Figure 13: Face operations.
any walk in W. Then, the face move operation replaces P in W by the unique subpath of C between the endpoints of P that is edge-disjoint from P.
Face Pull. Applicable to (W, f) if C is a subwalk Q of W. Then, the face pull operation replaces Q in W by a single occurrence of the first vertex in Q.
Face Push. Applicable to (W, f) if (i) no edge in E(C) belongs to any walk in W, and (ii) there exist two consecutive edges e, e' in W with common vertex v ∈ V(C) (where W visits e first, and v is visited between e and e') and an order (clockwise or counter-clockwise) to enumerate the edges incident to v starting at e such that the two edges of E(C) incident to v are enumerate between e and e', and for any pair of consecutive edges of W' for all W' ∈ W incident to v, it does not hold that one is enumerated between e and the two edges of E(C) while the other is enumerated between e' and the two edges of E(C). Let $\tilde{e}$ be the first among the two edges of E(C) that is enumerated. Then, the face push operation replaces the occurrence of v between e and e' in W by the traversal of C starting at $\tilde{e}$.
We verify that the application of a single operation results in a weak linkage.
Observation 5.2. Let $G$ be a triangulated plane graph with a weak linkage $W$, and a face $f$ that is not the outer face. Let $W \in W$ with a discrete homotopy operation applicable to $(W, f)$. Then, the result of the application is another weak linkage aligned to $W$.
Then, discrete homotopy is defined as follows.
Definition 5.4 (Discrete Homotopy). Let $G$ be a triangulated plane graph with weak linkages $W$ and $W'$. Then, $W$ is discretely homotopic to $W'$ if there exists a finite sequence of discrete homotopy operations such that when we start with $W$ and apply the operations in the sequence one after another, every operation is applicable, and the final result is $W'$.
We verify that discrete homotopy gives rise to an equivalence relation.
Lemma 5.1. Let $G$ be a triangulated plane graph with weak linkages $W, W'$ and $W''$. Then, (i) $W$ is discretely homotopic to itself, (ii) if $W$ is discretely homotopic to $W'$, then $W'$ is discretely homotopic to $W$, and (iii) if $W$ is discretely homotopic to $W'$ and $W'$ is discretely homotopic to $W'',$ then $W$ is discretely homotopic to $W''$.
Proof. Statement (i) is trivially true. The proof of statement (ii) is immediate from the observation that each discrete homotopy operation has a distinct inverse. Indeed, every face move operation is invertible by a face move operation (applied to the same walk and cycle). Additionally, every face pull operation is invertible by a face push operation (applied to the same walk and cycle), and vice versa. Hence, given the sequence of operations to transform $W$ to $W'$, say $\phi$, the sequence of operations to transform $W'$ to $W$ is obtained by first writing the