A Note On Acyclic Token Sliding Reconfiguration Graphs of Independent Sets

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 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 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 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 𝖳𝖲_k(G)-graphs.