On Reconfiguration Graphs of Independent Sets Under Token Sliding

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 TS - ( 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 TS k -reconfiguration graph of G belongs to some graph class 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 P (including s -partitedness, planarity, Eulerianity, girth, and the clique’s size), whether the corresponding TS - ( TS k -) reconfiguration graph of G also satisfies P , and vice versa. Additionally, we give a decomposition result for splitting a TS k -reconfiguration graph into smaller pieces.