The Paired Domination Number of Cubic Graphs

TL;DR

This paper proves Goddard and Henning's conjecture that the paired domination number of cubic graphs is at most 4n/7, confirming the bound for graphs with minimum degree 3, excluding the Petersen graph.

Contribution

The paper establishes the conjecture for cubic graphs, providing a significant advance in understanding paired domination in graphs with degree constraints.

Findings

Proves the conjecture for cubic graphs.

Confirms the bound of 4n/7 for paired domination number.

Excludes the Petersen graph from the bound.

Abstract

Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G, denoted by {\gamma}pr(G), is the size of its smallest paired dominating set. Goddard and Henning conjectured that {\gamma}pr(G) {\leq} 4n/7 holds for every graph G with {\delta}(G) {\geq} 3, except the Petersen Graph. In this paper, we prove this conjecture for cubic graphs.