec{G}$. It is easy to verify that $h$ is indeed a homology between $\psi$ and $\phi$, that is, for any edge $e \in A(\vec{G})$ it holds that $\psi(e) = h(f_1)^{-1} \cdot \phi(e) \cdot h(f_2)$, where $f_1$ and $f_2$ are the faces on the left and the right of $e$ with respect to its orientation. This proves the case where $\ell = 1$.
+
+Now for $\ell > 1$, consider the weak linkage $\tilde{W}^*$ obtained from $\tilde{W}$ after applying the sequence $O_1, O_2, \dots, O_{\ell-1}$. Then by the induction hypothesis, we can assume that the flow associated with $\tilde{W}^*$, say $\psi^*$ is homologous to $\phi$. Further, applying $O_\ell$ to $\tilde{W}^*$ gives us $\tilde{W}'$, and hence the flows $\psi^*$ and $\phi$ are also homologous. Hence, by Observation 3.1 the flows $\phi$ and $\psi$ are homologous. $\square$
+
+¹¹That is, the underlying undirected graph of $\vec{C}$ is a cycle.

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+Having Corollary 3.1 and Lemma 5.2 at hand, we prove the following theorem.
+
+**Lemma 5.3.** There exists a polynomial-time algorithm that, given an instance $(G, S, T, g, k)$ of Planar Disjoint Paths where $G$ is triangulated, a sensible weak linkage $\mathcal{W}$ in $G$ and a subset $X \subseteq E(G)$, either finds a solution of $(G - X, S, T, g, k)$ or determines that no solution of $(G - X, S, T, g, k)$ is discretely homotopic to $\mathcal{W}$ in $G$.
+
+*Proof.* We first convert the given instance of Planar Disjoint Paths into an instance of Directed Planar Disjoint Paths as follows. We convert the graph $G$ into the digraph $\tilde{G}$, as described earlier. Then we construct $\tilde{X}$ from $X$ by picking the two arcs of opposite orientation for each edge in $X$. Then we convert the sensible weak linkage $\mathcal{W}$ into a weak linkage $\tilde{\mathcal{W}}$ in $\tilde{G}$. Finally, we obtain the flow $\phi$ in $\tilde{G}$ associated with $\tilde{\mathcal{W}}$. Next, we apply Corollary 3.1 to the instance $(\tilde{G}, S, T, g, k)$, $\tilde{X}$ and $\phi$. Then either it returns a solution $\hat{\mathcal{P}}$ that is disjoint from $\tilde{X}$, or that there is no solution that is homologous to $\phi$ and disjoint from $\tilde{X}$. In the first case, $\hat{\mathcal{P}}$ can be easily turned into a solution $\mathcal{P}$ for the undirected input instance that is disjoint from $X$. In the second case, we can conclude that the undirected input instance has no solution that is discretely homotopic to $\mathcal{W}$. Indeed, if this were not the case, then consider a solution $\mathcal{P}$ to $(G - X, S, T, g, k)$ that is discretely homotopic to $\mathcal{W}$. Then we have a solution $\hat{\mathcal{P}}$ to the directed instance that is disjoint to $\tilde{X}$. Hence, by Lemma 5.2, the flow associated with $\hat{\mathcal{P}}$ is homologous to $\phi$, the flow associated with $\tilde{\mathcal{W}}$. Hence, $\hat{\mathcal{P}}$ is a solution to the instance $(\tilde{G}, S, T, g, k)$ that is disjoint from $\tilde{X}$ and whose flow is homologous to $\phi$. But this is a contradiction to Corollary 3.1. □
+
+As a corollary to this lemma, we derive the following result.
+
+**Corollary 5.1.** There exists a polynomial-time algorithm that, given an instance $(G, S, T, g, k)$ of Planar Disjoint Paths and a sensible weak linkage $\mathcal{W}$ in $H_G$, either finds a solution of $(G, S, T, g, k)$ or decides that no solution of $(G, S, T, g, k)$ is discretely homotopic to $\mathcal{W}$ in $H_G$.
+
+*Proof.* Consider the instance $(H_G, S, T, g, k)$ along with the set $X = E(H_G) \setminus E(G)$ of forbidden edges. We then apply Lemma 5.3 to $(H_G, S, T, g, k)$, $X$ and $\mathcal{W}$ (note that $H_G$ is triangulated). If we obtain a solution to this instance, then it is also a solution in $G$ since it traverses only edges in $E(H_G) \setminus X = E(G)$. Else, we correctly conclude that there is no solution of $(H_G - X, S, T, g, k)$ (and hence also of $(G, S, T, g, k)$) that is discretely homotopic to $\mathcal{W}$ in $H_G$. □
+
+# 6 Construction of the Backbone Steiner Tree
+
+In this section, we construct a tree that we call a backbone Steiner tree $R = R^3$ in $H_G$. Recall that $H_G$ is the radial completion of $G$ enriched with $4|V(G)| + 1$ parallel copies for each edge. These parallel copies will not be required during the construction of $R$, and therefore we will treat $H_G$ as having just one copy of each edge. Hence, we can assume that $H_G$ is a simple planar graph, and then $E(H_G) = O(n)$ where $n$ is the number of vertices in $G$. We denote $H = H_G$ when $G$ is clear from context. The tree $R$ will be proven to admit the following property: if the input instance is a Yes-instance, then it admits a solution $\mathcal{P} = (P_1, \dots, P_k)$ that is discretely homotopic to a weak linkage $\mathcal{W} = (W_1, \dots, W_k)$ in $H$ aligned with $\mathcal{P}$ that uses at most $2^{\mathcal{O}(k)}$ edges parallel to those in $R$, and none of the edges not parallel to those in $R$. We use the term Steiner tree to refer to any subtree of $H$ whose set of leaves is precisely $S \cup T$. To construct the backbone Steiner tree $R = R^3$, we start with an arbitrary Steiner tree $R^1$ in $H$. Then over several steps, we modify the tree to satisfy several useful properties.
+
+## 6.1 Step I: Initialization
+
+We initialize $R^1$ to be an arbitrarily chosen Steiner tree. Thus, $R^1$ is a subtree of $H$ such that $V_{=1}(R^1) = S \cup T$. The following observation is immediate from the definition of a Steiner tree.

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+**Observation 6.1.** Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths. Let $R'$ be a Steiner tree. Then, $|V_1(R')| = 2k$ and $|V_{\ge 3}(R')| \le 2k - 1$.
+
+Before we proceed to the next step, we claim that every vertex of $H$ is, in fact, “close” to the vertex set of $R_1$. For this purpose, we need the following proposition by Jansen et al. [23].
+
+**Proposition 6.1 (Proposition 2.1 in [23zia]).** Let $G$ be a plane graph and with disjoint subsets $X, Y \subseteq V(G)$ such that $G[X]$ and $G[Y]$ are connected graphs and $\mathrm{rdist}_G(X, Y) = d \ge 2$. For any $r \in \{0, 1, \dots, d-1\}$, there is a cycle $C$ in $G$ such that all vertices $u \in V(C)$ satisfy $\mathrm{rdist}_G(X, \{u\}) = r$, and such that $V(C)$ separates $X$ and $Y$ in $G$.
+
+Additionally, we need the following simple observation.
+
+**Observation 6.2.** Let $G$ be a triangulated plane graph. Then, for any pair of vertices $u, v \in V(G)$, $\mathrm{dist}_G(u, v) = \mathrm{rdist}_G(u, v)$.
+
+*Proof.* Let $\mathrm{rdist}_G(u, v) = t$, and consider a sequence of vertices $u = x_1, x_2, \dots, x_{t+1} = v$ that witnesses this fact—then, every two consecutive vertices in this sequence have a common face. Since $G$ is triangulated, we have that $\{x_i, x_{i+1}\} \in E(G)$ for every two consecutive vertices $x_i, x_{i+1}$, $1 \le i \le t$. Hence, $x_1, x_2, \dots, x_{t+1}$ is a walk from $u$ to $v$ in $G$ with $t$ edges, and therefore $\mathrm{dist}_G(u, v) \le \mathrm{rdist}_G(u, v)$. Conversely, let $\mathrm{dist}_G(u, v) = \ell$; then, there is a path with $\ell$ edges from $u$ to $v$ in $G$, which gives us a sequence of vertices $u = y_1, y_2, \dots, y_{\ell+1} = v$ where each pair of consecutive vertices forms an edge in $G$. Since $G$ is planar, each such pair of consecutive vertices $y_i, y_{i+1}$, $1 \le i \le \ell$, must have a common face. Therefore, $\mathrm{rdist}_G(u, v) \le \mathrm{dist}_G(u, v)$. $\square$
+
+It is easy to see that Observation 6.2 is not true for general plane graph. However, this
+observation will be useful for us because the graph $H$, where we construct the backbone Steiner
+tree, is triangulated. We now present the promised claim, whose proof is based on Proposition
+6.1, Observation 6.2 and the absence of long sequences of $S \cup T$-free concentric cycles in good
+instances. Here, recall that $c$ is the fixed constant in Corollary 4.1. We remark that, for the
+sake of clarity, throughout the paper we denote some natural numbers whose value depends on
+$k$ by notations of the form $\alpha_{\text{subscript}}(k)$ where the subscript of $\alpha$ hints at the use of the value.
+
+**Lemma 6.1.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R'$ be a Steiner tree. For every vertex $v \in V(H)$, it holds that $\mathrm{dist}_H(v, V(R')) \le \alpha_{\mathrm{dist}}(k) := 4 \cdot 2^{ck}$.
+
+*Proof.* Suppose, by way of contradiction, that $\mathrm{dist}_H(v^*, V(R')) > \alpha_{\mathrm{dist}}(k)$ for some vertex $v^* \in V(H)$. Since $H$ is the (enriched) radial completion of $G$, it is triangulated. By Observation 6.2, $\mathrm{rdist}_H(u,v) = \mathrm{dist}_H(u,v)$ for any pair of vertices $u,v \in V(H)$. Thus, $\mathrm{rdist}_H(v^*, V(R')) > \alpha_{\mathrm{dist}}(k)$. By Proposition 6.1, for any $r \in \{0, 1, \dots, \alpha_{\mathrm{dist}}(k)\}$, there is a cycle $C_r$ in $H$ such that all vertices $u \in V(C_r)$ satisfy $\mathrm{rdist}_H(v^*, u) = \mathrm{dist}_H(v^*, u) = r$, and such that $V(C_r)$ separates $\{v^*\}$ and $V(R')$ in $H$. In particular, these cycles must be pairwise vertex-disjoint, and each one of them contains either (i) $v^*$ in its interior (including the boundary) and $V(R')$ in its exterior (including the boundary), or (ii) $v^*$ in its exterior (including the boundary) and $V(R')$ in its interior (including the boundary). We claim that only case (i) is possible. Indeed, suppose by way of contradiction that $C_i$, for some $r \in \{0, 1, \dots, \alpha_{\mathrm{dist}}(k)\}$, contains $v^*$ in its exterior and $V(R')$ in its interior. Because the outer face of $H$ contains a terminal $t^* \in T$ and $t^* \in V(R')$, we derive that $t^* \in V(C_i)$. Thus, $\mathrm{rdist}_H(v^*, t^*) = i \le \alpha_{\mathrm{dist}}(k)$. However, because $t^* \in V(R')$, this is a contradiction to the supposition that $\mathrm{dist}_H(v^*, V(R')) > \alpha_{\mathrm{dist}}(k)$. Thus, our claim holds true. From this, we conclude that $C = (C_0, C_1, \dots, C_{\alpha_{\mathrm{dist}}(k)})$ is a $V(R')$-free sequence of concentric cycles in $H$. Since $S \cup T \subseteq V(R')$, it is also $S \cup T$-free.
+
+Consider some odd integer $r \in \{1, 2, \dots, \alpha_{\text{dist}(k)}\}$. Note that every vertex $u \in V(C_r)$ that does not belong to $V(G)$ lies in some face $f$ of $G$, and that the two neighbors of $u$ in $C_r$ must belong to the boundary of $f$ (by the definition of radial completion). Moreover, each of the

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+vertices on the boundary of $f$ is at distance (in $H$) from $u$ that is the same, larger by one or
+smaller by one, than the distance of $f$ from $u$, and hence none of these vertices can belong to
+any $C_i$ for $i > r + 1$ as well as $i < r + 1$. For every $r \in \{1, 2, \dots, \alpha_{\text{dist}}(k) - 1\}$ such that $r$
+mod $3 = 1$, define $C'_r$ as some cycle contained in the closed walk obtained from $C_r$ by replacing
+every vertex $u \in V(C_r) \setminus V(G)$, with neighbors $x, y$ on $C_r$, by a path from $x$ to $y$ on the boundary
+of the face of $G$ that corresponds to $u$. In this manner, we obtain an $S \cup T$-free sequence of
+concentric cycles in $G$ whose length is at least $2^{ck}$. However, this contradicts the supposition
+that $(G, S, T, g, k)$ is good.
+□
+
+## 6.2 Step II: Removing Detours
+
+In this step, we modify the Steiner tree to ensure that there exist no “shortcuts” via vertices outside the Steiner tree. This property will be required in subsequent steps to derive additional properties of the Steiner tree. To formulate this, we need the following definition (see Fig. 6).
+
+**Definition 6.1 (Detours in Trees).** A subtree *T* of a graph *G* has a detour if there exist two vertices *u*, *v* ∈ *V*<sub>≥3</sub>(*T*) ∪ *V*<sub>=1</sub>(*T*) that are near each other, and a path *P* in *G*, such that
+
+1. *P* is shorter than $\mathbf{path}_T(u, v)$, and
+
+2. one endpoint of *P* belongs to the connected component of $T - V(\mathbf{path}_T(u,v)) \setminus \{u,v\}$ that contains *u*, and the other endpoint of *P* belongs to the connected component of $T - V(\mathbf{path}_T(u,v)) \setminus \{u,v\}$ that contains *v*.
+
+Such vertices *u*, *v* and path *P* are said to witness the detour. Moreover, if *P* has no internal vertex from (*V*(*T*) \ *V*(\*path<sub>*T*</sub>(*u*, *v*)) ) ∪ {*u*, *v*} and its endpoints do not belong to *V*<sub>=1</sub>(*T*) \ {*u*, *v*}, then *u*, *v* and *P* are said to witness the detour compactly.
+
+We compute a witness for a detour as follows. Note that this lemma also implies that, if there exists a detour, then there exists a compact witness rather than an arbitrary one.
+
+**Lemma 6.2.** There exists an algorithm that, given a good instance (G, S, T, g, k) of Planar Disjoint Paths and a Steiner tree R', determines in time O(k²·n) whether R' has a detour. In case the answer is positive, it returns u, v and P that witness the detour compactly.
+
+*Proof.* Let $Q = \mathbf{path}_{R'}(u, v) - \{u, v\}$ for some two vertices $u, v \in V_{\ge 3}(T) \cup V_{=1}(T)$ that are near each other. Then, $R' - V(Q)$ contains precisely two connected components: $R'_u$ and $R'_v$ that contain $u$ and $v$, respectively. Consider a path $P$ of minimum length between vertices $x \in V(R'_u)$ and $y \in V(R'_v)$ in $H$, over all choices of $x$ and $y$. Further, we choose $P$ so that contains as few vertices of $(S \cup T) \setminus \{u, v\}$ as possible. Suppose that $|E(P)| \le |E(\mathbf{path}_{R'}(u, v))| - 1$. Then, we claim that $P$ is a compact detour witness. To prove this claim, we must show that (i) no internal vertex of $P$ lies in $(V(R') \setminus V(\mathbf{path}_{R'}(u, v))) \cup \{u, v\} = V(R'_u) \cup V(R'_v)$, and (ii) the endpoints of $P$ do not lie in $V_{=1}(R') \setminus \{u, v\} = (S \cup T) \setminus \{u, v\}$. The first property follows directly from the choice of $P$. Indeed, if $P$ were a path from $x \in V(R'_u)$ to $y \in V(R'_v)$, which contained an internal vertex $z \in V(R'_u)$, then the subpath $P'$ of $P$ with endpoints $z$ and $y$ is a strictly shorter path from $V(R'_u)$ to $V(R'_v)$ (the symmetric argument holds when $z \in V(R'_v)$).
+
+For the second property, we give a proof by contradiction. To this end, suppose that some terminal $w \in (S \cup T) \setminus \{u, v\}$ belongs to $P$. Necessarily, $w \in V(R'_u) \cup V(R'_v)$ (by the definition of a Steiner tree). Without loss of generality, suppose that $w \in V(R'_u)$. By the first property, $w$ must be an endpoint of $P$. Let $z \in V(R'_v)$ be the other endpoint of $P$. Because the given instance is good, $w$ has degree 1 in $G$, thus we can let $n(w)$ denote its unique neighbor in $G$. Observe that $w$ lies on only one face of $G$, which contains both $w$ and $n(w)$. Hence, $w$ is adjacent to exactly two vertices in $H: n(w)$ and a vertex $f(w) \in V(H) \setminus V(G)$. Furthermore, $\{n(w), f(w)\} \in E(H)$, i.e. $w, n(w)$ and $f(w)$ form a triangle in $H$. Thus, $P$ contains exactly one

