Fair Domination in Graphs

TL;DR

This paper introduces the concept of fair dominating sets in graphs, explores their properties, and establishes bounds on the fair domination number for various classes of graphs, including connected graphs, maximal outerplanar graphs, and trees.

Contribution

It defines the fair domination number and provides bounds and characterizations for different graph classes, including constructions achieving these bounds.

Findings

For connected graphs of order n ≥ 3 with no isolated vertices, fd(G) ≤ n - 2.

In maximal outerplanar graphs, fd(G) < 17n/19.

For trees of order n ≥ 2, fd(T) ≤ n/2, with equality only for the corona of a tree.

Abstract

A fair dominating set in a graph $G$ (or FD-set) is a dominating set $S$ such that all vertices not in $S$ are dominated by the same number of vertices from $S$; that is, every two vertices not in $S$ have the same number of neighbors in $S$. The fair domination number, $fd(G)$, of $G$ is the minimum cardinality of a FD-set. We present various results on the fair domination number of a graph. In particular, we show that if $G$ is a connected graph of order $n \ge 3$ with no isolated vertex, then $fd(G) \le n - 2$, and we construct an infinite family of connected graphs achieving equality in this bound. We show that if $G$ is a maximal outerplanar graph, then $fd(G) < 17n/19$. If $T$ is a tree of order $n \ge 2$, then we prove that $fd(T) \le n/2$ with equality if and only if $T$ is the corona of a tree.