On Unique Neighborhoods in Bipartite and Expander Graphs

TL;DR

This paper investigates the property of unique neighborhoods in bipartite and expander graphs, exploring how neighborhood structures can serve as unique identifiers for nodes in these graph classes.

Contribution

It provides new insights into the conditions and properties that ensure neighborhood uniqueness in bipartite and expander graphs.

Findings

Characterization of neighborhood uniqueness in bipartite graphs

Analysis of neighborhood distinguishability in expander graphs

Implications for graph identification and symmetry breaking

Abstract

An undirected graph is said to have \emph{unique neighborhoods} if any two distinct nodes have also distinct sets of neighbors. In this way, the connections of a node to other nodes can characterize a node like an "identity", irrespectively of how nodes are named, as long as two nodes are distinguishable. We study the uniqueness of neighborhoods in (random) bipartite graphs, and expander graphs.