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+of $n(w)$ or $f(w)$ (otherwise, we can obtain a strictly shorter path connecting $w$ and the other endpoint of $P$ that contradicts the choice of $P$). Let $a(w) \in \{n(w), f(w)\}$ denote the neighbor of $w$ in $P$, and note that, by the first property, $a(w) \notin V(R'_u)$. Note that it may be the case that $a(w) = z$. Since $w$ is a leaf of $R'$, exactly one of $n(w)$ and $f(w)$ is adjacent to $w$ in $R'$, and we let $b(w)$ denote this vertex. Because $w \neq u$, we have that $V(R'_u)$ contains but is not equal to $\{w\}$, and therefore $b(w) \in V(R'_v)$. In turn, by the first property, this means that $a(w) \neq b(w)$ (because otherwise $a(w) \neq z$ and hence it is an internal vertex of $P$, which cannot belong to $V(R'_u)$). Because $w, a(w)$ and $b(w)$ form a triangle in $H$, we obtain a path $P' \neq P$ in $H$ by replacing $w$ with $b(w)$ in $P$. Observe that $P'$ connects the vertex $b(w) \in V(R'_u)$ to the vertex $z \in V(R'_v)$. Furthermore, because $|E(P')| = |E(P)|$, and $P'$ contains strictly fewer vertices of $(S \cup T) \setminus \{u, v\}$ compared to $P$, we contradict the choice of $P$. Therefore, $P$ also satisfies the second property, and we conclude that $u, v, P$ compactly witness a detour in $R'$.
+
+We now show that a compact detour in $R'$ can be computed in $\mathcal{O}(k^2 \cdot n)$ time. First, observe that if there is a detour witnessed by some $u, v$ and $P$, then $u, v \in V_{\ge 3}(R') \cup V_{=1}(R')$. By Observation 6.1, $|V_{\ge 3}(R') \cup V_{=1}(R')| \le 4k$. Therefore, there are at most $16k^2$ choices for the vertices $u$ and $v$. We consider each choice, and test if there is detour for it in linear time as follows. Fix a choice of distinct vertices $u, v \in V_{\ge 3}(R') \cup V_{=1}(R')$, and check if they are near each other in $R'$ in $\mathcal{O}(|V(R)|)$ time by validating that each internal vertex of $\textbf{path}_{R'}(u, v)$ has degree 2. If they are not near each other, move on to the next choice. Otherwise, consider the path $Q = \textbf{path}_{R'}(u, v) - \{u, v\}$, and the trees $R'_u$ and $R'_v$ of $R' - V(Q)$ that contain $u$ and $v$, respectively. Now, consider the graph $\tilde{H}$ derived from $H$ by first deleting $(V(Q) \cup S \cup T) \setminus \{u, v\}$ and then introducing a new vertex $r$ adjacent to all vertices in $V(R'_u)$. We now run a breadth first search (BFS) from $r$ in $\tilde{H}$. This step takes $\mathcal{O}(n)$ time since $|E(\tilde{H})| = \mathcal{O}(n)$ (because $H$ is planar). From the BFS-tree, we can easily compute a shortest path $P$ between a vertex $x \in V(R'_u)$ and a vertex $y \in V(R'_v)$. Observe that $V(P) \cap (S \cup T) \subseteq \{u, v\}$ by the construction of $\tilde{H}$. If $|E(P)| < |E(\textbf{path}_{R'}(u, v))|$, then we output $u, v, P$ as a compact witness of a detour in $R'$. Else, we move on to the next choice of $u$ and $v$. If we fail to find a witness for all choices of $u$ and $v$, then we output that $R'$ has no detour. Observe that the total running time of this process is bounded by $\mathcal{O}(k^2 \cdot n)$. This concludes the proof. $\square$
+
+Accordingly, as long as $R^1$ has a detour, compactly witnessed by some vertices $u, v$ and a path $P$, we modify it as follows: we remove the edges and the internal vertices of $\textbf{path}_{R^1}(u, v)$, and add the edges and the internal vertices of $P$. We refer to a single application of this operation as *undetouring* $R^1$. For a single application, because we consider compact witnesses rather than arbitrary ones, we have the following observation.
+
+**Observation 6.3.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths with a Steiner tree $R'$. The result of undetouring $R'$ is another Steiner tree with fewer edges than $R'$.
+
+*Proof.* Consider a compact detour witness $(u,v,P)$ of $R'$. Then, $|E(P)| < |E(\textbf{path}_{R'}(u,v))|$. Let $Q = \textbf{path}_{R'}(u,v) - \{u,v\}$, and let $R'_u$ and $R'_v$ be the two trees of $R' - V(Q)$ that contain $u$ and $v$, respectively. Consider the graph $\tilde{R}$ obtained from $(R' - V(Q)) \cup P$ by iteratively removing any leaf vertex that does not lie in $S \cup T$. We claim that the graph $\tilde{R}$ that result from undetouring $R'$ (with respect to $(u,v,P)$) is a Steiner tree with strictly fewer edges than $R'$. Clearly, $\tilde{R}$ is connected because $P$ reconnects the two trees $R'_u$ and $R'_v$ of $R' - V(Q)$. Further, as $P$ contains no internal vertex from $(V(R') \setminus V(\textbf{path}_{R'}(u,v))) \cup \{u,v\} = V(R'_u) \cup V(R'_v)$, and $R'_u$ and $R'_v$ are trees, $\tilde{R}$ is cycle-free. Additionally, all the vertices in $S \cup T$ are present in $\tilde{R}$ by construction and they remain leaves due to the compactness of the witness. Hence, $\tilde{R}$ is a Steiner tree in $G$. Because $|E(P)| < |E(\textbf{path}_{R'}(u,v))|$, it follows that $\tilde{R}$ contains fewer edges than $R'$. $\square$
+
+Initially, $R^1$ has at most $n-1$ edges. Since every iteration decreases the number of edges (by Observation 6.3) and can be performed in time $\mathcal{O}(k^2 \cdot n)$ (by Lemma 6.2), we obtain the following result.

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+Figure 14: Separators and flows for long degree-2 paths.
+
+**Lemma 6.3.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths with a Steiner tree $R'$. An exhaustive application of the operation undetouring $R'$ can be performed in time $O(k^2 \cdot n^2)$, and results in a Steiner tree that has no detour.
+
+We denote the Steiner tree obtained at the end of Step II by $R^2$.
+
+## 6.3 Step III: Small Separators for Long Paths
+
+We now show that any two parts of $R^2$ that are “far” from each other can be separated by small separators in $H$. This is an important property used in the following sections to show the existence of a “nice” solution for the input instance. Specifically, we consider a “long” maximal degree-2 path in $R^2$ (which has no short detours in $H$), and show that there are two separators of small cardinality, each “close” to one end-point of the path. The main idea behind the proof of this result is that, if it were false, then the graph $H$ would have had large treewidth (see Proposition 6.2), which contradicts that $H$ has bounded treewidth (by Corollary 4.1). We first define the threshold that determines whether a path is long or short.
+
+**Definition 6.2 (Long Paths in Trees).** Let $G$ be a graph with a subtree $T$. A subpath of $T$ is $k$-long if its length is at least $\alpha_{\text{long}}(k) := 10^4 \cdot 2^{ck}$, and $k$-short otherwise.
+
+As $k$ will be clear from context, we simply use the terms long and short. Towards the computation of two separators for each long path, we also need to define which subsets of $V(R^2)$ we would like to separate.
+
+**Definition 6.3 ($P'_u, P''_u, A_{R^2,P,u}$ and $B_{R^2,P,u}$).** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour. For any long maximal degree-2 path $P$ of $R^2$ and for each endpoint $u$ of $P$, define $P'_u$, $P''_u$ and $A_{R^2,P,u}$, $B_{R^2,P,u} \subseteq V(R^2)$ as follows.
+
+* $P'_u$ (resp. $P''_u$) is the subpath of $P$ consisting of the $\alpha_{\text{pat}}(k) := 100 \cdot 2^{ck}$ (resp. $\alpha_{\text{pat}}(k)/2 = 50 \cdot 2^{ck}$) vertices of $P$ closest to $u$.
+
+* $A_{R^2,P,u}$ is the union of $V(P''_u)$ and the vertex set of the connected component of $R^2$ - $(V(P'_u) \setminus \{u\})$ containing $u$.
+
+* $B_{R^2,P,u} = V(R^2) \setminus (A_{R^2,P,u} \cup V(P'_u))$.
+
+For each long maximal degree-2 path $P$ of $R^2$ and for each endpoint $u$ of $P$, we compute a “small” separator $\text{Sep}_{R^2}(P, u)$ as follows. Let $A = A_{R^2,P,u}$ and $B = B_{R^2,P,u}$. Then, compute a subset of $V(H) \setminus (A \cup B)$ of minimum size that separates $A$ and $B$ in $H$, and denote it by $\text{Sep}_{R^2}(P, u)$ (see Fig. 14). Since $A \cap B = \emptyset$ and there is no edge between a vertex in $A$ and a vertex in $B$ (because $R^2$ has no detours), such a separator exists. Moreover, it can be computed

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+in time $O(n|\mathbf{Sep}_{R^2}(P, u)|)$: contract each set among A and B into a single vertex and then obtain a minimum vertex $s-t$ cut by using Ford-Fulkerson algorithm.
+
+To argue that the size of $\mathbf{Sep}_{R^2}(P, u)$ is upper bounded by $2^{O(k)}$, we make use of the following proposition due to Bodlaender et al. [5].
+
+**Proposition 6.2 (Lemma 6.11 in [5]).** Let $G$ be a plane graph, and let $H$ be its radial completion. Let $t \in \mathbb{N}$. Let $C, Z, C_1, Z_1$ be disjoint subsets of $V(H)$ such that
+
+1. $H[C]$ and $H[C_1]$ are connected graphs,
+
+2. $Z$ separates $C$ from $Z_1 \cup C_1$ and $Z_1$ separates $C \cup Z$ from $C_1$ in $H$,
+
+3. $\text{dist}_H(Z, Z_1) \geq 3t + 4$, and
+
+4. $G$ contains $t+2$ pairwise internally vertex-disjoint paths with one endpoint in $C \cap V(G)$ and the other endpoint in $C_1 \cap V(G)$.
+
+Then, the treewidth of $G[V(M) \cap V(G)]$ is larger than $t$ where $M$ is the union of all connected components of $H \setminus (Z \cup Z_1)$ having at least one neighbor in $Z$ and at least one neighbor in $Z_1$.
+
+Additionally, the following immediate observation will come in handy.
+
+**Observation 6.4.** Let $G$ be a plane graph. Let $H$ be the radial completion of $G$, and let $H'$ be the radial completion of $H$. Then, for all $u, v \in V(H)$, $\text{dist}_H(u, v) \leq \text{dist}_{H'}(u, v)$.
+
+*Proof.* Note that $H$ is triangulated. Thus, for all $u, v \in V(H)$ and path $P$ in $H'$ between $u$ and $v$, we can obtain a path between $u$ and $v$ whose length is not longer than the length of $P$ by replacing each vertex $w \in V(H') \setminus V(H)$ by at most one vertex of the boundary of the face in $H$ that $w$ represents. $\square$
+
+We now argue that $\mathbf{Sep}_{R^2}(P, u)$ is small.
+
+**Lemma 6.4.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $\mathbb{R}^2$ be a Steiner tree that has no detour, $P$ be a long maximal degree-2 path of $\mathbb{R}^2$, and $u$ be an endpoint of $P$. Then, $|\mathbf{Sep}_{\mathbb{R}^2}(P, u)| \leq \alpha_{\text{sep}}(k) := \frac{7}{2} \cdot 2^{ck} + 2$.
+
+*Proof.* Denote $P' = P_u', P'' = P_u''$, $A = A_{R^2,P,u}$ and $B = B_{R^2,P,u}$. Recall that $H$ is the radial completion of $G$ (enriched with parallel edges), and let $H'$ denote the radial completion of $H$. Towards an application of Proposition 6.2, define $C = A$, $C_1 = B$, $Z = N_{H'}(C)$, $Z_1 = N_{H'}(C_1)$ and $t = \frac{7}{2} \cdot 2^{ck}$. Since $\mathbb{R}^2$ is a subtree of $H$, it holds that $H[C]$ and $H[C_1]$ are connected, and therefore $H'[C]$ and $H'[C_1]$ are connected as well. From the definition of $Z$ and $Z_1$, it is immediate that $Z$ separates $C$ from $Z_1 \cup C_1$ and $Z_1$ separates $C \cup Z$ from $C_1$ in $H$. Clearly, $C \cap C_1 = \emptyset$, $C \cap Z = \emptyset$, and $C_1 \cap Z_1 = \emptyset$. We claim that, in addition, $Z \cap C_1 = \emptyset$, $Z_1 \cap C = \emptyset$ and $Z \cap Z_1 = \emptyset$. To this end, it suffices to show that $\text{dist}_{H'}(Z, Z_1) \geq 3t + 4$. Indeed, because $Z = N_{H'}(C)$ and $Z_1 = N_{H'}(C_1)$, we have that each inequality among $Z \cap C_1 \neq \emptyset$, $Z_1 \cap C = \emptyset$ and $Z \cap Z_1 = \emptyset$, implies that $\text{dist}_{H'}(Z, Z_1) \leq 2$.
+
+Lastly, we show that $\text{dist}_{H'}(Z, Z_1) \geq 3t + 4$. As $\text{dist}_{H'}(C, C_1) \leq \text{dist}_H(Z, Z_1) + 2$, it suffices to show that $\text{dist}_{H'}(C, C_1) \geq 3t + 6$. Because $C \cup C_1 \subseteq V(H)$, Observation 6.4 implies that $\text{dist}_{H'}(C, C_1) \geq \text{dist}_H(C, C_1)$. Hence, it suffices to show that $\text{dist}_H(C, C_1) \geq 3t + 6$. However, $\text{dist}_H(C, C_1) \geq |E(P')| - |E(P'')|$ since otherwise we obtain a contradiction to the supposition that $\mathbb{R}^2$ has no detour. This means that $\text{dist}_H(C, C_1) \geq \alpha_{\text{pat}}(k)/2 - 1 \geq 3t + 6$ as required.
+
+Recall that $\mathbf{Sep}_{\mathbb{R}^2}(P, u)$ is a subset of $V(H) \setminus (C \cup C'_1)$ of minimum size that separates $C$ and $C_1$ in $H$. We claim that $\left|\mathbf{Sep}_{\mathbb{R}^2}(P, u)\right| \leq \alpha_{\text{sep}}(k)$. Suppose, by way of contradiction, that $\left|\mathbf{Sep}_{\mathbb{R}^2}(P, u)\right| > \alpha_{\text{sep}}(k) = t + 2$. By Menger's theorem, the inequality $\left|\mathbf{Sep}_{\mathbb{R}^2}(P, u)\right| > \alpha_{\text{sep}}(k)$ implies that $H$ contains $t+2$ pairwise internally vertex-disjoint paths with one endpoint in $C \subseteq V(H)$ and the other endpoint in $C_1 \subseteq V(H)$. From this, we conclude that all of the conditions in the premise of Proposition 6.2 are satisfied. Thus, the treewidth of $H[V(M) \cap V(H)]$ is larger

