Dominance complexes and vertex cover numbers of graphs

TL;DR

This paper explores the relationship between the dominance complex of a graph and its vertex cover number, establishing a new lower bound connecting topological and combinatorial graph properties.

Contribution

It introduces a novel bound linking the connectivity of the dominance complex to the vertex cover number of a graph.

Findings

Connectivity of dominance complex plus 2 bounds vertex cover number from below

Establishes a new connection between topological and combinatorial graph invariants

Provides insights into the structure of dominating sets and vertex covers

Abstract

The dominance complex $D(G)$ of a simple graph $G = (V,E)$ is the simplicial complex consisting of the subsets of $V$ whose complements are dominating. We show that the connectivity of $D(G)$ plus $2$ is a lower bound for the vertex cover number $\tau(G)$ of $G$.