Reconstruction of C_4-free graphs from the set of closed neighborhoods and digital convexity

TL;DR

This paper proves that all C4-free graphs can be reconstructed solely from their sets of closed neighborhoods, extending previous results and linking digital convexity to graph reconstructibility.

Contribution

It demonstrates that C4-free graphs are reconstructible from the set of closed neighborhoods, strengthening prior multiset-based results and connecting digital convexity to graph reconstruction.

Findings

Reconstruction from the set of closed neighborhoods is possible for C4-free graphs.

Reconstruction from digitally convex sets is equivalent to from closed neighborhoods.

All graphs with girth at least five are reconstructible from digitally convex sets.

Abstract

Fomin, Kratochv\'il, Lokshtanov, Mancini, and Telle showed that every $C_{4}$-free graph is reconstructible from the \emph{multiset} of closed neighborhoods. We strengthen their result proving that every $C_{4}$-free graph is reconstructible from the \emph{set} of closed neighborhoods. This extends the work of Lafrance et al.\ by showing that all $C_{4}$-free graphs, and hence all graphs of girth at least five, are reconstructible from their digitally convex sets. A subset $S$ of vertices in a graph $G$ is digitally convex if, for every vertex $v \notin S$, there is a private neighbor of $v$. We establish that reconstruction from digitally convex sets is equivalent to reconstruction from the set of closed neighborhoods.