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+Figure 8: Rollback spirals.
+
+of times it “winds around” $P$ inside the ring (see Fig. 9). At least intuitively, it should be clear that winding numbers and non-rollback spirals are related. In particular, each ring can only have $2^{\mathcal{O}(k)}$ visitors and crossings subpaths (because the size of each separator is $2^{\mathcal{O}(k)}$), and we only have $\mathcal{O}(k)$ rings to deal with. Thus, it is possible to show that if the winding number of every crossing subpath is upper bounded by $2^{\mathcal{O}(k)}$, then the total number of non-rollback spirals is upper bounded by $2^{\mathcal{O}(k)}$ as well. The main tool we employ to bound the winding number of every crossing path is the following known result (rephrased to simplify the overview).
+
+**Proposition 2.2 ([14]).** Let $G$ be a graph embedded in a ring with a crossing path $P$. Let $\mathcal{P}$ and $Q$ be two collections of vertex-disjoint crossings paths of the same size. (A path in $\mathcal{P}$ can intersect a path in $Q$, but not another path in $\mathcal{P}$.) Then, $G$ has a collection of crossing paths $\mathcal{P}'$ such that (i) for every path in $\mathcal{P}$, there is a path in $\mathcal{P}'$ with the same endpoints and vice versa, and (ii) the maximum difference between (the absolute value of) the winding numbers with respect to $P$ of any path in $\mathcal{P}$ and any path in $Q$ is at most 6.
+
+To see the utility of Proposition 2.2, suppose momentarily that none of our rings has visitors. Then, if we could ensure that for each of our rings, there is a collection $Q$ of vertex-disjoint paths of maximum size such that the winding number of each path in $Q$ is a constant, Proposition 2.2 would have the following implication: if there is a solution, then we can modify it within each ring to obtain another solution such that each crossing subpath of each of its paths will have a constant winding number (under the supposition that the rings are disjoint, which we will deal with later in the overview), see Fig. 9. Our situation is more complicated due to the existence of visitors—we need to ensure that the replacement $\mathcal{P}'$ does not intersect them. On a high-level, this situation is dealt with by first showing how to ensure that visitors do not “go too deep” into the ring on either side of it. Then, we consider an “inner ring” where visitors do not exist, on which we can apply Proposition 2.2. Afterwards, we are able to bound the winding number of each crossing path by $2^{\mathcal{O}(k)}$ (but not by a constant) in the (normal) ring.
+
+**Modifying $R$ within Rings.** To ensure the existence of the aforementioned collection $Q$ for each ring, we need to modify $R$. To this end, consider a long path $P$ with separators $S_u$ and $S_v$, and let $\mathcal{P}'$ be the subpath of $P$ inside the ring defined by the two separators. We compute a maximum-sized collection of vertex-disjoint paths $\text{Flow}(u, v)$ such that each of them has one endpoint in $S_u$ and the other in $S_v$.³ Then, we prove a result that roughly states the following.
+
+**Lemma 2.2.** There is a path $\mathcal{P}^*$ in the ring defined by $S_u$ and $S_v$ with the same endpoints as $\mathcal{P}'$ crossing each path in $\text{Flow}(u, v)$ at most once. Moreover, $\mathcal{P}^*$ is computable in linear time.
+
+³This flow has an additional property: there is a tight collection of $\mathcal{C}(u, v)$ of concentric cycles separating $S_u$ and $S_v$ such that paths in $\text{Flow}(u, v)$ do not “oscillate” too much between any two cycles in the collection. Such a maximum flow is said to be *minimal* with respect to $\mathcal{C}(u, v)$.

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+than $t$ where $M$ is the union of all connected components of $H' \setminus (Z \cup Z_1)$ having at least one
+neighbor in $Z$ and at least one neighbor in $Z_1$. However, $H[V(M) \cap V(H)]$ is a subgraph of
+$H$, which means that the treewidth of $H$ is also larger than $t$. By Proposition 3.2, this implies
+that the treewidth of $G$ is larger than $2^{ck}$. This contradicts the supposition that $(G, S, T, g, k)$
+is good. From this, we conclude that $\lvert \mathrm{Sep}_{R^2}(P, u) \rvert \le \alpha_{\mathrm{sep}}(k)$. $\square$
+
+Recall that $\mathbf{Sep}_{R^2}(P, u)$ is computable in time $\mathcal{O}(n|\mathbf{Sep}_{R^2}(P, u)|)$. Thus, by Lemma 6.4, we obtain the observation below. We remark that the reason we had to argue that the separator is small is not due to this observation, but because the size bound will be crucial in later sections.
+
+**Observation 6.5.** Let $(G, S, T, g, k)$ be a good instance of *Planar Disjoint Paths*. Let $R^2$ be a Steiner tree that has no detour, $P$ be a long maximal degree-2 path of $R^2$, and $u$ be an endpoint of $P$. Then, $\mathbf{Sep}_{R^2}(P, u)$ can be computed in time $2^{\mathcal{O}(k)n}$.
+
+Moreover, we have the following immediate consequence of Proposition 3.1.
+
+**Observation 6.6.** Let $(G, S, T, g, k)$ be a good instance of *Planar Disjoint Paths*. Let $R^2$ be a Steiner tree that has no detour, $P$ be a long maximal degree-2 path of $R^2$, and $u$ be an endpoint of $P$. Then, $H[\mathbf{Sep}_{R^2}(P, u)]$ is a cycle.
+
+## 6.4 Step IV: Internal Modification of Long Paths
+
+In this step, we replace the “middle” of each long maximal degree-2 path $P = \text{path}_{R^2}(u, v)$ of $R^2$ by a different path $P^*$. This “middle” is defined by the two separators obtained in the previous step. Let us informally explain the reason behind this modification. In Section 7 we will show that, if the given instance $(G, S, T, g, k)$ admits a solution (which is a collection of disjoint paths connecting $S$ and $T$), then it also admits a “nice” solution that “spirals” only a few times around parts of the constructed Steiner tree. This requirement is crucial, since it is only such solutions $\mathcal{P}$ that are discretely homotopic to weak linkages $\mathcal{W}$ in $H$ aligned with $\mathcal{P}$ that use at most $2^{\mathcal{O}(k)}$ edges parallel to those in $R$, and none of the edges not parallel to those in $R$. To ensure the existence of nice solutions, we show how an arbitrary solution can be rerouted to avoid too many spirals. This rerouting requires a collection of vertex-disjoint paths between $\mathbf{Sep}_{R^2}(P, u)$ and $\mathbf{Sep}_{R^2}(P, v)$ which itself does not spiral around the Steiner tree. The replacement of $P$ by $P^*$ in the Steiner tree, described below, will ensure this property.
+
+To describe this modification, we first need to assert the statement in the following simple lemma, which partitions every long maximal degree-2 path $P$ of $R^2$ into three parts (see Fig. 14).
+
+**Lemma 6.5.** Let $(G, S, T, g, k)$ be a good instance of *Planar Disjoint Paths*. Let $R^2$ be a Steiner tree with no detour, and $P$ be a long maximal degree-2 path of $R^2$ with endpoints $u$ and $v$. Then, there exist vertices $u' = u'_P \in \mathbf{Sep}_{R^2}(P, u) \cap V(P)$ and $v' = v'_P \in \mathbf{Sep}_{R^2}(P, v) \cap V(P)$ such that:
+
+1. The subpath $P_{u,u'}$ of $P$ with endpoints $u$ and $u'$ has no internal vertex from $\mathbf{Sep}_{R^2}(P, u) \cup \mathbf{Sep}_{R^2}(P, v)$, and $\alpha_{\text{pat}}(k)/2 \le |V(P_{u,u'})| \le \alpha_{\text{pat}}(k)$. Additionally, the subpath $P_{v,v'}$ of $P$ with endpoints $v$ and $v'$ has no internal vertex from $\mathbf{Sep}_{R^2}(P, u) \cup \mathbf{Sep}_{R^2}(P, v)$, and $\alpha_{\text{pat}}(k)/2 \le |V(P_{v,v'})| \le \alpha_{\text{pat}}(k)$.
+
+2. Let $P_{u',v'}$ be the subpath of $P$ with endpoints $u'$ and $v'$. Then, $P = P_{u,u'} - P_{u',v'} - P_{v',v}$.
+
+*Proof.* We first prove that there exists a vertex $u' \in \text{Sep}_{R^2}(P, u) \cap V(P)$ such that the subpath $P_{u,u'}$ of $P$ between $u$ and $u'$ has no internal vertex from $\text{Sep}_{R^2}(P, u) \cup \text{Sep}_{R^2}(P, v)$. To this end, let $P' = P'_u$, and let $\tilde{P}$ denote the subpath of $P$ that consists of the $\alpha_{\text{pat}}(k)+1$ vertices of $P$ that are closest to $u$. Let $A = A_{R^2,P,u}$ and $B = B_{R^2,P,u}$. Recall that $\text{Sep}_{R^2}(P, u) \subseteq V(H) \setminus (A \cup B)$ separates $A$ and $B$ in $H$. Since $\tilde{P}$ is a path with the endpoint $u$ in $A$ and the other endpoint in $B$, it follows that $\text{Sep}_{R^2}(P, u) \cap V(\tilde{P}) \neq \emptyset$. Accordingly, let $u'$ denote the vertex of $P'$ closest

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+to $u$ that belongs to $\mathrm{Sep}_{R^2}(P, u)$. Then, $u' \in \mathrm{Sep}_{R^2}(P, u) \cap V(P)$ and the subpath $P_{u,u'}$ of $P$
+between $u$ and $u'$ has no internal vertex from $\mathrm{Sep}_{R^2}(P, u)$. As the number of vertices of $P_{u,u'}$
+is between those of $P''_u$ and $P'$, the inequalities $\alpha_{\mathrm{pat}}(k)/2 \le |V(P_{u,u'})| \le \alpha_{\mathrm{pat}}(k)$ follow. It
+remains to argue that $P_{u,u'}$ has no internal vertex from $\mathrm{Sep}_{R^2}(P, v)$. Because $\mathrm{Sep}_{R^2}(P, v) \subseteq
+V(H) \setminus (A_{R^2,P,v} \cup B_{R^2,P,v})$, the only vertices of $P$ that $\mathrm{Sep}_{R^2}(P, v)$ can possibly contain are the
+$\alpha_{\mathrm{pat}}(k)$ vertices of $P$ that are closest to $v$. Since $P$ is long, none of these vertices belongs to $P'$,
+and hence $P_{u,u'}$ (which is a subpath of $P'$) has no internal vertex from $\mathrm{Sep}_{R^2}(P, v)$.
+
+Symmetrically, we derive the existence of a vertex $v' \in \text{Sep}_{R^2}(P, v) \cap V(P)$ such that the subpath $P_{v,v'}$ of $P$ between $v$ and $v'$ has no internal vertex from $\text{Sep}_{R^2}(P, u) \cup \text{Sep}_{R^2}(P, v)$.
+
+Lastly, we prove that $P = P_{u,u'} - P_{u',v'} - P_{v',v}$. Since $P_{u,u'}, P_{u',v'}$ and $P_{v',v}$ are subpaths of $P$
+such that $V(P) = V(P_{u,u'}) \cup V(P_{u',v'}) \cup V(P_{v,v'})$, it suffices to show that (i) $V(P_{u,u'}) \cap V(P_{u',v'}) = \{u'\}$, (ii) $V(P_{v,v'}) \cap V(P_{u',v'}) = \{v'\}$, and (iii) $V(P_{u,u'}) \cap V(P_{v,v'}) = \emptyset$. Because $P$ is long and
+$|V(P_{u,u'})|, |V(P_{v,v'})| \le \alpha_{\text{pat}}(k)$, it is immediate that item (iii) holds. For item (i), note that
+$V(P_{u,u'}) \cap V(P_{u',v'})$ can be a strict superset of $\{u'\}$ only if $P_{u',v'}$ is a subpath of $P_{u,u'}$; then,
+$v' \in V(P_{u,u'})$, which means that $P_{u,u'}$ has an internal vertex from $\text{Sep}_{R^2}(P, v)$ and results in a
+contradiction. Thus, item (i) holds. Symmetrically, item (ii) holds as well. $\square$
+
+In what follows, when we use the notation $u'_p$, we refer to the vertex in Lemma 6.5. Before
+we describe the modification, we need to introduce another notation and make an immediate
+observation based on this notation.
+
+**Definition 6.4** ($\tilde{A}_{R^2,P,u}$). Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour, $P$ be a long maximal degree-2 path of $R^2$, and $u$ be an endpoint of $P$. Then, $\tilde{A}_{R^2,P,u} = (V(P_{u,u'_p}) \setminus \{u'_p\}) \cup A_{R^2,p,u}$.
+
+**Observation 6.7.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour, and $P$ be a long maximal degree-2 path of $R^2$ with endpoints $u$ and $v$. Then, there exists a single connected component $C_{R^2,P,u}$ in $H - (\text{Sep}_{R^2}(u, P) \cup \text{Sep}_{R^2}(v, P))$ that contains $\tilde{A}_{R^2,P,u}$ and a different single connected component $C_{R^2,P,v}$ in $H - (\text{Sep}_{R^2}(u, P) \cup \text{Sep}_{R^2}(v, P))$ that contains $\tilde{A}_{R^2,P,v}$.
+
+We proceed to describe the modification. For brevity, let $S_u = \text{Sep}_{R^2}(P, u)$ and $S_v = \text{Sep}_{R^2}(P, v)$. Recall that there is a terminal $t^* \in T$ that lies on the outer face of $H$ (and $G$). By Observation 6.6, $S_v$ and $S_v$ induce two cycles in $H$, and $t^*$ lies in the exterior of both these cycles. Assume w.l.o.g. that $u$ lies in the interior of both $S_u$ and $S_v$, while $v$ lies in the exterior of both $S_u$ and $S_v$. Then, $S_u$ belongs to the strict interior of $S_v$. We construct a sequence of concentric cycles between $S_u$ and $S_v$ as follows.
+
+**Lemma 6.6.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree with no detour, and $P$ be a long maximal degree-2 path of $R^2$ with endpoints $u$ and $v$. Let $S_u = \text{Sep}_{R^2}(P, u)$ and $S_v = \text{Sep}_{R^2}(P, v)$, where $S_u$ lies in the strict interior of $S_v$. Then, there is a sequence of concentric cycles $C(u, v) = (C_1, C_2, \dots, C_p)$ in $G$ of length $p \ge 100\alpha_{\text{sep}}(k)$ such that $S_u$ is in the strict interior of $C_1$ in $H$, $S_v$ is in the strict exterior of $C_p$ in $H$, and there is a path $\eta$ in $H$ with one endpoint $v_0 \in S_u$ and the other endpoint $v_{p+1} \in S_v$, such that the intersection of $V(\eta)$ with $V(G) \cup S_u \cup S_v$ is $\{v_0, v_1, \dots, v_{p+1}\}$ for some $v_i \in V(C_i)$ for every $i \in \{1, \dots, p\}$. Furthermore, $C(u, v)$ can be computed in linear time.
+
+*Proof.* Towards the computation of *C*(u, v), delete all vertices that lie in the strict interior of S<sub>u</sub> or in the strict exterior of S<sub>v</sub>, as well as all vertices of V(H) \ (V(G) ∪ S<sub>u</sub> ∪ S<sub>v</sub>). Denote the resulting graph by G<sup>+</sup><sub>u,v</sub>, and note that it has a plane embedding in the “ring” defined by H[S<sub>u</sub>] and H[S<sub>v</sub>]. Observe that S<sub>u</sub>, S<sub>v</sub> ⊆ V(G<sup>+</sup><sub>u,v</sub>), where S<sub>v</sub> defines the outer face of the embedding of G<sup>+</sup><sub>u,v</sub>. Thus, any cycle in this graph that separates S<sub>u</sub> and S<sub>v</sub> must contain S<sub>u</sub> in its interior and S<sub>v</sub> in its exterior. Furthermore, G<sub>u,v</sub> = G<sup>+</sup><sub>u,v</sub> − V(H) is an induced subgraph of G, which

