The isomorphism problem for generalized Baumslag–Solitar groups with one mobile edge

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. Every $GBS$ group is the fundamental group $\pi_1(\mathbb A)$ of some graph labeled $\mathbb A$. This paper deals with the isomorphism problem for $GBS$ groups, which is the problem of determining whether $\pi_1(\mathbb A)\cong\pi_1(\mathbb B)$ for two given graphs labeled $\mathbb A$ and $\mathbb B$. We describe an algorithm that decides this problem for the case where one of the labeled graphs has one mobile edge.