Admissible slides for generalized Baumslag–Solitar groups

A generalized Baumslag-Solitar group ($GBS$ group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. Any $GBS$ group is isomorphic to fundamental group $\pi_1(\mathbb{A})$ of some labeled graph $\mathbb{A}$. Slide is a transformation of labeled graphs. Slides play an important role in isomorphism problem for GBS groups. Given an edge $e$ with label $\lambda$ and $\alpha\in\mathbb{Q}$. In this paper we describe an algorithm that checks if there exists a cycle $p$ such that after slide $e$ over $p$ label $\lambda$ multiplies by $\alpha$ or not. If such cycle exists then the algorithm finds one of them.