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.

翻译：对于图 $H$ 和整数 $k\ge 1$，**令牌滑动重构图** $\mathsf{TS}_k(H)$ 和**令牌跳跃重构图** $\mathsf{TJ}_k(H)$ 以 $H$ 的 $k$-团作为顶点，当两个顶点对应的团可以通过将一个顶点替换为相邻的非成员（对于 $\mathsf{TS}_k(H)$）或任意非成员（对于 $\mathsf{TJ}_k(H)$）而相互转换时，这两个顶点相邻。对于目标图 $G$，我们研究可行性集合 $\mathcal{K}^{\mathsf{TS}}(G)$ 和 $\mathcal{K}^{\mathsf{TJ}}(G)$，它们分别包含所有满足 $G$ 同构于某个图 $H$ 的 $\mathsf{TS}_k(H)$ 或 $\mathsf{TJ}_k(H)$ 的整数 $k$。我们确定了完全图、路径、圈、完全二部图、书图、友谊图及其补图的精确可行性集合，并给出了所有约翰逊图的完全分类。