Near-domination in graphs

A vertexandnbsp;uandnbsp;of a graph andldquo;t-dominatesandrdquo; a vertexandnbsp;vandnbsp;if there are at mostandnbsp;tandnbsp;vertices different fromandnbsp;u, vandnbsp;that are adjacent toandnbsp;vandnbsp;and not toandnbsp;u; and a graph is andldquo;t-dominatingandrdquo; if for every pair ofandnbsp;distinct vertices, one of themandnbsp;t-dominates the other. Our main result says that if a graph isandnbsp;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 Erdős-Hajnal conjecture.</p

Near-domination in graphs

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 Erdős-Hajnal conjecture.</p