internal vertices (if any exists) do not belong to $R$ and which contains at least one edge that is not parallel to an edge of $R$. The set of sequences of $W$ is denoted by $\text{Seq}(W)$. For a weak linkage $W$, the set of sequences of $W$ is defined as $\text{Seq}(W) = \bigcup_{W' \in W} \text{Seq}(W')$.
Notice that the set of sequences of a walk does not necessarily form a partition of the walk because the walk can traverse edges parallel to the edges of $R$ and these edges do not belong to any sequence. Moreover, for sensible weak linkages, the endpoints of every sequence belong to $R$. To deal only with sequences that are paths, we need the following definition.
Definition 8.13 (Well-Behaved Weak Linkage). Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. A weak linkage $W$ is well-behaved if every sequence in $\text{Seq}(W)$ is a path or a cycle.
When we will push sequences one-by-one, we ensure that the current sequence to be pushed can be handled by the cycle move operation in Definition 8.11. To this end, we define the notion of an innermost sequence, based on another notion called a projecting cycle (see Fig. 20). We remark that this cycle will not necessarily be the one on which we apply a cycle move operation, since this cycle might contain in its interior edges of some walks of the weak linkage.
Definition 8.14 (Projecting Cycle). Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $W$ be a sensible well-behaved weak linkage, and $S \in \text{Seq}(W)$. The projecting cycle of $S$ is the cycle $C$ formed by $S$ and the subpath $P$ of $R$ between the endpoints of $S$. Additionally, $\text{Volume}(S)$ denotes the number of faces enclosed by the projecting cycle of $S$.
Now, we define the notion of an innermost sequence.
Definition 8.15 (Innermost Sequence). Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $W$ be a sensible well-behaved weak linkage, and $S \in \text{Seq}(W)$. Then, $S$ is innermost if every edge in $E(W)$ that is drawn in the interior of its projecting cycle is parallel to some edge of $R$.
We now argue that, unless the set of sequences is empty, there must exist an innermost one.
Lemma 8.4. Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $W$ be a sensible well-behaved weak linkage such that $\text{Seq}(W) \neq \emptyset$. Then, there exists an innermost sequence in $\text{Seq}(W)$.
Proof. Let $S \in \text{Seq}(W)$ be a sequence that minimizes $\text{Volume}(S)$. We claim that $S$ is innermost. Suppose, by way of contradiction, that this claim is false. Then, there exists an edge $e \in E(W)$ that is drawn in the interior of the projecting cycle of $S$ and is not parallel to any edge of $R$. Thus, $e$ belongs to some sequence $S' \in \text{Seq}(W)$. Because $W$ is well-behaved, $S$ and $S'$ are vertex disjoint. This implies that the projecting cycle of $S'$ is contained in the interior of the projecting cycle of $S$. Because $H$ is triangulated, this means that $\text{Volume}(S') < \text{Volume}(S)$, which is a contradiction to the choice of $S$. $\square$
When we push the sequence onto $R$, we need to ensure that we have enough copies of each edge of $R$ to do so. To this end, we need the following definition.
Definition 8.16 (Shallow Weak Linkage). Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $W$ be a sensible well-behaved weak linkage. Then, $W$ is shallow if for every edge $e_0 \in E(R)$, the following condition holds. Let $\ell$ (resp. $h$) be the number of sequences $S \in \text{Seq}(W)$ whose projecting cycle encloses $e_1$ (resp. $e_{-1}$). Then, $e_i$ is not used by $W$ for every $i \in {-n, -n+1, \dots, -n+\ell-1} \cup {0} \cup {n-h+1, n-h+2, \dots, n}$.
To ensure that we make only cycle moves/pulls as in Definition 8.11, we do not necessarily push the sequence at once, but gradually shrink the area enclosed by its projecting cycle.¹⁸
¹⁸Instead, we could have also always pushed a sequence at once by defining moves and pulls for closed walks, which we find somewhat more complicated to analyze formally.