Fair coalition in graphs
Saeid Alikhani, Abbas Jafari, Maryam Safazadeh

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 be a simple graph. A dominating set of is a subset such that every vertex not in is adjacent to at least one vertex in . The cardinality of a smallest dominating set of , denoted by , is the domination number of . For , a -fair dominating set (-set) in , is a dominating set such that for every vertex . A fair dominating set in is a -set for some integer . We consider -sets and define a fair coalition in a graph as a pair of disjoint subsets that satisfy the following conditions: (a) neither nor constitutes a -fair dominating set of , and (b) constitutes a -fair dominating set of . A fair coalition partition of a graph is a partition of its…
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Taxonomy
TopicsAdvanced Graph Theory Research · Game Theory and Voting Systems · Complexity and Algorithms in Graphs
