A Note on Acyclic Token Sliding Reconfiguration Graphs of Independent Sets

Abstract

We continue the study of Token Sliding (reconfiguration) graphs of independent sets initiated by the authors in an earlier paper [Graphs Comb. 39.3, 59, 2023]. Two of the topics in that paper were to study which graphs \(G\) are Token Sliding graphs and which properties of a graph are inherited by a Token Sliding graph. In this paper, we continue this study specializing in the case of when \(G\) and/or its Token Sliding graph \(\mathsf{TS}_k(G)\) is a tree or forest, where \(k\) is the size of the independent sets considered. We consider two problems. The first is to find necessary and sufficient conditions on \(G\) for \(\mathsf{TS}_k(G)\) to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be a Token Sliding graph. For the first problem, we give a forbidden subgraph characterization for the cases of \(k=2,3\). For the second problem, we show that for every \(k\)-ary tree \(T\) there is a graph \(G\) for which \(\mathsf{TS}_{k+1}(G)\) is isomorphic to \(T\). A number of other results are given along with a join operation that aids in the construction of \(\mathsf{TS}_k\)-graphs.