Figure 21: An illustration of Lemma 8.5
Definition 8.17 (Shrinking Cycle). Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $\mathcal{W}$ be a sensible well-behaved weak linkage, and $S \in \text{Seq}(\mathcal{W})$ with an endpoint $v \in V(R)$. Then, a cycle $C$ in $H$ is a shrinking cycle for $(S, v)$ if it has no edge of $R$ in its interior and it can be partitioned into three paths where the first has at least one edge and the last has at most one edge: (i) a subpath $P_1$ of $S$ with $v$ as an endpoint; (ii) a subpath $P_2$ from the other endpoint $u$ of $P_1$ to a vertex on $R$ that consists only of edges drawn in the strict interior of the projecting cycle of $S$ and no vertex of $S$ apart from $u$; (iii) a subpath $P_3$ that has $v$ as an endpoint and whose edge (if one exists) is either not parallel to any edge of $R$ or it is the $i$-th copy of an edge parallel to some edge of $R$ for some $i \in {-n+\ell-1, n-h+1}$, where $\ell$ and $h$ are as in Definition 8.16.
With respect to shrinking cycles, we prove two claims. First, we assert their existence.
Lemma 8.5. Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $\mathcal{W}$ be a sensible well-behaved weak linkage, and $S \in \text{Seq}(\mathcal{W})$ with an endpoint $v \in V(R)$. Then, there exists a shrinking cycle for $(S, v)$.
Proof. Let $e$ be an edge in $S$ incident to $v$ (if there are two such edges, when $S$ is a cycle, pick one of them arbitrarily), and denote the other endpoint of $e$ by $u$. Because $H$ is triangulated, $e$ belongs to the boundary $B$ of a face $f$ of $H$ in the interior of $C$ such that $B$ is a cycle that consists of only two or three edges. If $B$ does not contain any vertex of $V(R) \cup V(S)$ besides $u$ and $v$, then it is clearly a shrinking cycle (see Fig. 21). Thus, we now suppose that $B$ is a cycle on three edges whose third vertex, $w$, belongs to $V(R) \cup V(S)$. If $w \in V(S)$, then the cycle that consists of the subpath of $S$ from $v$ to $w$ and the edge in $E(B)$ between $v$ and $w$ is also clearly a shrinking cycle (see Fig. 21). Thus, we now also suppose that $w \in V(R)$.
We further distinguish between two cases. First, suppose that $w$ is not adjacent to $v$ on $R$. In this case, $B$ does not enclose any edge of $R$ as well as any edge parallel to an edge of $R$. Moreover, $B$ can be partitioned into $P_1, P_2$ and $P_3$ that are each a single edge, where $P_1$ consists of the edge in $B$ between $v$ and $u$, $P_2$ consists of the edge in $B$ between $u$ and $w$, and $P_3$ consists of the edge in $B$ between $w$ and $v$, thereby complying with Definition 8.17. Thus, $B$ is a shrinking cycle for $(S, v)$. Now, suppose that $w$ is adjacent to $v$ on $R$. Then, define $P_1, P_2$ and $P_3$ similarly to before except that to $P_3$, we do not take the edge of $B$ between $v$ and $w$ but its parallel $i$-th copy where $i \in {-n+\ell-1, n-h+1}$ such that $\ell$ and $h$ are as in Definition 8.16. The choice of whether $i = -n+\ell-1$ or $i = n-h+1$ is made so that the cycle $B'$ consisting of $P_1, P_2$ and $P_3$ does not enclose any edge of $R$. (Such a choice necessarily exists, see Fig. 21). □
Now, we prove that making a cycle move/pull operation using a shrinking cycle is valid and maintains some properties of weak linkages required for our analysis.
Lemma 8.6. Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $\mathcal{W}$ be a sensible, well-behaved, shallow and outer-terminal weak linkage, and $S \in \text{Seq}(\mathcal{W})$ be innermost with an endpoint $v \in V(R) \setminus {t^*}$. Let $C$ be a shrinking cycle for $(S, v)$ that encloses as many faces as possible. Then, the cycle move/pull operation is applicable to $(W, C)$ where $W \in \mathcal{W}$ is the walk having $S$ as a sequence. Furthermore, the resulting weak linkage