Neighborhood Reconstruction and Cancellation of Graphs

Richard H Hammack,Cristina Mullican

doi:10.37236/6676

Abstract

We connect two seemingly unrelated problems in graph theory.Any graph $G$ has a neighborhood multiset $\mathscr{N}(G)= \{N(x) \mid x\in V(G)\}$ whose elements are precisely the open vertex-neighborhoods of $G$. In general there exist non-isomorphic graphs $G$ and $H$ for which $\mathscr{N}(G)=\mathscr{N}(H)$. The neighborhood reconstruction problem asks the conditions under which $G$ is uniquely reconstructible from its neighborhood multiset, that is, the conditions under which $\mathscr{N}(G)=\mathscr{N}(H)$ implies $G\cong H$. Such a graph is said to be neighborhood-reconstructible.The cancellation problem for the direct product of graphs seeks the conditions under which $G\times K\cong H\times K$ implies $G\cong H$. Lovász proved that this is indeed the case if $K$ is not bipartite. A second instance of the cancellation problem asks for conditions on $G$ that assure $G\times K\cong H\times K$ implies $G\cong H$ for any bipartite~$K$ with $E(K)\neq \emptyset$. A graph $G$ for which this is true is called a cancellation graph.We prove that the neighborhood-reconstructible graphs are precisely the cancellation graphs. We also present some new results on cancellation graphs, which have corresponding implications for neighborhood reconstruction. We are particularly interested in the (yet-unsolved) problem of finding a simple structural characterization of cancellation graphs (equivalently, neighborhood-reconstructible graphs).

Highlights

PreliminariesAn an edge (x, y) ∈ E(G) is denoted xy
We assume our reader is at least somewhat familiar with direct products
Two types of questions have been asked about neighborhood multisets

Summary

Preliminaries

An an edge (x, y) ∈ E(G) is denoted xy. The open neighborhood of a vertex x ∈ V (G) is the set NG(x) = {y ∈ V (G) | xy ∈ E(G)}, which we may denote as N (x) when this is unambiguous. By G ∼= H we mean that G and H are isomorphic. An isomorphism from G to itself is called an automorphism of G. The direct product of two graphs G and H is the graph G×H with vertices V (G)×V (H) and edges E(G × H) = {(x, x′)(y, y′) | xy ∈ E(G) and x′y′ ∈ E(H)}. If G is bipartite, G × K2 = G + G, as illustrated on the right of Figure 1

Neighborhood reconstruction

Cancellation

Further Results