Fair coalition in graphs

TL;DR

This paper introduces the concept of fair coalition partitions in graphs, focusing on 1-fair dominating sets and their combinations, and explores the fair coalition number for various graph classes.

Contribution

It defines fair coalition partitions in graphs and initiates the study of the fair coalition number, providing initial results for specific graph types.

Findings

Defined fair coalition and fair coalition number in graphs

Calculated fair coalition number for some specific graphs

Established foundational concepts for future research

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 $k\geq 1$. We consider $1FD$-sets and define a fair coalition in a graph $G$ as a pair of disjoint subsets $A_1, A_2 \subseteq A$ that satisfy the following conditions: (a) neither $A_1$ nor $A_2$ constitutes a $1$-fair dominating set of $G$, and (b) $A_1\cup A_2$ constitutes a $1$-fair dominating set of $G$. A fair coalition partition of a graph $G$ is a partition $\Upsilon = \{A_1,A_2,\ldots,A_k\}$ of its vertex set such that every set $A_i$ of $\Upsilon$ is either a singleton $1$-fair dominating set of $G$, or is not a $1$-fair dominating set of $G$ but forms a fair coalition with another non-$1$-fair dominating set $A_j\in \Upsilon$. We define the fair coalition number of $G$ as the maximum cardinality of a fair coalition partition of $G$, and we denote it by $\mathcal{C}_f(G)$. We initiate the study of the fair coalition in graphs and obtain $\mathcal{C}_f(G)$ for some specific graphs.