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+consists of all vertices of $G$ that, in $H$, lie in the strict exterior of $S_u$ and in the strict interior of $S_v$ simultaneously, or lie in $S_v \cup S_v$. In particular, any cycle of $G_{u,v}$ is also a cycle in $G$.
+
+Now, $\mathcal{C}(u, v)$ is computed as follows. Start with an empty sequence, and the graph $G_{u,v} - (S_u \cup S_v)$. As long as there is a cycle in the current graph such that all vertices of $S_u$ are in the strict interior of $C$ with respect to $H$, remove vertices of degree at most 1 in the current graph until no such vertices remain, and append the outer face of the current graph as a cycle to the constructed sequence. It is clear that this process terminates in linear time, and that by the above discussion, it constructs a sequence of concentric cycles $\mathcal{C}(u, v) = (C_1, C_2, \dots, C_p)$ in $G$ such that $S_u$ is in the strict interior of $C_1$ in $H$, $S_v$ is in the strict exterior of $C_p$ in $H$.
+
+To assert the existence of a path $\eta$ in $H$ with one endpoint $v_0 \in S_u$ and the other endpoint $v_{p+1} \in S_v$, such that the intersection of $V(\eta)$ with $V(G) \cup S_u \cup S_v$ is $\{v_0, v_1, \dots, v_{p+1}\}$ for some $v_i \in V(C_i)$ for every $i \in \{1, \dots, p\}$, we require the following claim.
+
+**Claim 6.1.** Let $C_{p+1} = H[S_v]$. For every $i \in \{1, 2, \dots, p\}$ and every vertex $w \in V(C_i)$, there exists a vertex $w' \in V(C_{i+1})$ such that $w$ and $w'$ lie on a common face in $G_{u,v}^+$. Moreover, there exist vertices $w \in S_u$ and $w' \in V(C_1)$ that lie on a common face in $G_{u,v}^+$.
+
+*Proof of Claim 6.1.* Consider $i \in \{1, 2, \dots, p\}$ and a vertex $w \in V(C_i)$. We claim that $\mathrm{rdist}(w, V(C_{i+1})) \le 1$, i.e. there must be $w' \in V(C_{i+1})$ such that $w, w'$ have a common face in $G_{u,v}^+$. By way of contradiction, suppose that $\mathrm{rdist}(w, V(C_{i+1})) \ge 2$. Then, by Proposition 6.1, there is a cycle $C$ that separates $w$ and $V(C_{i+1})$ in $G_{u,v}^+$ such that $\mathrm{rdist}(w, w'') = 1$ for every vertex $w'' \in V(C)$. Here, $w$ lies in the strict interior of $C$, and $C$ lies in the strict interior of $C_{i+1}$. Further, $C$ is vertex disjoint from $C_{i+1}$, since $\mathrm{rdist}(w, w') \ge 2$ for every $w' \in V(C_{i+1})$. Now, consider the outer face of $G[V(C_i) \cup V(C)]$. By the construction of $C_i$, this outer face must be $C_i$. However, $w \in V(C_i)$ cannot belong to it, hence we reach a contradiction.
+
+For the second part, we claim that $\mathrm{rdist}(S_u, V(C_1)) \le 1$, i.e. there must be $w \in S_u$ and $w' \in V(C_1)$ such that $w, w'$ have a common face in $G_{u,v}^+$. By way of contradiction, suppose that $\mathrm{rdist}(S_u, V(C_1)) \ge 2$. Then, by Proposition 6.1, there is a cycle $C$ that separates $S_u$ and $V(C_1)$ in $G_{u,v}^+$ such that $\mathrm{rdist}(S_u, w'') = 1$ for every vertex $w'' \in V(C)$. Further, $C$ is vertex disjoint from $C_1$, since $\mathrm{rdist}(S_u, w') \ge 2$ for every $w' \in V(C_1)$. However, this is a contradiction to the termination condition of the construction of $\mathcal{C}(u,v). ◦$
+
+Having this claim, we construct $\eta$ as follows. Pick vertices $v_0 \in S_u$ and $v_1 \in V(C_1)$ that lie on a common face in $G_{u,v}^+$. Then, for every $i \in \{2, \dots, p+1\}$, pick a vertex $v_i \in V(C_i)$ such that $v_{i-1}$ and $v_i$ lie on a common face in $G_{u,v}^+$. Thus, for every $i \in \{0, 1, \dots, p\}$, we have that $v_i$ and $v_{i+1}$ are either adjacent in $H$ or there exists a vertex $u_i \in V(H) \setminus V(G)$ such that $u_i$ is adjacent to both $v_i$ and $v_{i+1}$. Because $\mathcal{C}(u,v) = (C_1, C_2, \dots, C_p)$ is a sequence of concentric cycles in $G$ such that $S_u$ is in the strict interior of $C_1$ and $S_v$ is in the strict exterior of $C_p$, the $u_i$'s are distinct. Thus, $\eta = v_0 - u_0 - v_1 - u_1 - v_2 - u_2 - \dots - v_p - u_p - v_{p+1}$, where undefined $u_i$'s are dropped, is a path as required.
+
+Finally, we argue that $p \ge 100 \cdot \alpha_{\mathrm{sep}}(k)$. Note that $100\alpha_{\mathrm{sep}}(k) = 100(\frac{7}{2} \cdot 2^{ck} + 2) \le 400 \cdot 2^{ck}$, thus it suffices to show that $p \ge 400 \cdot 2^{ck}$. To this end, we obtain a lower bound on the radial distance between $S_u$ and $S_v$ in $G_{u,v}^+$. Recall that $|S_u|, |S_v| \le \alpha_{\mathrm{sep}}(k) = \frac{7}{2} \cdot 2^{ck} + 2 \le 4 \cdot 2^{ck}$. Let $P = \mathrm{path}_{R^2}(u,v)$, and recall that its length is at least $\alpha_{\mathrm{long}}(k) = 10^4 2^{ck}$. Since $R^2$ has not detour, $P$ is a shortest path in $H$ between $u$ and $v$, thus for any two vertices in $V(P)$, the subpath of $P$ between them is a shortest path between them. Now, recall the vertices $u' = u'_P$, $v' = v'_P$ (defined in Lemma 6.5), and denote the subpath between them by $P'$. By construction, $|E(P')| \ge \alpha_{\mathrm{long}}(k) - 2 \cdot \alpha_{\mathrm{pat}}(k) = (10^4 - 200) \cdot 2^{ck}$.
+
+We claim that the radial distance between $S_u$ and $S_v$ in $G_{u,v}^+$ is at least $|E(P')|/2 - |S_u| - |S_v|$. Suppose not, and consider a sequence of vertices in $G_{u,v}^+$ that witnesses this fact: $x_1, x_2, x_3, \dots, x_{p-1}, x_p \in V(G_{u,v}^+)$ where $x_1 \in S_u$, $x_p \in S_v$, $p < |E(P')|/2 - |S_v| - |S_u|$, and every two consecutive vertices lie on a common face. Consider a shortest such sequence, which visits each face of $G_{u,v}^+$ at most once. In particular, $x_1$ and $x_p$ are the only vertices of $S_u \cup S_v$.

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+structural results, such as the algorithm for Disjoint Paths [37], Minor Testing [37] (given two undirected graphs, *G* and *H* on *n* and *k* vertices, respectively, the goal is to check whether *G* contains *H* as a minor), the structural decomposition [38] and the Excluded Grid Theorem [40]. Unfortunately, all of these results suffer from such bad hidden constants and dependence on the parameter *k* that they have gotten their own term–“galactic algorithms” [31].
+
+It is the hope of many researchers that, in time, algorithms and structural results from Graph Minors can be more algorithmically efficient, perhaps even practically applicable. Substantial progress has been made in this direction, examples include the simpler decomposition theorem of Kawarabayashi and Wollan [28], the faster algorithm for computing the structural decomposition of Grohe et al. [22], the improved unique linkage theorem of Kawarabayashi and Wollan [27], the linear excluded grid theorem on minor free classes of Demaine and Hajiaghayi [16], paving the way for the theory of Bidimensionality [15], and the polynomial grid minor theorem of Chekuri and Chuzhoy [6]. The algorithm for Disjoint Paths is a cornerstone of the entire Graph Minor Theory, and a vital ingredient in the $g(k)n^3$-time algorithm for Minor Testing. Therefore, efficient algorithms for Disjoint Paths and Minor Testing are necessary and crucial ingredients in an algorithmically efficient Graph Minors theory. This makes obtaining $2^{\text{poly}(k)}n^{\mathcal{O}(1)}$ time algorithms for Disjoint Paths and Minor Testing a tantalizing and challenging goal.
+
+Theorem 1.1 is a necessary basic step towards achieving this goal—a $2^{\text{poly}(k)}n^{\mathcal{O}(1)}$ time algorithms for Disjoint Paths on general graphs also has to handle planar inputs, and it is easy to give a reduction from Planar Disjoint Paths to Minor Testing in such a way that a $2^{\text{poly}(k)}n^{\mathcal{O}(1)}$ time algorithm for Minor Testing would imply a $2^{\text{poly}(k)}n^{\mathcal{O}(1)}$ time algorithms for Planar Disjoint Paths. In addition to being a necessary step in the formal sense, there is strong evidence that an efficient algorithm for the planar case will be useful for the general case as well—indeed the algorithm for Disjoint Paths of Robertson and Seymour [37] relies on topology and essentially reduces the problem to surface-embedded graphs. Thus, an efficient algorithm for Planar Disjoint Paths represents a speed-up of the base case of the algorithm for Disjoint Paths of Robertson and Seymour. Coupled with the other recent advances [6, 15, 16, 22, 27, 28], this gives some hope that $2^{\text{poly}(k)}n^{\mathcal{O}(1)}$ time algorithms for Disjoint Paths and Minor Testing may be within reach.
+
+**Known Techniques and Obstacles in Designing a 2<sup>poly</sup>(<i>k</i>) Algorithm.** All known algorithms for both Disjoint Paths and Planar Disjoint Paths have the same high level structure. In particular, given a graph <i>G</i> we distinguish between the cases of <i>G</i> having “small” or “large” treewidth. In case the treewidth is large, we distinguish between two further cases: either <i>G</i> contains a “large” clique minor or it does not. This results in the following case distinctions.
+
+1. **Treewidth is small.** Let the treewidth of *G* be *w*. Then, we use the known dynamic programming algorithm with running time $2^{\mathcal{O}(w \log w)} n^{\mathcal{O}(1)}$ [41] to solve the problem. It is important to note that, assuming the Exponential Time Hypothesis (ETH), there is no algorithm for Disjoint Paths running in time $2^{\mathcal{O}(w \log w)} n^{\mathcal{O}(1)}$ [32], nor an algorithm for Planar Disjoint Paths running in time $2^{\mathcal{O}(w)} n^{\mathcal{O}(1)}$ [4].
+
+2. **Treewidth is large and *G* has a large clique minor.** In this case, we use the good routing property of the clique to find an irrelevant vertex and delete it without changing the answer to the problem. Since this case will not arise for graphs embedded on a surface or for planar graphs, we do not discuss it in more detail.
+
+3. **Treewidth is large and *G* has no large clique minor.** Using a fundamental structure theorem for minors called the “flat wall theorem”, we can conclude that *G* contains a large planar piece of the graph and a vertex *v* that is sufficiently insulated in the middle of it. Applying the unique linkage theorem [39] to this vertex, we conclude that it is irrelevant and remove it. For planar graphs, one can use the unique linkage theorem of Adler et al. [2]. In particular, we use the following result:
+
+Any instance of Disjoint Paths consisting of a planar graph with treewidth at least $82k^{3/2}2^k$ and $k$ terminal pairs contains a vertex $v$ such that every solution

