than $t$ where $M$ is the union of all connected components of $H' \setminus (Z \cup Z_1)$ having at least one neighbor in $Z$ and at least one neighbor in $Z_1$. However, $H[V(M) \cap V(H)]$ is a subgraph of $H$, which means that the treewidth of $H$ is also larger than $t$. By Proposition 3.2, this implies that the treewidth of $G$ is larger than $2^{ck}$. This contradicts the supposition that $(G, S, T, g, k)$ is good. From this, we conclude that $\lvert \mathrm{Sep}{R^2}(P, u) \rvert \le \alpha{\mathrm{sep}}(k)$. $\square$
Recall that $\mathbf{Sep}{R^2}(P, u)$ is computable in time $\mathcal{O}(n|\mathbf{Sep}{R^2}(P, u)|)$. Thus, by Lemma 6.4, we obtain the observation below. We remark that the reason we had to argue that the separator is small is not due to this observation, but because the size bound will be crucial in later sections.
Observation 6.5. Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour, $P$ be a long maximal degree-2 path of $R^2$, and $u$ be an endpoint of $P$. Then, $\mathbf{Sep}_{R^2}(P, u)$ can be computed in time $2^{\mathcal{O}(k)n}$.
Moreover, we have the following immediate consequence of Proposition 3.1.
Observation 6.6. Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour, $P$ be a long maximal degree-2 path of $R^2$, and $u$ be an endpoint of $P$. Then, $H[\mathbf{Sep}_{R^2}(P, u)]$ is a cycle.
6.4 Step IV: Internal Modification of Long Paths
In this step, we replace the “middle” of each long maximal degree-2 path $P = \text{path}{R^2}(u, v)$ of $R^2$ by a different path $P^*$. This “middle” is defined by the two separators obtained in the previous step. Let us informally explain the reason behind this modification. In Section 7 we will show that, if the given instance $(G, S, T, g, k)$ admits a solution (which is a collection of disjoint paths connecting $S$ and $T$), then it also admits a “nice” solution that “spirals” only a few times around parts of the constructed Steiner tree. This requirement is crucial, since it is only such solutions $\mathcal{P}$ that are discretely homotopic to weak linkages $\mathcal{W}$ in $H$ aligned with $\mathcal{P}$ that use at most $2^{\mathcal{O}(k)}$ edges parallel to those in $R$, and none of the edges not parallel to those in $R$. To ensure the existence of nice solutions, we show how an arbitrary solution can be rerouted to avoid too many spirals. This rerouting requires a collection of vertex-disjoint paths between $\mathbf{Sep}{R^2}(P, u)$ and $\mathbf{Sep}_{R^2}(P, v)$ which itself does not spiral around the Steiner tree. The replacement of $P$ by $P^*$ in the Steiner tree, described below, will ensure this property.
To describe this modification, we first need to assert the statement in the following simple lemma, which partitions every long maximal degree-2 path $P$ of $R^2$ into three parts (see Fig. 14).
Lemma 6.5. Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree with no detour, and $P$ be a long maximal degree-2 path of $R^2$ with endpoints $u$ and $v$. Then, there exist vertices $u' = u'P \in \mathbf{Sep}{R^2}(P, u) \cap V(P)$ and $v' = v'P \in \mathbf{Sep}{R^2}(P, v) \cap V(P)$ such that:
The subpath $P_{u,u'}$ of $P$ with endpoints $u$ and $u'$ has no internal vertex from $\mathbf{Sep}{R^2}(P, u) \cup \mathbf{Sep}{R^2}(P, v)$, and $\alpha_{\text{pat}}(k)/2 \le |V(P_{u,u'})| \le \alpha_{\text{pat}}(k)$. Additionally, the subpath $P_{v,v'}$ of $P$ with endpoints $v$ and $v'$ has no internal vertex from $\mathbf{Sep}{R^2}(P, u) \cup \mathbf{Sep}{R^2}(P, v)$, and $\alpha_{\text{pat}}(k)/2 \le |V(P_{v,v'})| \le \alpha_{\text{pat}}(k)$.
Let $P_{u',v'}$ be the subpath of $P$ with endpoints $u'$ and $v'$. Then, $P = P_{u,u'} - P_{u',v'} - P_{v',v}$.
Proof. We first prove that there exists a vertex $u' \in \text{Sep}{R^2}(P, u) \cap V(P)$ such that the subpath $P{u,u'}$ of $P$ between $u$ and $u'$ has no internal vertex from $\text{Sep}{R^2}(P, u) \cup \text{Sep}{R^2}(P, v)$. To this end, let $P' = P'u$, and let $\tilde{P}$ denote the subpath of $P$ that consists of the $\alpha{\text{pat}}(k)+1$ vertices of $P$ that are closest to $u$. Let $A = A_{R^2,P,u}$ and $B = B_{R^2,P,u}$. Recall that $\text{Sep}{R^2}(P, u) \subseteq V(H) \setminus (A \cup B)$ separates $A$ and $B$ in $H$. Since $\tilde{P}$ is a path with the endpoint $u$ in $A$ and the other endpoint in $B$, it follows that $\text{Sep}{R^2}(P, u) \cap V(\tilde{P}) \neq \emptyset$. Accordingly, let $u'$ denote the vertex of $P'$ closest