A Note On Acyclic Token Sliding Reconfiguration Graphs of Independent Sets
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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.
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