Connected Dominating Sets in Graphs With Stability Number Three

TL;DR

This paper studies connected dominating sets in specific graphs with independence number three and no induced C7 cycles, establishing bounds and introducing a new graph invariant related to Hadwiger's number.

Contribution

It introduces a new invariant h(G) based on connected dominating sets and provides bounds for graphs with independence number three and no induced C7 cycles.

Findings

Existence of a connected dominating set with at most 4 vertices in the specified graphs.

The invariant h(G) is at most the number of Hadwiger minors.

For these graphs, h(G) is at least one-fourth of the total number of vertices.

Abstract

In the special case of graphs G of independence number a(G)=3 without induced chordless cycles C7 it is shown that exists connected dominating set D of vertices with number of vertices n(D)<=4. Using the concept of connected dominating sets, we defined a new invariant h(G) that does not exceed the number of Hadwiger. For the considered graphs it is shown that h(G)>=n(G)/4.