Figure 14: Separators and flows for long degree-2 paths.
Lemma 6.3. Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths with a Steiner tree $R'$. An exhaustive application of the operation undetouring $R'$ can be performed in time $O(k^2 \cdot n^2)$, and results in a Steiner tree that has no detour.
We denote the Steiner tree obtained at the end of Step II by $R^2$.
6.3 Step III: Small Separators for Long Paths
We now show that any two parts of $R^2$ that are “far” from each other can be separated by small separators in $H$. This is an important property used in the following sections to show the existence of a “nice” solution for the input instance. Specifically, we consider a “long” maximal degree-2 path in $R^2$ (which has no short detours in $H$), and show that there are two separators of small cardinality, each “close” to one end-point of the path. The main idea behind the proof of this result is that, if it were false, then the graph $H$ would have had large treewidth (see Proposition 6.2), which contradicts that $H$ has bounded treewidth (by Corollary 4.1). We first define the threshold that determines whether a path is long or short.
Definition 6.2 (Long Paths in Trees). Let $G$ be a graph with a subtree $T$. A subpath of $T$ is $k$-long if its length is at least $\alpha_{\text{long}}(k) := 10^4 \cdot 2^{ck}$, and $k$-short otherwise.
As $k$ will be clear from context, we simply use the terms long and short. Towards the computation of two separators for each long path, we also need to define which subsets of $V(R^2)$ we would like to separate.
Definition 6.3 ($P'u, P''u, A{R^2,P,u}$ and $B{R^2,P,u}$). 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. For any long maximal degree-2 path $P$ of $R^2$ and for each endpoint $u$ of $P$, define $P'u$, $P''u$ and $A{R^2,P,u}$, $B{R^2,P,u} \subseteq V(R^2)$ as follows.
$P'u$ (resp. $P''u$) is the subpath of $P$ consisting of the $\alpha{\text{pat}}(k) := 100 \cdot 2^{ck}$ (resp. $\alpha{\text{pat}}(k)/2 = 50 \cdot 2^{ck}$) vertices of $P$ closest to $u$.
$A_{R^2,P,u}$ is the union of $V(P''_u)$ and the vertex set of the connected component of $R^2$ - $(V(P'_u) \setminus {u})$ containing $u$.
$B_{R^2,P,u} = V(R^2) \setminus (A_{R^2,P,u} \cup V(P'_u))$.
For each long maximal degree-2 path $P$ of $R^2$ and for each endpoint $u$ of $P$, we compute a “small” separator $\text{Sep}{R^2}(P, u)$ as follows. Let $A = A{R^2,P,u}$ and $B = B_{R^2,P,u}$. Then, compute a subset of $V(H) \setminus (A \cup B)$ of minimum size that separates $A$ and $B$ in $H$, and denote it by $\text{Sep}_{R^2}(P, u)$ (see Fig. 14). Since $A \cap B = \emptyset$ and there is no edge between a vertex in $A$ and a vertex in $B$ (because $R^2$ has no detours), such a separator exists. Moreover, it can be computed