On Reconfiguration Graph of Independent Sets under Token Sliding
share
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, and connected split graphs) 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.
READ FULL TEXT