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.