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+in this sequence. Then, we can extend this sequence on both sides to derive another sequence of vertices of $G_{u,v}^+$ starting at $u'$ and ending at $v'$ such that the prefix of the new sequence is a path in $G_{u,v}^+[S_u]$ from $u'$ to $x_1$, the midfix is $x_1, x_2, x_3, \dots, x_{p-1}, x_p$, and the suffix is a path in $G_{u,v}^+[S_v]$ from $x_p$ to $v'$. Further, the length of the new sequence of vertices is smaller than $|E(P')|/2$. Hence, the radial distance between $u'$ and $v'$ in $H' = H[V(G) \cup S_u \cup S_v]$ (the graph derived from $G_{u,v}^+$ by reintroducing the vertices of $G$ that lie inside $S_u$ or outside $S_v$) is smaller than $|E(P')|/2$, and let it be witnessed by a sequence $Q[u', v']$. As $H$ is the radial completion of $G$, observe that $Q[u', v']$ gives rise to a path $Q$ in $H$ between $u'$ and $v'$ of length smaller than $|E(P')|$. However, then $u', v'$ and $Q$ witness a detour in $\mathbb{R}^2$, which is a contradiction. Hence, the radial distance between $S_u$ and $S_v$ in $G_{u,v}^+$ is at least
+
+$$ 
+\begin{aligned}
+& \frac{(10^4 - 200)}{2} \cdot 2^{ck} - (|S_u| + |S_v|) \\
+& \geq 4900 \cdot 2^{ck} - 2\alpha_{\text{sep}}(k) \\
+& = 4900 \cdot 2^{ck} - (7 \cdot 2^{ck} + 4) \geq 400 \cdot 2^{ck}.
+\end{aligned}
+$$
+
+Now, observe that $S_u$ and $S_v$ are connected sets in $G_{u,v}^+$ and $S_v$ forms the outer-face of $G_{u,v}^+$. Then, by Proposition 6.1, we obtain a collection of at least $400 \cdot 2^{ck}$ disjoint cycles in $G_{u,v}^+$, where each cycle separates $S_u$ and $S_v$. Note that these cycles are disjoint from $S_u \cup S_v$, and hence they lie in $G$. Moreover, each of them contains $S_u$ in its strict interior, and $S_v$ in its strict exterior. Thus, it is clear that the sequence $\mathcal{C}(u,v)$ computed above must contain at least $400 \cdot 2^{ck} > 100\alpha_{\text{sep}}(k)$ cycles. $\square$
+
+Recall the graph $G_{u,v}$, which is an induced subgraph of $G$, which consists of all vertices of $G$ that, in $H$, lie in the strict exterior of $S_u$ and in the strict interior of $S_v$ simultaneously, or lie in $S_u \cup S_v$. With $\mathcal{C}(u,v)$ at hand, we compute a maximum size collection of disjoint paths from $S_u$ to $S_v$ in $G_{u,v}$ that minimizes the number of edges it traverses outside $E(\mathcal{C}(u,v))$. In this observation, the implicit assumption that $\ell \le \alpha_{\text{sep}}(k)$ is justified by Lemma 6.4.
+
+**Observation 6.8.** Let the maximum flow between $S_u \cap V(G)$ and $S_v \cap V(G)$ in $G_{u,v}$ be $\ell \le \alpha_{\text{sep}}(k)$. Given the sequence $\mathcal{C}(u,v)$ of Lemma 6.6, a collection $\text{Flow}_{\mathbb{R}^2}(u,v)$ of $\ell$ vertex-disjoint paths in $G_{u,v}$ from $S_u \cap V(G)$ to $S_v \cap V(G)$ that minimizes $|E(\text{Flow}_{\mathbb{R}^2}(u,v)) \setminus E(\mathcal{C}(u,v))|$ is computable in time $\mathcal{O}(n^{3/2}\log^3 n)$.
+
+*Proof.* We determine $\ell \le \alpha_{\text{sep}}(k)$ in time $2^{\mathcal{O}(k)n}$ by using Ford-Fulkerson algorithm. Next, we define a weight function $\bar{w}$ on $E(G_{u,v})$ as follows:
+
+$$ w(e) = \begin{cases} 0 & \text{if } e \in E(\mathcal{C}(u,v)) \\ 1 & \text{otherwise} \end{cases} $$
+
+We now compute a minimum cost flow between $S_u \cap V(G)$ and $S_v \cap V(G)$ of value $\ell$ in $G_{u,v}$ under the weight function $\bar{w}$. This can be done in time $\mathcal{O}(n^{3/2}\log^3 n)$ by [13, Theorem 1], as the cost of such a flow is bounded by $\mathcal{O}(n)$. Clearly, the result is a collection $\text{Flow}_{\mathbb{R}^2}(u,v)$ of $\ell$ vertex-disjoint paths from $S_u \cap V(G)$ to $S_v \in V(G)$ minimizing $|E(\text{Flow}_{\mathbb{R}^2}(u,v)) \setminus E(\mathcal{C}(u,v))|$. $\square$
+
+Having $\text{Flow}_{\mathbb{R}^2}(u,v)$ at hand, we proceed to find a certain path between $u'_P$ and $v'_P$ that will be used to replace $P_{u'_P, v'_P}$. We remark that all vertices and edges of $\text{Flow}_{\mathbb{R}^2}(u,v)$ lie between $S_u$ and $S_v$ in the plane embedding of $H$. The definition of this path is given by the following lemma, and the construction will make it intuitively clear that the paths in $\text{Flow}_{\mathbb{R}^2}(u,v)$ do not “spiral around” the Steiner tree once we replace $P_{u'_P, v'_P}$ with $P_{u'_P, v'_P}^*$. Note that in the lemma, we consider a path $\bar{P}$ in $H$, while the paths in $\text{Flow}_{\mathbb{R}^2}(u,v)$ are in $G$.

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+**Lemma 6.7.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $\mathbb{R}^2$ be a Steiner tree with no detour, and $P$ be a long maximal degree-2 path of $\mathbb{R}^2$, with endpoints $u$ and $v$. Let $u' = u'_P$ and $v' = v'_P$. Then, there exists a path $P_{u',v'}^*$ in $H - (V(C_{R^2,P,u}) \cup V(C_{R^2,P,u}))$ between $u'$ and $v'$ with the following property: there do not exist three vertices $x, y, z \in V(P_{u',v'}^*)$ such that (i) $\text{dist}_{P_{u',v'}^*}(u', x) < \text{dist}_{P_{u',v'}^*}(u', y) < \text{dist}_{P_{u',v'}^*}(u', z)$, and (ii) there exist a path in $\text{Flow}_{\mathbb{R}^2}(u, v)$ that contains $x$ and $z$ and a different path in $\text{Flow}_{\mathbb{R}^2}(u, v)$ that contains $y$. Moreover, such a path $P_{u',v'}^*$ can be computed in time $\mathcal{O}(n)$.
+
+*Proof.* Let $P_{u',v'}^*$ be a path in $H - (V(C_{\mathbb{R}^2,P,u}) \cup V(C_{\mathbb{R}^2,P,u}))$ between $u'$ and $v'$ that minimizes the number of paths $Q \in \text{Flow}_{\mathbb{R}^2}(u,v)$ for which there exist at least one triple $x, y, z \in V(P_{u',v'}^*)$ that has the two properties in the lemma and $x, z \in V(Q)$. Due to the existence of $P_{u',v'}$, such a path $P_{u',v'}^*$ exists. We claim that this path $P_{u',v'}^*$ has no triple $x, y, z \in V(P_{u',v'}^*)$ that has the two properties in the lemma. Suppose, by way of contradiction, that our claim is false, and let $x, y, z \in V(P_{u',v'}^*)$ be a triple that has the two properties in the lemma. Note that, when traversed from $u'$ to $v'$, $P_{u',v'}^*$ first visits $x$, then visits $y$ and afterwards visits $z$. Let $Q'$ be the path in $\text{Flow}_{\mathbb{R}^2}(u,v)$ that contains $x$ and $z$. Let $x'$ and $z'$ be the first and last vertices of $Q'$ that are visited by $P_{u',v'}^*$. Then, replace the subpath of $P_{u',v'}^*$ between $x'$ and $z'$ by the subpath of $Q'$ between $x'$ and $z'$. This way we obtain a path $P'$ in $H - (V(C_{\mathbb{R}^2,P,u}) \cup V(C_{\mathbb{R}^2,P,u}))$ between $u'$ and $v'$ for which there exist fewer paths $Q \in \text{Flow}_{\mathbb{R}^2}(u,v)$, when compared to $P_{u',v'}^*$, for which there exists at least one triple $x, y, z \in V(P')$ that has the two properties in the lemma and such that $x, z \in V(Q)$. As we have reached a contradiction, we conclude that our initial claim is correct. While the proof is existential, it can clearly be turned into a linear-time algorithm. $\square$
+
+The following is a direct corollary of the above lemma.
+
+**Corollary 6.1.** For each path $Q \in \text{Flow}_{\mathbb{R}^2}(u,v)$, there are at most two edges in $P_{u',v'}^*$ such that one endpoint of the edge lies in $V(Q)$ and the other lies in $V(G) \setminus V(Q)$.
+
+Having Lemma 6.7 at hand, we modify $\mathbb{R}^2$ as follows: for every long maximal degree-2 path $P$ of $\mathbb{R}^2$ with endpoints $u$ and $v$, replace $P_{u'_P, v'_P}$ by $P_{u'_P, v'_P}^*$. Denote the result of this modification by $\mathbb{R} = \mathbb{R}^3$. We refer to $\mathbb{R}^3$ as a *backbone Steiner tree*. Let us remark that the backbone Steiner tree $\mathbb{R}^3$ is always accompanied by the separators $\text{Sep}_{\mathbb{R}^2}(P, u)$ and $\text{Sep}_{\mathbb{R}^2}(P, v)$, and the collection $\text{Flow}_{\mathbb{R}^2}(u, v)$ for every long maximal degree-2 path $\mathcal{P} = \text{path}_{\mathbb{R}^2}(u, v)$ of $\mathbb{R}^2$. These separators and flows will play crucial role in our algorithm. In the following subsection, we will prove that $\mathbb{R}^3$ is indeed a Steiner tree, and in particular it is a tree. Additionally and crucially, we will prove that the separators computed previously remain separators. Let us first conclude the computational part by stating the running time spent so far. From Lemma 6.3, Observations 6.5 and 6.8, and by Lemma 6.7, we have the following result.
+
+**Lemma 6.8.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Then, a backbone Steiner tree $\mathcal{R}$ can be computed in time $2^{\mathcal{O}(k)n^{3/2}\log^3 n}$.
+
+## 6.5 Analysis of $\mathbb{R}^3$ and the Separators $\text{Sep}_{\mathbb{R}^2}(P, u)$
+
+Having constructed the backbone Steiner tree $\mathbb{R}^3$, we turn to analyse its properties. Among other properties, we show that useful properties of $\mathbb{R}^2$ also transfer to $\mathbb{R}^3$. We begin by proving that the two separators of each long maximal degree-2 path $P$ of $\mathbb{R}^2$ partition $V(\mathcal{H})$ into five “regions”, and that the vertices in each region are all close to the subtree of $\mathcal{R}$ that (roughly) belongs to that region. Specifically, the regions are $V(C_{\mathbb{R}^2,P,u})$, $\text{Sep}_{\mathbb{R}^2}(P, u)$, $V(C_{\mathbb{R}^2,P,v})$, $\text{Sep}_{\mathbb{R}^2}(P, v)$, and $V(\mathcal{H}) \setminus (V(C_{\mathbb{R}^2,P,u}) \cup V(C_{\mathbb{R}^2,P,v}) \cup \text{Sep}_{\mathbb{R}^2}(P, u) \cup \text{Sep}_{\mathbb{R}^2}(P, v))$, and our claim is as follows.
+
+**Lemma 6.9.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $\mathbb{R}^2$ be a Steiner tree that has no detour, $P$ be a long maximal degree-2 path of $\mathbb{R}^2$, and $u$ and $v$ be its endpoints.

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+1. For all $w \in \text{Sep}_{R^2}(P, u)$, it holds that $\text{dist}_H(w, u'_P) \le \alpha_{\text{sep}}(k)$.
+
+2. For all $w \in \text{Sep}_{R^2}(P, v)$, it holds that $\text{dist}_H(w, v'_P) \le \alpha_{\text{sep}}(k)$.
+
+3. For all $w \in V(C_{R^2,P,u})$, it holds that $\text{dist}_H(w, \tilde{A}_{R^2,P,u} \cup \{u'_P\}) \le \alpha_{\text{dist}}(k) + \alpha_{\text{sep}}(k)$.
+
+4. For all $w \in V(C_{R^2,P,v})$, it holds that $\text{dist}_H(w, \tilde{A}_{R^2,P,v} \cup \{v'_P\}) \le \alpha_{\text{dist}}(k) + \alpha_{\text{sep}}(k)$.
+
+5. For all $w \in V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}) \cup \text{Sep}_{R^2}(P, u) \cup \text{Sep}_{R^2}(P, v))$, it holds that $\text{dist}_H(w, V(R^2) \setminus (\tilde{A}_{R^2,P,u} \cup \tilde{A}_{R^2,P,v})) \le \alpha_{\text{dist}}(k) + \alpha_{\text{sep}}(k)$.
+
+*Proof.* First, note that Conditions 1 and 2 follow directly from Lemma 6.4 and Observation 6.6.
+
+For Condition 3, consider some vertex $w \in V(C_{R^2,P,u})$. By Lemma 6.1, $\text{dist}_H(w, V(R^2)) \le \alpha_{\text{dist}}(k)$. Thus, there exists a path $Q$ in $H$ with $w$ as one endpoint and the other endpoint $x$ in $V(R^2)$ such that the length of $Q$ is at most $\alpha_{\text{dist}}(k)$. In case $x \in \tilde{A}_{R^2,P,u} \cup \{u'_P\}$, we have that $\text{dist}_H(w, \tilde{A}_{R^2,P,u}) \le \alpha_{\text{dist}}(k)$, and hence the condition holds. Otherwise, by the definition of $\text{Sep}_{R^2}(P, u)$, the path $Q$ must traverse at least one vertex from $\text{Sep}_{R^2}(P, u)$. Thus, $\text{dist}_H(w, \text{Sep}_{R^2}(P, u)) \le \alpha_{\text{dist}}(k)$. Combined with Condition 1, we derive that $\text{dist}_H(w, \tilde{A}_{R^2,P,u} \cup \{u'_P\}) \le \alpha_{\text{dist}}(k) + \alpha_{\text{sep}}(k)$. The proof of Condition 4 is symmetric.
+
+The proof of Condition 5 is similar. Consider some vertex $w \in V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}) \cup \text{Sep}_{R^2}(P, u) \cup \text{Sep}_{R^2}(P, v))$. As before, there exists a path $Q$ in $H$ with $w$ as one endpoint and the other endpoint $x$ in $V(R^2)$ such that the length of $Q$ is at most $\alpha_{\text{dist}}(k)$. In case $x \in \tilde{A}_{R^2,P,u} \cup \{u'_P\}$, we are done. Otherwise, the path $Q$ must traverse at least one vertex from $\text{Sep}_{R^2}(P, u) \cup \text{Sep}_{R^2}(P, v)$. Specifically, if $x \in V(C_{R^2,P,u})$, then it must traverse at least one vertex from $\text{Sep}_{R^2}(P, u)$, and otherwise $x \in V(C_{R^2,P,v})$ and it must traverse at least one vertex from $\text{Sep}_{R^2}(P, v)$. Combined with Conditions 1 and 2, we derive that $\text{dist}_H(w, V(R^2) \setminus (\tilde{A}_{R^2,P,u} \cup \tilde{A}_{R^2,P,v})) \le \alpha_{\text{dist}}(k) + \alpha_{\text{sep}}(k)$. $\square$
+
+An immediate corollary of Lemma 6.9 concerns the connectivity of the “middle region” as follows. (This corollary can also be easily proved directly.)
+
+**Corollary 6.2.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour, $P$ be a long maximal degree-2 path of $R^2$, and $u$ and $v$ be its endpoints. Then, $H[V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))]$ is a connected graph.
+
+*Proof.* By Lemma 6.9 and the definition of $\text{Sep}_{R^2}(P, u)$ and $\text{Sep}_{R^2}(P, v)$, for every vertex in $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))$, the graph $H$ has a path from that vertex to some vertex in $\text{Sep}_{R^2}(P, u) \cup \text{Sep}_{R^2}(P, v)$ that lies entirely in $H[V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))]$. Thus, the corollary follows from Observation 6.6. $\square$
+
+Next, we utilize Lemma 6.9 and Corollary 6.2 to argue that the “middle regions” of different long maximal degree-2 paths of $R^2$ are distinct. Recall that we wish to reroute a given solution to be a solution that “spirals” only a few times around the Steiner tree. This lemma allows us to independently reroute the solution in each of these “middle regions”. In fact, we prove the following stronger statement concerning these regions. The idea behind the proof of this lemma is that if it were false, then $R^2$ admits a detour, which is contradiction.
+
+**Lemma 6.10.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour. Additionally, let $P$ and $\hat{P}$ be two distinct long maximal degree-2 paths of $R^2$. Let $u$ and $v$ be the endpoints of $P$, and $\hat{u}$ and $\hat{v}$ be the endpoints of $\hat{P}$. Then, one of the two following conditions holds:
+
+• $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v})) \subseteq V(C_{R^2,\hat{P},\hat{u}}).$
+
+• $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v})) \subseteq V(C_{R^2,\hat{P},\hat{v}}).$

