On Reconfiguration Graphs of Independent Sets under Token Sliding

TL;DR

This paper investigates the structure and properties of reconfiguration graphs of independent sets under token sliding, focusing on their classification within various graph classes and properties, and provides decomposition results.

Contribution

It characterizes when reconfiguration graphs belong to specific graph classes and properties, and introduces a decomposition method for these graphs.

Findings

Reconfiguration graphs can be classified within various graph classes.

Certain properties of the original graph are preserved in the reconfiguration graph.

A decomposition approach for reconfiguration graphs is established.

Abstract

An independent set of a graph $G$ is a vertex subset $I$ such that there is no edge joining any two vertices in $I$. Imagine that a token is placed on each vertex of an independent set of $G$. The $\mathsf{TS}$- ($\mathsf{TS}_k$-) reconfiguration graph of $G$ takes all non-empty independent sets (of size $k$) as its nodes, where $k$ is some given positive integer. Two nodes are adjacent if one can be obtained from the other by sliding a token on some vertex to one of its unoccupied neighbors. This paper focuses on the structure and realizability of these reconfiguration graphs. More precisely, we study two main questions for a given graph $G$: (1) Whether the $\mathsf{TS}_k$-reconfiguration graph of $G$ belongs to some graph class $\mathcal{G}$ (including complete graphs, paths, cycles, complete bipartite graphs, connected split graphs, maximal outerplanar graphs, and complete graphs minus one edge) and (2) If $G$ satisfies some property $\mathcal{P}$ (including $s$-partitedness, planarity, Eulerianity, girth, and the clique's size), whether the corresponding $\mathsf{TS}$- ($\mathsf{TS}_k$-) reconfiguration graph of $G$ also satisfies $\mathcal{P}$, and vice versa. Additionally, we give a decomposition result for splitting a $\mathsf{TS}_k$-reconfiguration graph into smaller pieces.