On Reconfiguration Graphs of Independent Sets Under Token Sliding

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 $${\textsf{TS}}$$ - ( $${\textsf{TS}}_k$$ -) 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 $${\textsf{TS}}_k$$ -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 $${\textsf{TS}}$$ - ( $${\textsf{TS}}_k$$ -) reconfiguration graph of G also satisfies $${\mathcal {P}}$$ , and vice versa. Additionally, we give a decomposition result for splitting a $${\textsf{TS}}_k$$ -reconfiguration graph into smaller pieces.