The overlap number of a graph

TL;DR

This paper investigates the overlap number of various graph classes, establishing exact values for some and proving NP-completeness for related problems, thus advancing understanding of graph representations.

Contribution

It determines the overlap numbers for specific graph classes and proves the NP-completeness of extension and minimization problems in overlap representations.

Findings

Overlap numbers for cliques and bipartite graphs are characterized.

Exact overlap numbers for paths, cycles, and caterpillars are established.

NP-completeness of extension and limited containment problems is proved.

Abstract

An overlap representation is an assignment of sets to the vertices of a graph in such a way that two vertices are adjacent if and only if the sets assigned to them overlap. The overlap number of a graph is the minimum number of elements needed to form such a representation. We find the overlap numbers of cliques and complete bipartite graphs by relating the problem to previous research in combinatorics. The overlap numbers of paths, cycles, and caterpillars are also established. Finally, we show the NP-completeness of the problems of extending an overlap representation and finding a minimum overlap representation with limited containment.