Minimum Vector Rank and Complement Critical Graphs

TL;DR

This paper explores the concept of minimum vector rank in graphs, defining critical and complement critical graphs, establishing conditions for certain classes, and linking these to the Graph Complement Conjecture.

Contribution

It introduces the notion of complement critical graphs, provides conditions for their identification, and connects these concepts to the open Graph Complement Conjecture.

Findings

Characterization of complement critical graphs

Conditions for minimum vector rank in specific graph classes

Support for the Graph Complement Conjecture

Abstract

Given a graph G, a real orthogonal representation of G is a function from its set of vertices to R^d such that two vertices are mapped to orthogonal vectors if and only if they are not neighbors. The minimum vector rank of a graph is the smallest dimension d for which such a representation exists. This quantity is closely related to the minimum semidefinite rank of G, which has been widely studied. Considering the minimum vector rank as an analogue of the chromatic number, this work defines critical graphs as those for which the removal of any vertex decreases the minimum vector rank; and complement critical graphs as those for which the removal of any vertex decreases the minimum vector rank of either the graph or its complement. It establishes necessary and sufficient conditions for certain classes of graphs to be complement critical, in the process calculating their minimum vector rank. In addition, this work demonstrates that complement critical graphs form a sufficient set to prove the Graph Complement Conjecture, which remains open.