Enumeration of accurate dominating sets

TL;DR

This paper investigates the concept of accurate dominating sets in graphs, introduces the accurate domination number, and counts such sets in specific graph classes to deepen understanding of their properties.

Contribution

It introduces the accurate domination number and provides enumeration results for accurate dominating sets in certain graph families.

Findings

Derived the accurate domination number for specific graphs

Counted accurate dominating sets in particular graph classes

Enhanced understanding of accurate domination in graph theory

Abstract

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ 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 $G$. A dominating set $D$ is an accurate dominating set of $G$, if no $|D|$-element subset of $V\setminus D$ is a dominating set of $G$. The accurate domination number, $\gamma_a(G)$, is the cardinality of a smallest accurate dominating set $D$. In this paper, after presenting preliminaries, we count the number of accurate dominating sets of some specific graphs.