On Realizing Reconfiguration Graphs of Cliques

TL;DR

This paper investigates the conditions under which certain reconfiguration graphs of cliques, specifically Token Sliding and Token Jumping graphs, are isomorphic to given target graphs, providing exact classifications for various graph families.

Contribution

It characterizes the feasibility sets of clique reconfiguration graphs for several well-known graph classes, offering complete classifications for Johnson graphs.

Findings

Determined the exact feasibility sets for complete graphs, paths, cycles, and other families.

Provided complete classifications for all Johnson graphs.

Analyzed the structure of reconfiguration graphs for various graph classes.

Abstract

For a graph $H$ and an integer $k\ge 1$, the \emph{Token Sliding reconfiguration graph} $\mathsf{TS}_k(H)$ and the \emph{Token Jumping reconfiguration graph} $\mathsf{TJ}_k(H)$ have as vertices the $k$-cliques of $H$, with two vertices adjacent when one clique is obtained from the other by replacing one vertex with an adjacent non-member, and respectively by an arbitrary non-member. For a target graph $G$, we study the feasibility sets $\mathcal{K}^{\mathsf{TS}}(G)$ and $\mathcal{K}^{\mathsf{TJ}}(G)$, consisting of all integers $k$ for which $G$ is isomorphic to $\mathsf{TS}_k(H)$ and $\mathsf{TJ}_k(H)$, respectively, for some graph $H$. We determine the exact feasibility sets for complete graphs, paths, cycles, complete bipartite graphs, book graphs, friendship graphs, and their complements, and give complete classifications for all Johnson graphs.