Paired domination in graphs with minimum degree four

TL;DR

This work investigates bounds on the paired domination number $\gamma_{\rm pr}(G)$ for graphs with minimum degree at least $4$. The authors introduce a colored-graph method, leveraging a weight function on a boundary-colored graph $G_D$ and the concept of $D$-desirable sets to iteratively extend a PD-set while guaranteeing a minimum weight decrease. They prove the key bound $\gamma_{\rm pr}(G) \le \frac{10}{17}n$ (i.e., $<0.5883n$) for graphs of order $n$, improving prior results that achieved at best $<0.634567n$ for $\delta(G) \ge 3$. The paper also discusses conjectures aiming for a $4/7$-type bound when $\delta(G) \ge 4$, as well as related questions in bipartite and regular graph settings, highlighting the ongoing effort to tighten bounds on paired domination in sparse graphs. All mathematical notation is expressed with $...$ delimiters throughout.

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 $δ(G) \ge 4$, then $\gpr(G) \le \frac{10}{17}n < 0.5883 n$.

Figures (1)

• Figure 1: The graph $H_8$

Theorems & Definitions (17)

• Theorem 1
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• Definition 1
• Claim 1
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• Claim 7