Almost Every Graph Is Reconstructible from Its Token Graphs

Abstract

Let $F_k(G)$ denote the $k$-token graph of a finite graph $G$. The graph $F_k(G)$ is a spanning subgraph of $J(n,k)$, but its vertices are not given their $k$-subset labels. We show that, for fixed-density random graphs, the Johnson adjacency relation can be determined from common-neighbor counts. For every fixed $p\in (0,1)$ and $G_n\sim G(n,p)$, the probability that $G_n$ is reconstructible from $F_k(G_n)$ for every $1\le k\le n-1$ is at least $1-(1+o(1))n(1-p{)}^{n-1}$. In particular, asymptotically almost every labeled $n$-vertex graph $G$ is reconstructible from $F_k(G)$ for every nontrivial rank $2\le k\le n-2$. The same density-one conclusion for isomorphism classes is outlined in Remark.