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+Figure 15: Illustration of Lemma 6.10.
+
+*Proof.* We first prove that the intersection of $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))$ with $V(H) \setminus (V(C_{R^2,\hat{P},\hat{u}}) \cup V(C_{R^2,\hat{P},\hat{v}}))$ is empty. To this end, suppose by way of contradiction that there exists a vertex $w$ in this intersection. By Lemma 6.9, the inclusion of $w$ in both $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))$ and $V(H) \setminus (V(C_{R^2,\hat{P},\hat{u}}) \cup V(C_{R^2,\hat{P},\hat{v}}))$ implies that the two following inequalities are satisfied:
+
+• $\mathrm{dist}_H(w, V(R^2) \setminus (\tilde{A}_{R^2,P,u} \cup \tilde{A}_{R^2,P,v})) \leq \alpha_{\mathrm{dist}}(k) + \alpha_{\mathrm{sep}}(k).$
+
+• $\mathrm{dist}_H(w, V(R^2) \setminus (\tilde{A}_{R^2,\hat{P},\hat{u}} \cup \tilde{A}_{R^2,\hat{P},\hat{v}})) \leq \alpha_{\mathrm{dist}}(k) + \alpha_{\mathrm{sep}}(k).$
+
+From this, we derive the following inequality:
+
+$$
+\mathrm{dist}_H(V(R^2) \setminus (\tilde{A}_{R^2,P,u} \cup \tilde{A}_{R^2,P,v}), V(R^2) \setminus (\tilde{A}_{R^2,\hat{P},\hat{u}} \cup \tilde{A}_{R^2,\hat{P},\hat{v}})) \leq 2(\alpha_{\mathrm{dist}}(k) + \alpha_{\mathrm{sep}}(k)).
+$$
+
+In particular, this means that there exist vertices $x \in V(P_{u'_P, v'_P})$ and $y \in V(\hat{P}_{\hat{u}'_P, \hat{v}'_P})$ and a path $Q$ in $H$ between them whose length is at most $2(\alpha_{\text{dist}}(k) + \alpha_{\text{sep}}(k))$. Note that the unique path in $R^2$ between $x$ and $y$ traverses exactly one vertex in $\{u, v\}$. Suppose w.l.o.g. that this vertex is $u$. Then, consider the walk $W$ (which might be a path) obtained by traversing $P$ from $v$ to $x$ and then traversing $Q$ from $x$ to $y$. Now, notice that
+
+$$
+\begin{align*}
+|E(P)| - |E(W)| &\geq |E(P'_{u,u'_P})| - 2(\alpha_{\text{dist}}(k) + \alpha_{\text{sep}}(k)) \\
+&\geq \alpha_{\text{pat}}(k)/2 - 2(\alpha_{\text{dist}}(k) + \alpha_{\text{sep}}(k)) \\
+&= 50 \cdot 2^{ck} - 2(4 \cdot 2^{ck} + \frac{7}{2} \cdot 2^{ck} + 2) > 0.
+\end{align*}
+$$
+
+Here, the inequality $|E(P'_{u,u'_P})| \geq \alpha_{\text{pat}}(k)/2$ followed from Lemma 6.5. As $|E(P)| - |E(W)| > 0$, we have that $u, v$ and any subpath of the walk $W$ between $u$ and $v$ witness that $R^2$ has a detour (see Fig. 15). This is a contradiction, and hence we conclude that the intersection of $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))$ with $V(H) \setminus (V(C_{R^2,\hat{P},\hat{u}}) \cup V(C_{R^2,\hat{P},\hat{v}}))$ is empty.
+
+Having proved that the intersection is empty, we know that
+
+$$
+V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v})) \subseteq V(C_{R^2,\hat{P},\hat{u}}) \cup V(C_{R^2,\hat{P},\hat{v}}).
+$$
+
+Thus, it remains to show that $V(H)\setminus(V(C_{R^2,P,u})\cup V(C_{R^2,P,v}))$ cannot contain vertices from both $V(C_{R^2,\hat{P},\hat{u}})$ and $V(C_{R^2,\hat{P},\hat{v}})$. Since $H[V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))]$ is a connected graph (by Corollary 6.2), if $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))$ contains vertices from both $V(C_{R^2,\hat{P},\hat{u}})$ and $V(C_{R^2,\hat{P},\hat{v}})$, then it must also contain at least one vertex from $\mathrm{Sep}_{R^2}(\hat{P}, \hat{u}) \cup \mathrm{Sep}_{R^2}(\hat{P}, \hat{v}) \subseteq V(H) \setminus (V(C_{R^2,\hat{P},\hat{u}}) \cup V(C_{R^2,\hat{P},\hat{v}}))$, which we have already shown to be impossible. Thus, the proof is complete. □

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+We are now ready to prove that $R^3$ is a Steiner tree.
+
+**Lemma 6.11.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour, and $R^3$ be the subgraph constructed from $R^2$ in Step IV. Then, $R^3$ is a Steiner tree with the following properties.
+
+* $R^3$ has the same set of vertices of degree at least 3 as $R^2$.
+
+* Every short maximal degree-2 path $P$ of $R^2$ is a short maximal degree-2 path of $R^3$.
+
+* For every long maximal degree-2 path $P$ of $R^2$ with endpoints $u$ and $v$, the paths $P_{u,u'_P}$ and $P_{v,v'_P}$ are subpaths of the maximal degree-2 path of $R^3$ with endpoints $u$ and $v$.
+
+*Proof.* To prove that $R^3$ is a Steiner tree, we only need to show that $R^3$ is acyclic. Indeed, the construction of $R^3$ immediately implies that it is connected and has the same set of degree-1 vertices as $R^2$, which together with an assertion that $R^3$ is acyclic, will imply that it is a Steiner tree. The other properties in the lemma are immediate consequences of the construction of $R^3$. By its construction, to show that $R^3$ is acyclic, it suffices to prove two conditions:
+
+* For every long maximal degree-2 path $P$ of $R^2$ with endpoints $u$ and $v$, it holds that
+  $$V(P_{u'_{P}, v'_{P}}^{*}) \cap (V(R^2) \setminus V(P_{u'_{P}, v'_{P}})) = \emptyset.$$
+
+* For every two distinct long maximal degree-2 paths $P$ and $\hat{P}$ of $R^2$, it holds that
+  $$V(P_{u'_{P}, v'_{P}}^{*}) \cap V(\hat{P}_{\hat{u}'', \hat{v}''}) = \emptyset,$$
+  where $u$ and $v$ are the endpoints of $P$, $\hat{u}$ and $\hat{v}$ are the endpoints of $\hat{P}$,
+  $$u' = u'_{P}, v' = v'_{P}, \hat{u}' = \hat{u}'_{\hat{P}}, \text{ and } \hat{v}' = \hat{v}'_{\hat{P}}.$$
+
+The first condition follows directly from the fact that $P_{u'_{P}, v'_{P}}^{*}$ is a path in $H - (V(C_{R^2,P,u}) \cup V(C_{R^2,P,u}))$ while $V(R^2) \setminus V(P_{u'_{P}, v'_{P}})$ $\subseteq$ $V(C_{R^2,P,u}) \cup V(C_{R^2,P,u})$.
+
+For the second condition, note that Lemma 6.10 implies that $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v})) \subseteq V(C_{R^2,\hat{P},\hat{u}}) \cup V(C_{R^2,\hat{P},\hat{v}})$. Thus, we have that $V(P_{u'_{P},v'_{P}}^{*}) \subseteq V(C_{R^2,\hat{P},\hat{u}}) \cup V(C_{R^2,\hat{P},\hat{v}})$. However,
+$$V(\hat{P}_{\hat{u}',\hat{v}'}) \cap (V(C_{R^2,\hat{P},\hat{u}}) \cup V(C_{R^2,\hat{P},\hat{v}})) = \emptyset, \text{ and hence } V(P_{u'_{P},v'_{P}}^{*}) \cap V(\hat{P}_{\hat{u}',\hat{v}'}) = \emptyset. \quad \square$$
+
+We remark that $R^3$ might have detours. (These detours are restricted to $P_{u'_{P}, v'_{P}}$ for some long path $P = \text{path}_{R^3}(u, v)$ in $R^3$.) However, what is important for us is that we can still use the same small separators as before. To this end, we first define the appropriate notations, in particular since later we will like to address objects corresponding to $R^3$ directly (without referring to $R^2$). The validity of these notations follows from Lemma 6.11. Recall that $u'_{P}$ and $P_{u,u'_{P}}$ refer to the vertex and path in Lemma 6.5.
+
+**Definition 6.5 (Translating Notations of $R^2$ to $R^3$).** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ and $R^3$ be the Steiner trees constructed in Steps II and IV. For any long maximal degree-2 path $\hat{P}$ of $R^3$ and for each endpoint $u$ of $\hat{P}$:
+
+* Define $\hat{P}_{R^2}$ as the unique (long maximal degree-2) path in $R^2$ with the same endpoints as $\hat{P}$.
+
+* Let $P = \hat{P}_{R^2}$. Then, denote $u'_{\hat{P}} = u'_{P}$, $\hat{P}_{u,u'_{\hat{P}}} = P_{u,u'_{P}}$ and $\text{Sep}_{R^3}(\hat{P}, u) = \text{Sep}_{R^2}(P, u)$.
+
+* Define $A_{R^3, \hat{P}, u}^{*}$ as the union of $V(\hat{P}_{u, u'_{\hat{P}}})$ and the vertex set of the connected component of $R^3 - (V(\hat{P}_{u, u'_{\hat{P}}}) \setminus \{u\})$ containing $u$.
+
+* Define $B_{R^3, \hat{P}, u}^{*} = V(R^3) \setminus A_{R^3, \hat{P}, u}^{*}$.

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+Figure 16: Illustration of Lemma 6.12.
+
+In the context of Definition 6.5, note that by Lemma 6.5, $u' \in \text{Sep}_{R^3}(\hat{P}, u) \cap V(\hat{P})$ where $u' = u'_{\hat{P}}$, $\hat{P}_{u,u'}$ is the subpath of $\hat{P}$ between $u$ and $u'$, $\hat{P}_{u,u'}$ has no internal vertex from $\text{Sep}_{R^3}(\hat{P}, u) \cup \text{Sep}_{R^3}(\hat{P}, v)$, and $\alpha_{\text{pat}}(k)/2 \le |V(\hat{P}_{u,u'})| \le \alpha_{\text{pat}}(k)$. Additionally, note that $A^*_{R^3, \hat{P}, u}$ might not be equal to $A_{R^2, P, u}$ where $P = \hat{P}_{R^2}$. When $R^3$ is clear from context, we omit it from the subscripts.
+
+Now, let us argue why, in a sense, we can still use the same small separators as before.
+Recall that a backbone Steiner tree is a Steiner tree constructed in Step IV.
+
+**Lemma 6.12.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^3$ be a backbone Steiner tree. Additionally, let $\tilde{P}$ be a long maximal degree-2 path of $R^3$, and $u$ be an endpoint of $\tilde{P}$. Then, $\text{Sep}(\tilde{P}, u)$ separates $A^*_{\tilde{P},u}$ and $B^*_{\tilde{P},u}$ in $H$.
+
+*Proof*. Denote $P = \hat{P}_{R^2}$ where $R^2$ is the Steiner tree computed in Step II to construct $R^3$. Then, $\text{Sep}(\hat{P}, u)$ separates $V(C_{R^2,P,v})$ and $V(C_{R^2,P,u})$ in $H$. Thus, to prove that $\text{Sep}(\hat{P}, u)$ separates $A^*_{\hat{P},u}$ and $B^*_{\hat{P},u}$ in $H$, it suffices to show that $A^*_{\hat{P},u} \subseteq V(C_{R^2,P,u})$ and $A^*_{\hat{P},v} \subseteq V(C_{R^2,P,v})$ (because $B^*_{\hat{P},u} \setminus A^*_{\hat{P},v} \subseteq V(P^*_{u'_P,v'_P})$ and $P^*_{u'_P,v'_P} \cap V(C_{R^2,P,u} = \emptyset$ by the construction of $P^*_{u'_P,v'_P}$). We only prove that $A^*_{\hat{P},u} \subseteq V(C_{R^2,P,u})$. The proof of the other containment is symmetric.
+
+Clearly, $A_{R^2,P,u} \cap A^*_{\hat{P},v} \subseteq V(C_{R^2,P,u})$. Thus, due to Lemma 6.11, to show that $A^*_{\hat{P},\tilde{u}} \subseteq V(C_{R^2,P,u})$, it suffices to show the following claim: For every long maximal degree-2 path $\tilde{P}$ of $R^2$ whose vertex set is contained in $V(C_{R^2,P,u})$, it holds that the vertex set of $\tilde{P}_{\tilde{u}'\tilde{P},\tilde{v}'\tilde{P}}$ (computed by Lemma 6.7) is contained in $V(C_{R^2,P,u})$ as well, where $\tilde{u}$ and $\tilde{v}$ are the endpoints of $\tilde{P}$. We refer the reader to Fig. 16 for an illustration of this statement. For the purpose of proving it, consider some long maximal degree-2 path $\tilde{P}$ of $R^2$ whose vertex set is contained in $V(C_{R^2,P,u})$.
+
+By Lemma 6.10, we know that either $V(H) \setminus (V(C_{R^2,\tilde{P},\tilde{u}}) \cup V(C_{R^2,\tilde{P},\tilde{v}})) \subseteq V(C_{R^2,P,u})$ or $V(H) \setminus (V(C_{R^2,\tilde{P},\tilde{u}}) \cup V(C_{R^2,\tilde{P},\tilde{v}})) \subseteq V(C_{R^2,P,v})$. Moreover, by the definition of $\tilde{P}_{\tilde{u}'\tilde{P},\tilde{v}'\tilde{P}}$, its vertex set is contained in $V(H) \setminus (V(C_{R^2,\tilde{P},\tilde{u}}) \cup V(C_{R^2,\tilde{P},\tilde{v}}))$. Thus, to conclude the proof, it remains to rule out the possibility that $V(H) \setminus (V(C_{R^2,\tilde{P},\tilde{u}}) \cup V(C_{R^2,\tilde{P},\tilde{v}})) \subseteq V(C_{R^2,P,v})$. For this purpose, recall that we chose $\tilde{P}$ such that $V(\tilde{P}) \subseteq V(C_{R^2,P,u})$, and that $V(\tilde{P}) \cap V(C_{R^2,P,u}) \neq \emptyset$. Because $V(C_{R^2,P,u}) \cap V(C_{R^2,P,v}) = \emptyset$, we derive that the containment $V(H) \setminus (V(C_{R^2,\tilde{P},\tilde{u}}) \cup V(C_{R^2,\tilde{P},\tilde{v}})) \subseteq V(C_{R^2,P,v})$ is indeed impossible. □
+
+## 6.6 Enumerating Parallel Edges with Respect to $R^3$
+
+Recall that $H$ is enriched with $4n + 1$ parallel copies of each edge of the (standard) radial completion of $G$. While the copies did not play a role in the construction of $R$, they will be

