Near-domination in graphs

TL;DR

This paper investigates the properties of t-dominating graphs, showing they are structurally close to 0-dominating graphs, and explores related concepts and conjectures in graph theory.

Contribution

It introduces the concept of t-dominating graphs and proves they are close to 0-dominating graphs, extending understanding of domination properties in graphs.

Findings

t-dominating graphs are close to 0-dominating graphs

the analogous statement for digraphs does not hold

discusses connections with the Erdos-Hajnal conjecture

Abstract

A vertex u of a graph t-dominates a vertex v if there are at most t vertices different from u,v that are adjacent to v and not to u; and a graph is t-dominating if for every pair of distinct vertices, one of them t-dominates the other. Our main result says that if a graph is t-dominating, then it is close (in an appropriate sense) to being 0-dominating. We also show that an analogous statement for digraphs is false; and discuss some connections with the Erdos-Hajnal conjecture.