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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