Figure 20: A Sequence, its projecting cycle and a shrinking cycle.
between $u$ and $v$ and the path $P$, and the other cycle $C_2$ consists of the second subpath of $C$ between $u$ and $v$ and the path $P$. Notice that $C_1$ encloses less faces than $C$, and so does $C_2$. At least one of these two cycles, say, $C_1$, contains at least one edge of $Q$. Then, the cycle move operation is applicable to $(W, C_1)$. Indeed, let $\hat{Q}$ be the subpath of $Q$ that is a subpath of $C$, and notice that $E(P) \subseteq E(C_1) \subseteq E(C) \cup E(P)$ and $E(P) \cap E(W) = \emptyset$ (because the cycle move/pull operation is applicable to $(W, C)$). Therefore, $\hat{Q}$ is a subpath of $W$, $1 \le |E(\hat{Q})| \le |E(C_1)| - 1$, and no edge in $E(C_1) \setminus E(\hat{Q})$ belongs to any walk in $W$. Moreover, because $C_1$ belongs to the interior (including the boundary) of $C$, no edge drawn in the strict interior of $C$ belongs to any walk in $W$. Now, notice that after the application of the cycle move operation for $(W, C_1)$, $C_2$ also has at least one edge used by the walk $W'$ into which $W$ was modified—in particular, $E(P) \subseteq E(W')$. Moreover, consider the subpath (or subwalk that is a cycle) $Q'$ of $W'$ that results from the replacement of $\hat{Q}$ in $Q$ by the subpath of $C_1$ between the endpoints of $Q'$ that does not belong to $W$. Then, $Q'$ traverses some subpath (possible empty) of $C_1$ or $C_2$, then traverses $P$, and next traverses some other subpath of $C_1$ or $C_2$. So, the restriction of $Q'$ to $C_2$ is a non-empty path or cycle $Q^*$ that is a subwalk of $W'$. Furthermore, because $C_2$ is drawn in the interior of $C$ and the cycle move/pull operation is applicable to $(W, C)$, we have that no edge of $E(C_2) \setminus E(Q^*)$ or the strict interior of $C_2$ belongs to $E(W)$. Thus, the cycle move/pull operation is applicable to $(W', C_2)$. Now, the result of the application of this operation is precisely the result of the application of the original cycle move or pull operation applicable to $(W, C)$. To see this, observe that the edges of $E(C) \setminus E(W)$ that occur in $C_1$ along with $E(P)$ have replaced the edges of $E(C) \cap E(W)$ that occur in $C_1$ in the first operation, and the edges of $E(C) \setminus E(W)$ that occur in $C_2$ have replaced the edges of $E(C) \cap E(W)$ that occur in $C_1$ along with $E(P)$ in the second operation. Thus, by the inductive hypothesis with respect to $(W, C_1)$ and $(W', C_2)$, and because discrete homotopy is transitive, the claim follows.
Thus, it remains to prove that C has a path P fully drawn in its interior whose endpoints are two (distinct) vertices u, v ∈ V(C), and whose internal vertices and all of its edges do not belong to C. In case C has a chord (that is, an edge in G between two vertices of C that does not belong to C), then the chord is such a path P. Therefore, we now suppose that this is not the case. Then, C does not contain in its interior an edge parallel to an edge of C. In turn, because G is triangulated), when we consider some face f in the interior of C that contains an edge e of C, this face must be a triangle. Moreover, the vertex of f that is not incident to e cannot belong to C, since otherwise we obtain a chord in C. Thus, the subpath (that consists of two edges) of f between the endpoints of e that does not contain e is a path P with the above mentioned properties. □
In the process of pushing a solution onto R, we push parts of the solution one-by-one. We refer to these parts as sequences, defined as follows (see Fig. 20).
Definition 8.12 (Sequence). Let (G, S, T, g, k) be an instance of Planar Disjoint Paths, and R be a Steiner tree. Let W be a walk. Then, a sequence of W is a maximal subwalk of W whose