Domination Cover Number of Graphs

TL;DR

This paper introduces the domination cover number as a new way to evaluate dominating sets in graphs, exploring its properties and algorithms across various graph classes, with applications in problems like the art gallery problem.

Contribution

It defines the domination cover number, studies its properties in different graph classes, and proposes algorithms for trees and block graphs.

Findings

Introduces the domination cover number as a new graph parameter.

Provides algorithms for computing the domination cover number in trees and block graphs.

Highlights applications in real-world problems like the art gallery problem.

Abstract

A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in $O(1.7159^n)$ time. It is known that the number of minimal dominating sets for interval graphs and trees on $n$ vertices is at most $3^{n/3} \approx 1.4422^n$. In this paper, we introduce the domination cover number as a new criterion for evaluating the dominating sets in graphs. The domination cover number of a dominating set $D$, denoted by $\mathcal{C}_D(G)$, is the summation of the degrees of the vertices in $D$. Maximizing or minimizing this parameter among all minimal dominating sets have interesting applications in many real-world problems, such as the art gallery problem. Moreover, we investigate this concept for different graph classes and propose some algorithms for finding the domination cover number in trees, block graphs.