| Figure 19: Segments, segment groups and their labeling. | |
| Towards bounding the multiplicity of the pushed weak linkage we construct, and also an important ingredient on its own in the reconstruction in Section 9, we need to repeatedly eliminate U-turns in the weak linkage we deal with. Here, U-turns are defined as follows. | |
| **Definition 8.3 (U-Turn in Weak Linkage).** Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $\mathcal{W}$ be a weak linkage in $H$. Then, a U-turn in $\mathcal{W}$ is a pair of parallel edges $\{e, e'\}$ visited consecutively by some walk $W \in \mathcal{W}$ such that the strict interior of the cycle formed by $e$ and $e'$ does not contain the first or last edge of any walk in $\mathcal{W}$. We say that $\mathcal{W}$ is U-turn-free if it does not have any U-turn. | |
| Still, having a pushed weak linkage of low multiplicity and no U-turns does not suffice for faithful reconstruction due to ambiguity in which edge copies are being used. This ambiguity will be dealt with by the following definition. | |
| **Definition 8.4 (Canonical Weak Linkage).** Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $\mathcal{W}$ be a weak linkage in $H$ pushed onto $R$. Then, $\mathcal{W}$ is canonical if (i) for every edge $e_i \in E(\mathcal{W})$, $i \ge 1$ (ii) if $e_i \in E(\mathcal{W})$, then all the parallel edges $e_j$, for $1 \le j < i$, are also in $E(\mathcal{W})$. | |
| For brevity, we say that a weak linkage $\mathcal{W}$ in $H$ is simplified if it is sensible, pushed onto $R$, canonical, U-turn-free and has multiplicity upper bounded by $\alpha_{\text{mul}}(k) := 2\alpha_{\text{potential}}(k)$ where $\alpha_{\text{potential}}(k) = 2^{\mathcal{O}(k)}$ will be defined precisely in Lemma 8.2. For the process that simplifies a pushed weak linkage, we will maintain a property that requires multiplicity at most $2n$ as well as a relaxation of canonicity. This property is defined as follows. | |
| **Definition 8.5 (Extremal Weak Linkage).** Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $\mathcal{W}$ be a weak linkage in $H$ that is pushed onto $R$. Then, $\mathcal{W}$ is extremal if its multiplicity is at most $2n$ and for any two parallel edges $e_i, e_j \in E(\mathcal{W})$ where $i \ge 1$ and $j \le -1$, we have $(i-1) + |j+1| \ge 2n$. | |
| Additionally, we will maintain the following property. | |
| **Definition 8.6 (Outer-Terminal Weak Linkage).** Let $(G, S, T, g, k)$ be a nice instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $\mathcal{W}$ be a weak linkage in $H$. Then, $\mathcal{W}$ is outer-terminal if it uses exactly one edge incident to $t^*$. | |
| **Segments, Segment Groups and Potential.** To analyze the “complexity” of a weak linkage, we partition it into segments and segment groups, and then associate a potential function with it based on this partition. Intuitively, a segment of a walk is a maximal subwalk that does not cross $R$ (see Fig. 19). Formally, it is defined as follows. |