On Realizing Reconfiguration Graphs of Cliques
This work addresses the realizability problem for reconfiguration graphs of cliques, providing foundational classification results for specific graph classes.
The paper characterizes which graphs can be realized as token sliding or token jumping reconfiguration graphs of cliques, determining exact feasibility sets for several graph families including complete graphs, paths, cycles, and Johnson graphs.
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.