Graphs with equal domination and certified domination numbers

TL;DR

This paper investigates the relationships between various domination parameters in graphs, including domination, upper domination, certified domination, and upper certified domination numbers, providing new insights into their interconnections.

Contribution

It introduces and analyzes the relationships among four domination parameters, including the newly defined certified domination and upper certified domination numbers.

Findings

Established bounds between domination and certified domination numbers.

Characterized graphs where these parameters are equal.

Provided new inequalities relating the different domination numbers.

Abstract

A set $D$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex in $V_G-D$ is adjacent to at least one vertex in $D$. The domination number (upper domination number, respectively) of a graph $G$, denoted by $\gamma(G)$ ($\Gamma(G)$, respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of $G$. A subset $D\subseteq V_G$ is called a certified dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ has either zero or at least two neighbors in $V_G-D$. The cardinality of a~smallest (largest minimal, respectively) certified dominating set of $G$ is called the certified upper certified, respectively domination number of $G$ and is denoted by $\gamma_{\rm cer}(G)$ ($\Gamma_{\rm cer}(G)$, respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied.