Paired-Domination in Claw-Free Graphs with Minimum Degree at Least Four

Xinguo Cao1, Erfang Shan1,2
1School of Management, Shanghai University, Shanghai 200444, China
2Department of Mathematics, Shanghai University, Shanghai 200444, China

Abstract

A paired-dominating set of a graph \(G\) is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number is the minimum cardinality of a paired-dominating set of \(G\). In this paper, we investigate the paired-domination number in claw-free graphs with minimum degree at least four. We show that a connected claw-free graph \(G\) with minimum degree at least four has paired-domination number at most \(\frac{4}{7}\) its order.