Connected ($C_4$,Diamond)-free Graphs Are Uniquely Reconstructible from Their Token Graphs

A diamond is the graph that is obtained from removing an edge from the complete graph on $4$ vertices. A ($C_4$,diamond)-free graph is a graph that does not contain a diamond or a cycle on four vertices as induced subgraphs. Let $G$ be a connected ($C_4$,diamond)-free graph on $n$ vertices. Let $1 \le k \le n-1$ be an integer. The $k$-token graph, $F_k(G)$, of $G$ is the graph whose vertices are all the sets of $k$ vertices of $G$; two of which are adjacent if their symmetric difference is a pair of adjacent vertices in $G$. Let $F$ be a graph isomorphic to $F_k(G)$. In this paper we show that given only $F$, we can construct in polynomial time a graph isomorphic to $G$. Let $\operatorname{Aut}(G)$ be the automorphism group of $G$. We also show that if $k\neq n/2$, then $\operatorname{Aut}(G) \simeq \operatorname{Aut}(F_k(G))$; and if $k = n/2$, then $\operatorname{Aut}(G) \simeq \operatorname{Aut}(F_k(G)) \times \mathbb{Z}_2$.

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