Figure 1: Flow at a vertex and its reduction.
Figure 2: Two different ways of extracting a walk from a flow.
directed graphs. (In this context, undirected graphs are treated as directed graphs by replacing each edge by two parallel arcs of opposite directions.) Specifically, we denote an instance of Directed Planar Disjoint Paths as a tuple $(D, S, T, g, k)$ where $D$ is a directed plane graph, $S, T \subseteq V(D)$, $k = |S|$ and $g: S \to T$ is bijective. Then, a solution is a set $\mathcal{P}$ of pairwise vertex-disjoint directed paths in $D$ containing, for each vertex $s \in S$, a path directed from $s$ to $g(s)$.
In the language of flows, each arc of $D$ is assigned a word with letters in $T \cup T^{-1}$ (that is, we treat the set of vertices $T$ also as an alphabet), where $T^{-1} = {t^{-1} : t \in T}$. This collection of words is denoted by $(T \cup T^{-1})^*$ and let 1 denote the empty word. A word is reduced if, for all $t \in T$, the letters $t$ and $t^{-1}$ do not appear consecutively. Then, a flow is an assignment of reduced words to arcs that satisfies two constraints. First, when we concatenate the words assigned to the arcs incident to a vertex $v \notin S \cup T$ in clockwise order, where words assigned to ingoing arcs are reversed and their letters negated, the result (when reduced) is the empty word 1 (see Fig. 1). This is an algebraic interpretation of the standard flow-conservation constraint. Second, when we do the same operation with respect to a vertex $v \in S \cup T$, then when the vertex is in $S$, the result is $g(s)$ (rather than the empty word), and when it is in $T$, the result is $t$. There is a natural association of flows to solutions: for every $t \in T$, assign the letter $t$ to all arcs used by the path from $g^{-1}(t)$ to $t$.
Roughly speaking, Schrijver proved that if a flow $\phi$ is given along with the instance $(D, S, T, g, k)$, then in polynomial time we can either find a solution or determine that there is no solution “similar to $\phi$”. Specifically, two flows are homologous (which is the notion of similarity) if one can be obtained from the other by a set of “face operations” defined as follows.
Definition 2.1. Let $D$ be a directed plane graph with outer face $f$, and denote the set of faces of $D$ by $\mathcal{F}$. Two flows $\phi$ and $\psi$ are homologous if there exists a function $h : \mathcal{F} \to (T \cup T^{-1})^*$ such that (i) $h(f) = 1$, and (ii) for every arc $e \in A(D)$, $h(f_1)^{-1} \cdot \phi(e) \cdot h(f_2) = \psi(e)$ where $f_1$ and $f_2$ are the faces at the left-hand side and the right-hand side of $e$, respectively.
Then, a slight modification of Schrijver's theorem [42] readily gives the following corollary.
Corollary 2.1. There is a polynomial-time algorithm that, given an instance $(D, S, T, g, k)$ of Directed Planar Disjoint Paths, a flow $\phi$ and a subset $X \subseteq A(D)$, either finds a solution of $(D - X, S, T, g, k)$ or decides that there is no solution of it such that the “flow associated with it” and $\phi$ are homologous in $D$.
Discrete Homotopy and Our Objective. While the language of flows and homology can be used to phrase our arguments, it also makes them substantially longer and somewhat obscure because it brings rise to multiple technicalities. For example, different sets of non-crossing walks may correspond to the same flow (see Fig. 2). Instead, we define a notion of discrete homotopy, inspired by (standard) homotopy. Specifically, we deal only with collections of non-crossing and edge-disjoint walks, called weak linkages. Then, two weak linkages are discretely homotopic if