A Note On Acyclic Token Sliding Reconfiguration Graphs of Independent Sets

TL;DR

This work advances the study of token sliding reconfiguration graphs TS_k(G) by analyzing acyclicity when G is a tree or forest and by characterizing which trees/forests arise as TS_k-graphs. It provides forbidden-subgraph criteria for TS_k(G) to be a forest for k ∈ {2,3}, and introduces H-join and H-decomposition as a constructive tool to combine TS_k-graphs while preserving acyclicity. The authors prove that every k-ary tree is realizable as TS_{k+1}(G) for some G and show that any tree embeds into a TS_2-forest, illustrating broad realizability; they also investigate the D_{r,n,s} family to illuminate when these trees yield TS_k-graphs. These results establish a framework for building and decomposing TS_k-graphs and suggest conjectures for higher k, with potential implications for understanding the structure of reconfiguration spaces in combinatorial problems.

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.

Figures (12)

• Figure 1: A graph $G$ with $\mathsf{TS}_2(G) = D_{1,3,2}$. Each node $ab$ represents a size-$2$ stable set of $G$.
• Figure 2: A list $\mathcal{G}$ of $n$-vertex graphs $G$ ($4 \leq n \leq 7$) excluding $\overline{C_n}$ ($n \geq 5$) such that if $\mathsf{TS}_2(G^\prime)$ has no cycle then $G^\prime$ does not contain any member $G$ of $\mathcal{G}$ as an induced subgraph.
• Figure 3: Illustration for Proposition \ref{['keq2']}: Some trees $T^\prime$ containing $N[b]$ whose $\mathsf{TS}_2$-graphs have a cycle. Here $r$ is the number of children of $b$. Copies of $2K_2$ and $D_{2,2,2}$ are marked by red color.
• Figure 4: Illustration for Proposition \ref{['keq3']}: Some trees $T^\prime$ containing $N[b]$ whose $\mathsf{TS}_3$-graphs have a cycle. Here $r$ and $t$ are respectively the number of children of $b$ and its child $e$. Copies of $2K_2 + K_1$ and $D_{2,4,2}$ are marked by red color.
• Figure 5: The graphs $G_1$, $G_2$, $H(G_1, G_2)$, and their corresponding $\mathsf{TS}_2$-graphs. Here $\mathsf{TS}_2(H(G_1, G_2)) = \mathsf{TS}_2(G_1) \cup \mathsf{TS}_2(G_2)$.

Theorems & Definitions (47)

• Lemma 1
• proof
• Proposition 2
• proof
• Corollary 3
• proof
• Corollary 4
• proof
• Proposition 5
• proof