A Note On Acyclic Token Sliding Reconfiguration Graphs of Independent Sets

TL;DR

This paper investigates the conditions under which token sliding reconfiguration graphs of independent sets are trees or forests, providing characterizations and constructions for such graphs.

Contribution

It offers necessary and sufficient conditions for token sliding graphs to be trees or forests, including forbidden subgraph characterizations and construction methods.

Findings

Forbidden subgraph characterizations for k=2,3

Every k-ary tree can be realized as a token sliding graph for some G

Introduces a join operation to construct token sliding graphs

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.