Fair Sets of Some Class of Graphs

TL;DR

This paper introduces the concept of fair sets in graphs, characterizes their properties in specific graph classes, and identifies fair sets in common graph types, advancing understanding of equitable vertex subsets.

Contribution

It defines fair sets based on partiality, proves their connectivity in trees, characterizes block graphs, and identifies fair sets in various standard graphs and their Cartesian products.

Findings

Fair sets induce connected subgraphs in trees.

Block graphs are characterized by connected fair sets.

Fair sets are identified for several standard graph classes.

Abstract

Given a non empty set $S$ of vertices of a graph, the partiality of a vertex with respect to $S$ is the difference between maximum and minimum of the distances of the vertex to the vertices of $S$. The vertices with minimum partiality constitute the fair center of the set. Any vertex set which is the fair center of some set of vertices is called a fair set. In this paper we prove that the induced subgraph of any fair set is connected in the case of trees and characterise block graphs as the class of chordal graphs for which the induced subgraph of all fair sets are connected. The fair sets of $K_{n}$, $K_{m,n}$, $K_{n}-e$, wheel graphs, odd cycles and symmetric even graphs are identified. The fair sets of the Cartesian product graphs are also discussed.