For an integer $k\ge 0$ and a graph $G$, the \emph{token-sliding reconfiguration graph $\mathsf{TS}_k(G)$} has the independent $k$-sets of $G$ as vertices. Two vertices are adjacent if one token can slide along an edge of $G$ and the resulting $k$-set is still independent. We study the following realizability problem: for fixed $k\ge 2$, which graphs are isomorphic to $\mathsf{TS}_k(G)$ for some graph $G$? This inverse viewpoint asks which abstract state spaces can occur exactly under a local token rule. We give positive realizability results for the complement targets $\overline{K_n}$, $\overline{K_{m,n}}$, and $\overline{K_n-e}$, and we determine sharp cutoffs for complements of paths and cycles. We also prove a product formula for token-sliding graphs of disjoint unions and apply it to Cartesian products of complete graphs, paths, and cycles. For every grid $Γ_{m,n}=P_m\square P_n$ with $2\le m\le n$, we realize $Γ_{m,n}$ at token value $m+n-2$ and at every token value $k\ge 4$. At small token values, we prove that $C_4\square C_n$ is not a $\mathsf{TS}_2$-graph for $n\ge 4$, classify ladders $Γ_{2,n}$, and settle the first non-ladder grid: for $k\ge 2$, $Γ_{3,3}$ is realizable if and only if $k\ge 4$.

翻译：对整数$k\ge 0$和图$G$，定义\emph{Token-滑动重构图$\mathsf{TS}_k(G)$}：其顶点为$G$的独立$k$-集，两顶点相邻当且仅当某个Token可沿$G$的一条边滑动且所得$k$-集仍为独立集。我们研究如下可实现性问题：对固定$k\ge 2$，哪些图同构于某图$G$的$\mathsf{TS}_k(G)$？这种逆视角探究在局部Token规则下何种抽象状态空间能够精确出现。我们给出了补图目标$\overline{K_n}$、$\overline{K_{m,n}}$和$\overline{K_n-e}$的正向可实现性结果，并确定了路径与环的补图存在尖锐阈值。此外，我们证明了不相交并图的Token-滑动图的乘积公式，并将其应用于完全图、路径与环的笛卡尔积。对每个满足$2\le m\le n$的网格$Γ_{m,n}=P_m\square P_n$，我们在Token值$m+n-2$及所有$k\ge 4$时实现了$Γ_{m,n}$。在较小Token值下，我们证明$C_4\square C_n$（$n\ge 4$）不是$\mathsf{TS}_2$-图，分类了阶梯图$Γ_{2,n}$，并解决了首个非阶梯网格问题：对$k\ge 2$，$Γ_{3,3}$可实现的充要条件是$k\ge 4$。