Upper paired domination versus upper domination

TL;DR

This paper investigates the relationship between upper paired domination and upper domination numbers in graphs, characterizing specific graph classes where the upper paired domination number equals twice the upper domination number.

Contribution

It provides characterizations for bipartite and unicyclic graphs satisfying the equality, and explores conditions for graphs with restricted girth, including $C_3$-free cactus graphs.

Findings

Characterizations for bipartite graphs with $ ext{Gamma}_{pr}(G)= 2 ext{Gamma}(G)$.

Characterizations for unicyclic graphs with the same equality.

Results on graphs with girth at least 6 and $C_3$-free cactus graphs.

Abstract

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.