On accurate domination in graphs

TL;DR

This paper investigates the accurate domination number in graphs, characterizes trees where it equals the domination number, and compares it across various graph coronas, advancing understanding of domination parameters.

Contribution

It characterizes all trees with equal accurate and standard domination numbers and analyzes their behavior in different graph coronas.

Findings

Identified all trees with equal accurate and domination numbers.

Compared accurate and domination numbers in various graph coronas.

Provided insights into the structure of graphs with specific domination properties.

Abstract

A dominating set of a graph $G$ is a subset $D \subseteq V_G$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. The accurate domination number of $G$, denoted by $\gamma_{\rm a}(G)$, is the cardinality of a smallest set $D$ that is a dominating set of $G$ and no $|D|$-element subset of $V_G \setminus D$ is a dominating set of $G$. We study graphs for which the accurate domination number is equal to the domination number. In particular, all trees $G$ for which $\gamma_{\rm a}(G) = \gamma(G)$ are characterized. Furthermore, we compare the accurate domination number with the domination number of different coronas of a graph.