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+Figure 9: A solution winding in a ring (top), and the “unwinding” or it (bottom).
+
+Having $P^*$ at hand, we replace $P'$ by $P^*$. This is done for every maximal degree-2 path, and thus we complete the construction of $R$. However, at this point, it is not clear why after we perform these replacements, the separators considered earlier remain separators, or that we even still have a tree. Roughly speaking, a scenario as depicted in Fig. 10 can potentially happen. To show that this is not the case, it suffices to prove that there cannot exist a vertex that belongs to two different rings. Towards that, we apply another preprocessing operation: we ensure that the radial completion of $G$ does not have $2^{ck}$ (for some constant $c$) concentric cycles that contain no vertex in $S \cup T$ by using another result by Adler et al. [2]. Informally, a sequence of concentric cycles is a sequence of vertex-disjoint cycles where each one of them is contained inside the next one in the sequence. Having no such sequences, we prove the following.
+
+**Lemma 2.3.** Let $R'$ be any Steiner tree. For every vertex $v$, there exists a vertex in $V(R')$ whose distance to $v$ (in the radial completion of $G$) is $2^{ck}$ for some constant $c$.
+
+To see why intuitively this lemma is correct, note that if $v$ was “far” from $R'$ in the radial completion of $G$, then in $G$ itself $v$ is surrounded by a large sequence of concentric cycles that contain no vertex in $S \cup T$. Having Lemma 2.3 at hand, we show that if a vertex belongs to a certain ring, then it is “close” to at least one vertex of the restriction of $R$ to that ring. In turn, that means that if a vertex belongs to two rings, it can be used to exhibit a “short” path between one vertex in the restriction of $R$ to one ring and another vertex in the restriction of $R$ to the second ring. By choosing constants properly, this path is shown to exhibit a detour in $R$, and hence we reach a contradiction. (In this argument, we use the fact that for every vertex $u$, towards the computation of the separator, we considered a vertex $u'$ of distance $2^{c_1k}$ from $u$—this subpath between $u$ and $u'$ is precisely that subpath that we will shortcut.)

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+**Observation 6.10.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths with a backbone Steiner tree $R$. For every $v \in V(H)$, $\text{order}_v$ is an enumeration of $\hat{E}_R(v)$ in either clockwise or counter-clockwise order around $v$ (with a fixed start). Further, for any pair $e, e' \in E_R(v)$ such that $e$ occurs before $e'$ in $\text{order}_v$, the edges $e_0, e_1, \dots, e_{2n}$ occur before $e'_0, e'_1, \dots, e'_{2n}$.
+
+# 7 Existence of a Solution with Small Winding Number
+
+In this section we show that if the given instance admits a solution, then it admits a “nice solution”. The precise definition of nice will be in terms of “winding number” of the solution, which counts the number of times the solution spirals around the backbone steiner tree. Our goal is to show that there is a solution of small winding number.
+
+## 7.1 Rings and Winding Numbers
+
+Towards the definition of a ring, let us remind that $H$ is the triangulated plane multigraph obtained by introducing $4n + 1$ parallel copies of each edge to the radial completion of the input graph $G$. Hence, each face of $H$ is either a triangle or a 2-cycle.
+
+**Definition 7.1 (Ring).** Let $I_{\text{in}}$, $I_{\text{out}}$ be two disjoint cycles in $H$ such that the cycle $I_{\text{in}}$ is drawn in the strict interior of the cycle $I_{\text{out}}$. Then, $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ is the plane subgraph of $H$ induced by the set of vertices that are either in $V(I_{\text{in}}) \cup V(I_{\text{out}})$ or drawn between $I_{\text{in}}$ and $I_{\text{out}}$ (i.e. belong to the exterior of $I_{\text{in}}$ and the interior of $I_{\text{out}}$).
+
+We call $I_{\text{in}}$ and $I_{\text{out}}$ are the *inner* and *outer interfaces* of $\text{Ring}(I_{\text{in}}, I_{\text{out}})$. We also say that this ring is induced by $I_{\text{in}}$ and $I_{\text{out}}$. Recall the notion of self-crossing walks defined in Section 5. Unless stated otherwise, all walks considered here are *not self-crossing*. A walk $\alpha$ in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ is *traversing* the ring if one of its endpoints lies in $I_{\text{in}}$ and the other lies in $I_{\text{out}}$. A walk $\alpha$ is *visiting* the ring if both its endpoints together lie in either $I_{\text{in}}$ or in $I_{\text{out}}$; moreover $\alpha$ is an *inner visitor* if both its endpoints lie in $I_{\text{in}}$, and otherwise it is an *outer visitor*.
+
+**Definition 7.2 (Orienting Walks).** Fix an arbitrary ordering of all vertices in $I_{\text{in}}$ and another one for all vertices in $I_{\text{out}}$. Then for a walk $\alpha$ in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ with endpoints in $V(I_{\text{in}}) \cup V(I_{\text{out}})$, orient $\alpha$ from one endpoint to another as follows. If $\alpha$ is a traversing walk, then orient it from its endpoint in $I_{\text{in}}$ to its endpoint in $I_{\text{out}}$. If $\alpha$ is a visiting walk, then both its endpoints lie either in $I_{\text{in}}$ or in $I_{\text{out}}$; then, orient $\alpha$ from its smaller endpoint to its greater endpoint.
+
+Observe that if $\alpha$ is a traversing path in the ring, then the orientation of $\alpha$ also defines its left-side and right-side. These are required for the following definition.
+
+**Definition 7.3 (Winding Number of a Walk w.r.t. a Traversing Path).** Let $\alpha$ be an a walk in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ with endpoints in $V(I_{\text{in}}) \cup V(I_{\text{out}})$, and let $\beta$ be a traversing path in this ring, such that $\alpha$ and $\beta$ are edge disjoint. The winding number, $\overline{\text{WindNum}}(\alpha, \beta)$, of $\alpha$ with respect to $\beta$ is the signed number of crossings of $\alpha$ with respect to $\beta$. That is, while walking along $\alpha$ (according to the orientation in Definition 7.2, for each intersection of $\alpha$ and $\beta$ record +1 if $\alpha$ crosses $\beta$ from left to right, -1 if $\alpha$ crosses $\beta$ from right to left, and 0 if it does not cross $\beta$. Then, the winding number $\overline{\text{WindNum}}(\alpha, \beta)$ is the sum of the recorded numbers.
+
+Observe that if $\alpha$ and $\beta$ are edge-disjoint traversing paths, then both $\overline{\text{WindNum}}(\alpha, \beta)$ and $\overline{\text{WindNum}}(\beta, \alpha)$ are well defined. We now state some well-known properties of the winding number. We sketch a proof of these properties in Appendix A, using homotopy.
+
+**Proposition 7.1.** Let $\alpha, \beta$ and $\gamma$ be three edge-disjoint paths traversing $\text{Ring}(I_{\text{in}}, I_{\text{out}})$. Then,
+
+$$ (i) \overline{\text{WindNum}}(\beta, \gamma) = -\overline{\text{WindNum}}(\gamma, \beta). $$

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+$$
+(ii) \quad \left| \overline{\text{WindNum}}(\alpha, \beta) - \overline{\text{WindNum}}(\alpha, \gamma) \right| - \left| \overline{\text{WindNum}}(\beta, \gamma) \right| \le 1.
+$$
+
+We say that $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$ is *rooted* if it is equipped with some fixed path $\eta$ that is traversing it, called the *reference path* of this ring. In a rooted ring $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$, we measure all winding numbers with respect to $\eta$, hence we shall use the shorthand $\overline{\mathrm{WindNum}}(\alpha) = \overline{\mathrm{WindNum}}(\alpha, \eta)$ when $\eta$ is implicit or clear from context. Here, we implicitly assume that the walk $\alpha$ is edge disjoint from $\eta$. This requirement will always be met by the following assumptions: (i) $H$ is a plane multigraph where we have $4n + 1$ parallel copies of every edge, and we assume that the reference path $\eta$ consists of only the 0-th copy $e_0$; and (ii) whenever we consider the winding number of a walk $\alpha$, it will edge-disjoint from the reference curve $\eta$ as it will not contain the 0-th copy of any edge. (In particular, the walks of the (weak) linkages that we consider will always satisfy this property.)
+
+Note that any visitor walk in $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$ with both endpoints in $I_{\mathrm{in}}$ is discretely homotopic to a segment of $I_{\mathrm{in}}$, and similarly for $I_{\mathrm{out}}$. Thus, we derive the following observation.
+
+**Observation 7.1.** Let $\alpha$ be a visitor in $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$. Then, $|\overline{\mathrm{WindNum}}(\alpha)| \le 1$.
+
+Recall the notion of a weak linkage defined in Section 5, which is a collection of edge-disjoint non-crossing walks. When we use the term *weak linkage of order k* in $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$, we refer to a weak linkage such that each walk has both endpoints in $V(I_{\mathrm{in}}) \cup V(I_{\mathrm{out}})$. For brevity, we abuse the term ‘weak linkage’ to mean a weak linkage in a ring when it is clear from context. Note that every walk in a weak linkage $\mathcal{P}$ is an inner visitor, or an outer visitor, or a traversing walk. This partitions $\mathcal{P}$ into $P_{\mathrm{in}}, P_{\mathrm{out}}, P_{\mathrm{traverse}}$. A weak linkage is *traversing* if it consists only of traversing walks. Assuming that $\mathrm{Ring}(I_{\mathrm{in}}, I_{\mathrm{out}})$ is rooted, we define the *winding number* of a traversing weak linkage $\mathcal{P}$ as $\overline{\mathrm{WindNum}}(\mathcal{P}) = \overline{\mathrm{WindNum}}(P_1)$. Recall that any two walks in a weak linkage are non-crossing. Then as observed in [14, Observation 4.4],¹²
+
+$$
+|\overline{\text{WindNum}}(P_i) - \overline{\text{WindNum}}(\mathcal{P})| \le 1 \quad \text{for all } i = 1, \dots, k.
+$$
+
+The above definition is extended to any weak linkage $\mathcal{P}$ in the ring as follows: if there is no walk in $\mathcal{P}$ that traverses the ring, then $\overline{\text{WindNum}}(\mathcal{P}) = 0$, otherwise $\overline{\text{WindNum}}(\mathcal{P}) = \overline{\text{WindNum}}(\mathcal{P}_{\text{traverse}})$. Note that, two aligned weak linkages $\mathcal{P}$ and $\mathcal{Q}$ in the ring may have different winding numbers (with respect to any reference path). Replacing a linkage $\mathcal{P}$ with an aligned linkage $\mathcal{Q}$ having a “small” winding number will be the main focus of this section.
+
+Lastly, we define a labeling of the edges based on the winding number of a walk (this relation
+is made explicit in the observation that follows).
+
+**Definition 7.4.** Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths, and $H$ be the radial completion of $G$. Let $\alpha$ be a (not self-crossing) walk in $H$, and let $\beta$ be a path in $H$ such that $\alpha$ and $\beta$ are edge disjoint. Let us fix (arbitrary) orientations of $\alpha$ and $\beta$, and define the left and right side of the path $\beta$ with respect to its orientation. The labeling $\text{label}_\beta^\alpha$ of each ordered pair of consecutive edges, $(e, e') \in E_H(\alpha) \times E_H(\alpha)$ by $\{-1, 0, +1\}$ with respect to $\beta$, where $e$ occurs before $e'$ when traversing $\alpha$ according to its orientation is defined as follows.
+
+• The pair $(e, e')$ is labeled $+1$ if $e$ is on the left of $\beta$ while $e'$ is on the right of $\beta$.
+
+• else, $(e, e')$ is labeled $-1$ if $e$ is on the right while $e'$ is on the left of $\beta$;
+
+* otherwise *e* and *e'* are on the same side of *β* and (*e*, *e*') is labeled 0.
+
+Note that in the above labeling only pairs of consecutive edges may get a non-zero label,
+depending on how they cross the reference path. For the ease of notation, we extend the above
+
+<sup>12</sup>This inequality also follows from the second property of Proposition 7.1 by setting α to be the reference path,
+β = P<sub>1</sub> and γ = P<sub>i</sub> and noting that <span style="text-decoration: overline;">WindNum</span>(β, γ) = 0.

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+labeling function to all ordered pairs of edges in $\alpha$ (including pairs of non-consecutive edges), by
+labeling them 0. Then we have the following observation, when we restrict $\alpha$ to $\text{Ring}(I_{\text{in}}, I_{\text{out}})$
+and set $\beta$ to be the reference path of this ring.
+
+**Observation 7.2.** Let $\alpha$ be a (not self-crossing) walk in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ with reference path $\eta$. Then $|\overline{\text{WindNum}}(\alpha, \eta)| = |\sum_{(e,e')\in E(\alpha)\times E(\alpha)} \text{label}_{\eta}^{\alpha}(e, e')|.$
+
+## 7.2 Rerouting in a Ring
+
+We now address the question of rerouting a solution to reduce its winding number with respect to the backbone Steiner tree. As a solution is linkage in the graph $G$ (i.e. a collection of vertex disjoint paths), we first show how to reroute linkages within a ring. In the later subsections, we will apply this to reroute a solution in the entire plane graph. We remark that from now onwards, our results are stated and proved only for linkages (rather than weak linkages). Further, define a *linkage of order k in a Ring($I_{in}, I_{out}$)* as a collection of *k* vertex-disjoint paths in $G$ such that each of these paths belongs to *Ring($I_{in}, I_{out}$)* and its endpoints belong to $V(I_{in}) \cup V(I_{out})$. As before, we simply use the term 'linkage' when the ring is clear from context. We will use the following proposition proved by Cygan et al. [14] using earlier results of Ding et al. [17]. Its statement has been rephrased to be compatible with our notation.
+
+**Proposition 7.2 (Lemma 4.8 in [14]).** Let $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ be a rooted ring in $H$ and let $\mathcal{P}$ and $\mathcal{Q}$ be two traversing linkages of the same order in this ring. Then, there exists a traversing linkage $\mathcal{P}'$ in this ring that is aligned with $\mathcal{P}$ and such that $|\overline{\text{WindNum}}(\mathcal{P}') - \overline{\text{WindNum}}(\mathcal{Q})| \le 6$.
+
+The formulation of [14] concerns directed paths in directed graphs and assumes a fixed pat-
+tern of in/out orientations of paths that is shared by the linkages $\mathcal{P}, \mathcal{Q}$ and $\mathcal{P}'$. The undirected
+case (as expressed above) can be reduced to the directed one by replacing every undirected edge
+in the graph by two oppositely-oriented arcs with same endpoints, and asking for any orienta-
+tion pattern (say, all paths should go from $I_{\text{in}}$ to $I_{\text{out}}$). Moreover, the setting itself is somewhat
+more general, where rings and reference paths are defined by curves and (general) homotopy.
+
+**Rings with Concentric Cycles.** Let $C = (C_1, C_2, \dots, C_p)$ concentric sequence of cycles in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ (then, $C_i$ is in the strict interior of $C_{i+1}$ for $i \in \{1, 2, \dots, p-1\}$). If $I_{\text{in}}$ is in the strict interior of $C_1$ and $C_p$ is in the strict interior of $I_{\text{out}}$, then we say that $C$ is *encircling*. An encircling concentric sequence $C$ in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ is *tight* if every $C \in C$ is a cycle in $G$, and there exists a path $\eta$ in $H$ traversing $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ such that the set of internal vertices of $\eta$ contain exactly $|C|$ vertices of $V(G)$, one on each each cycle in $C$. Let us fix one such encircling tight sequence in the ring $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ along with the path $\eta$ witnessing the tightness. Then, we set the path $\eta$ as the reference path of the ring. Here, we assume w.l.o.g. that $\eta$ contains only the 0-th copy of each of the edges comprising it. Any paths or linkages that we subsequently consider will not use the 0-th copy of any edge, and hence their winding numbers (with respect to $\eta$) will be well-defined. This is because that they arise from $G$, and when we consider them in $H$, we choose a 'non-0-th' copy out of the $4n+1$ copies of any (required) edge.
+
+A linkage $\mathcal{P}$ in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ is minimal with respect to $\mathcal{C}$ if among the linkages aligned with $\mathcal{P}$, it minimizes the total number of edges traversed that do not lie on the cycles of $\mathcal{C}$. The following proposition is essentially Lemma 3.7 of [14].
+
+**Proposition 7.3.** Let $G$ be a plane graph, and with radial completion $H$. Let $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ be a rooted ring in $H$. Suppose $|I_{\text{in}}|, |I_{\text{out}}| \le l$, for some integer $l$. Further, let $C = (C_1, \dots, C_p)$ be an encircling tight concentric sequence of cycles in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$. Finally, let $\mathcal{P}$ be a linkage in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ that is minimal with respect to $C$. Then, every inner visitor of $\mathcal{P}$ intersects less than $10l$ of the first cycles in the sequence $(C_1, \dots, C_p)$, while every outer visitor of $\mathcal{P}$ intersects less than $10l$ of the last cycles in this sequence.

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+A proof of this proposition can be obtained by first ordering the collection of inner and outer visitors by their 'distance' from the inner and outer interfaces, respectively, and the 'containment' relation between the cycles formed by them with the interfaces. This gives a partial order on the set of inner visitors and the set of outer visitors. Then if the proposition does not hold for $\mathcal{P}$, then the above ordering and containment relation can be used to reroute these paths along a suitable cycle. This will contradict the minimality of $\mathcal{P}$, since the rerouted linkage is aligned with it but uses strictly fewer edges outside of $\mathcal{C}$. The main result of this section can be now formulated as follows. (Its formulation and proof idea are based on Lemma 8.31 and Theorem 6.45 of [14].)
+
+**Lemma 7.1.** Let $G$ be a plane graph with radial completion $H$. Let $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ be a ring in $H$. Suppose that $|I_{\text{in}}|, |I_{\text{out}}| \le l$ for some integer $l$, and that in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ there is an encircling tight concentric sequence of cycles $\mathcal{C}$ of size larger than $40l$. Let $\eta$ be a traversing path in the ring witnessing the tightness of $\mathcal{C}$, and fix $\eta$ as the reference path. Finally, let $\mathcal{P} = \mathcal{P}_{\text{traverse}} \uplus \mathcal{P}_{\text{visitor}}$ be a linkage in $G$, where $\mathcal{P}_{\text{traverse}}$ is a traversing linkage comprising the paths of $\mathcal{P}$ traversing $\text{Ring}(I_{\text{in}}, I_{\text{out}})$, while $\mathcal{P}_{\text{visitor}} = \mathcal{P} \setminus \mathcal{P}_{\text{traverse}}$ consists of the paths whose both endpoints lie in either $V(I_{\text{in}})$ or $V(I_{\text{out}})$. Further, suppose that $\mathcal{P}$ is minimal with respect to $\mathcal{C}$. Then, for every traversing linkage $\mathcal{Q}$ in $G$ that is minimal with respect to $\mathcal{C}$ such that every path in $\mathcal{Q}$ is disjoint from $\eta$ and $|\mathcal{Q}| \ge |\mathcal{P}_{\text{traverse}}|$, there is a traversing linkage $\mathcal{P}'_{\text{traverse}}$ in $G$ such that
+
+(a) $\mathcal{P}'_{\text{traverse}}$ is aligned with $\mathcal{P}_{\text{traverse}}$,
+
+(b) the paths of $\mathcal{P}'_{\text{traverse}}$ are disjoint from the paths of $\mathcal{P}_{\text{visitor}}$, and
+
+(c) $|\overline{\text{WindNum}}(\mathcal{P}'_{\text{traverse}}) - \overline{\text{WindNum}}(\mathcal{Q})| \le 60l + 6.$
+
+*Proof.* Let $\mathcal{C} = (C_1, \dots, C_p)$, where $p > 40l$. Recall that $\mathcal{C}$ is a collection of cycles in $G$, and the path $\eta$ that witnesses the tightness of $\mathcal{C}$ contains $|\mathcal{C}|$ vertices of $V(G)$, one on each cycle of $\mathcal{C}$. Let $v_i$ denote the vertex where $\eta$ intersects the cycle $C_i \in \mathcal{C}$ for all $i \in \{1, 2, \dots, p\}$. Since $\mathcal{P}$ is minimal with respect to $\mathcal{C}$, Proposition 7.3 implies that the paths in $\mathcal{P}_{\text{visitor}}$ do not intersect any of the cycles $C_{10l}, C_{10l+1}, \dots, C_{p-10l+1}$ (note that since $p > 40l$, this sequence of cycles is non-empty). Call a vertex $x \in V(\text{Ring}(I_{\text{in}}, I_{\text{out}}))$ in the ring *non-separated* if there exists a path from $x$ to $C_{10l}$ whose set of internal vertices is disjoint from $\bigcup_{P \in \mathcal{P}_{\text{visitor}}} V(P)$. Otherwise, we say that the vertex $x$ is *separated*. Observe that every path in $\mathcal{P}_{\text{traverse}}$ is disjoint from the paths in $\mathcal{P}_{\text{visitor}}$ and intersects $C_{10l}$, hence all vertices on the paths of $\mathcal{P}_{\text{traverse}}$ are non-separated. Let $X$ denote the set of all non-separated vertices in the ring, and consider the graph $H[X]$. Observe that $H[X]$ is an induced subgraph of $\text{Ring}(I_{\text{in}}, I_{\text{out}})$, since it is obtained from $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ by deleting the separated vertices. Further, observe that $H[X]$ is a ring of $H$. Indeed, the inner interface of $H[X]$ is the cycle $\hat{I}_{\text{in}}$ obtained as follows: let $\mathcal{P}_{\text{in}}$ be the set of inner visitors in $\mathcal{P}$; then, $\hat{I}_{\text{in}}$ is the outer face of the plane graph $H[V(I_{\text{in}}) \cup \bigcup_{P \in \mathcal{P}_{\text{in}}} V(P)]$. It is easy to verify that all vertices on $\hat{I}_{\text{in}}$ are non-separated, and any vertex of $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ that lies in the strict interior of this cycle is separated. We then symmetrically obtain the outer interface $\hat{I}_{\text{out}}$ of $H[X]$ from the set $\mathcal{P}_{\text{out}}$ of outer visitors of $\mathcal{P}$. Then, $H[X] = \text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$. Here, $\hat{I}_{\text{in}}$ is composed alternately of subpaths of $I_{\text{in}}$ and inner visitors from $\mathcal{P}_{\text{visitor}}$, and symmetrically for $\hat{I}_{\text{out}}$.
+
+Note that the paths of $\mathcal{P}_{\text{traverse}}$ are completely contained in $\text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$ and they all traverse this ring. Thus, $\mathcal{P}_{\text{traverse}}$ can be regarded also as a traversing linkage in $\text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$. While $\mathcal{P}_{\text{traverse}}$ may have a different winding number in $\text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$ than in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$, the difference is “small” as we show below. (Note that the two winding numbers in the following claim are computed in two different rings.)
+
+**Claim 7.1.** Let $P$ be a path in $G$ that is disjoint from all paths in $\mathcal{P}_{\text{visitor}}$, such that $P$ belongs to $\text{Ring}(I_{\text{in}}, I_{\text{out}})$ and traverses it. Then, $P$ also belongs to $\text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$, and $|\overline{\text{WindNum}}(P, \eta)| \le 20l$. ¹³
+
+¹³Here $\hat{\eta}$ is the reference path of $\text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$, which is a subpath of $\eta$ in this ring.

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+Figure 18: Illustration of Claim 7.2.
+
+*Proof.* Since $P$ traverses $\text{Ring}(I_{\text{in}}, I_{\text{out}})$, it must intersect the cycle $C_{10\ell}$. Therefore, as $P$ is disjoint from $\mathcal{P}_{\text{visitor}}$, all vertices in $V(P)$ are non-separated. Hence, $P$ is present in $\text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$. Next, observe that there are at most $20\ell$ vertices of $G$ that are visited by $\eta$ but not visited by $\hat{\eta}$; indeed, these are vertices in the intersection of $\eta$ with $I_{\text{in}}$, $I_{\text{out}}$ and the first and last $10\ell - 1$ cycles of $\mathcal{C}$. It follows that any path in $G$ has at most $20\ell$ more crossings with $\eta$ than with $\hat{\eta}$. Since each such crossing contributes +1 or -1 to the winding number of $P$ with respect to $\eta$, the winding numbers of $P$ with respect to $\eta$ and $\hat{\eta}$ differ by at most $20\ell$. $\diamond$
+
+We now turn our attention to the linkage $Q$. In essence, our goal is show that every path in $Q$ can be “trimmed” to a path traversing $\text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$ such that their winding numbers are not significantly different. First, however, we prove that the paths in $Q$ cannot “oscillate” too much in $\text{Ring}(I_{\text{in}}, I_{\text{out}})$, based on the supposition that $Q$ is minimal with respect to $\mathcal{C}$.
+
+**Claim 7.2.** Let $Q \in \mathcal{Q}$, and let $u \in V(Q)$ such that it also lies on an inner visitor from $\mathcal{P}_{\text{visitor}}$. Then, the prefix of $Q$ between its endpoint on $I_{\text{in}}$ and $u$ does not intersect the cycle $C_{20\ell}$.
+
+*Proof.* Suppose, for the sake of contradiction, that the considered prefix contains some vertex $v$ that lies on $C_{20\ell}$. Since $u$ lies on an inner visitor $P \in \mathcal{P}_{\text{visitor}}$ and Proposition 7.3 states that an inner visitor cannot intersect $C_{10\ell}$, we infer that on the infix of $Q$ between $v$ and $u$ there exists a vertex that lies on the intersection of $Q$ and $C_{10\ell}$. Let $a$ be the first such vertex. Similarly, on the prefix of $Q$ from its endpoint on $I_{\text{in}}$ to $v$ there exists a vertex that lies on the intersection of $Q$ and $C_{10\ell}$. Let $b$ be the last such vertex. Then the whole infix of $Q$ between $a$ and $b$ does not intersect $C_{10\ell}$ internally (see Fig. 18), and hence, apart from endpoints, completely lies in the exterior of $C_{10\ell}$. Call this infix $Q^*$.
+
+Now consider $\text{Ring}(C_{10\ell}, I_{\text{out}})$, the ring induced by $I_{\text{out}}$ and $C_{10\ell}$. Moreover, consider the graph $G'$ obtained from $G$ by removing all vertices that are not in $\text{Ring}(C_{10\ell}, I_{\text{out}})$ and edges that are not in the strict interior of $\text{Ring}(C_{10\ell}, I_{\text{out}})$; in particular, the edges of $C_{10\ell}$ are removed, but the vertices are not. Note that $G'$ is a subgraph of $\text{Ring}(C_{10\ell}, I_{\text{out}})$. Finally, let $C' = C \setminus \{C_1, C_2, \dots, C_{10\ell}\}$; then, $C'$ is an encircling tight sequence of concentric cycles in $\text{Ring}(C_{10\ell}, I_{\text{out}})$.
+
+Let $Q'$ be the linkage in $G$ obtained by restricting paths of $Q$ to $G'$. Here, a path in $Q$ may break into several paths in $Q'$ (that are its maximal subpaths contained in $G'$). Since $Q$ is minimal with respect to $\mathcal{C}$, it follows that $Q'$ is minimal with respect to $\mathcal{C}'$. Now, observe that $Q^*$ belongs to $Q'$, hence it is an inner visitor of $\text{Ring}(C_{10\ell}, I_{\text{out}})$. However, $Q^*$ intersects the first $10\ell+1$ concentric cycles $C_{10\ell+1}, \dots, C_{20\ell}$ in the family $\mathcal{C}'$, which contradicts Proposition 7.3. $\diamond$
+
+Clearly, an analogous claim holds for outer visitors and the cycle $C_{p-20\ell+1}$. We now proceed to our main claim about the restriction of $Q$ to $\text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$.
+
+**Claim 7.3.** For every path $Q \in \mathcal{Q}$, there exists a subpath $\tilde{Q}$ of $Q$ that traverses $\text{Ring}(\hat{I}_{\text{in}}, \hat{I}_{\text{out}})$ and such that $\left|\text{WindNum}(\tilde{Q}, \hat{\eta}) - \text{WindNum}(Q, \eta)\right| \le 40\ell$.