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 (arXiv:2203.16861). 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 on 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 $\mathsf{TS}_k(G)$-graphs.