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.