On the number of fair dominating sets of graphs

TL;DR

This paper investigates the concept of fair dominating sets in graphs, focusing on counting their number in specific graph classes and exploring their properties.

Contribution

It introduces the notion of $k$-fair dominating sets and provides enumeration results for these sets in particular graphs.

Findings

Counted the number of fair dominating sets in specific graphs

Established properties of $k$-fair dominating sets

Extended understanding of domination concepts 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 $G$.For $k \geq 1$, a $k$-fair dominating set($kFD$-set) in $G$, is a dominating set $S$ such that $|N(v) \cap D|=k$ for every vertex $ v \in V\setminus D$.A fair dominating set, in $G$ is a $kFD$-set for some integer $k\geq 1$.In this paper, after presenting preliminaries, we count the number of fair dominating sets of some specific graphs.