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,