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

翻译：图$G$的独立集是指顶点子集$I$，使得$I$中任意两个顶点之间均无边相连。假设在$G$的独立集每个顶点上放置一个令牌。$G$的$\mathsf{TS}$-（$\mathsf{TS}_k$-）重构图以所有非空独立集（大小为$k$的独立集）为节点，其中$k$为给定正整数。若两个节点可通过将一个令牌从其所在顶点滑动至某个未被占据的邻接顶点而相互转换，则称它们相邻。本文聚焦于这些重构图的结构与可实现性。具体而言，我们针对给定图$G$研究两个主要问题：（1）$G$的$\mathsf{TS}_k$-重构图是否属于某图类$\mathcal{G}$（包括完全图、路径、圈、完全二部图、连通分裂图、极大外平面图及删去一条边的完全图）；（2）若$G$满足某性质$\mathcal{P}$（包括$s$-分图性、平面性、欧拉性、围长及团的大小），则$G$对应的$\mathsf{TS}$-（$\mathsf{TS}_k$-）重构图是否也满足$\mathcal{P}$，反之亦然。此外，我们给出一个将$\mathsf{TS}_k$-重构图分解为更小片段的分解结果。