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

TL;DR

This paper proves that connected ($C_4$,diamond)-free graphs can be uniquely reconstructed from their token graphs in polynomial time, and establishes a relationship between their automorphism groups.

Contribution

It demonstrates polynomial-time reconstructibility of certain graphs from token graphs and characterizes their automorphism groups based on the token graph parameter.

Findings

Graphs are reconstructible from token graphs in polynomial time.

Automorphism groups of graphs are preserved or related via a direct product with Z_2.

Reconstruction is possible for all $k$ except when $k = n/2$.

Abstract

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 $\operatorname{Aut}(G) \simeq \operatorname{Aut}(F_k(G)) \times \mathbb{Z}_2$.