Paired domination in graphs with minimum degree four

TL;DR

This paper establishes an upper bound on the paired domination number for graphs with minimum degree four, showing it is at most approximately 58.83% of the total vertices.

Contribution

It provides a new upper bound on the paired domination number for graphs with minimum degree four, improving understanding of dominating sets in such graphs.

Findings

Paired domination number is at most 10/17 of the total vertices.

Bound applies to graphs with minimum degree at least four.

Result tightens previous bounds in the literature.

Abstract

A set $S$ of vertices in a graph $G$ is a paired dominating set if every vertex of $G$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ admits a perfect matching. The minimum cardinality of a paired dominating set of $G$ is the paired domination number $\gpr(G)$ of $G$. We show that if $G$ is a graph of order~$n$ and $\delta(G) \ge 4$, then $\gpr(G) \le \frac{10}{17}n < 0.5883 n$.