Dominating functions in graphs – regularity versus irregularity

Abstract

A vertex v in a graph G is said to dominate a vertex u if either u = v or $uv\in E(G)$ and a set S of vertices in G is a dominating set of G if every vertex of G is dominated by at least one vertex in S. Domination has been looked at in an equivalent way. A function $f:V(G)\to \{0,1\}$ is a dominating function of a graph G if $\sum _{u\in N[v]}f(u)\ge 1$ for every vertex v of G. We use dominating functions to investigate graphs all of whose vertices are dominated by the same number of vertices as well as those graphs whose vertices are dominated by as many different number of vertices as possible.

Keywords:

• Domination
• dominating function
• regular domination
• antirregular graphs
• irregular domination

AMS Subject Classifications:

• 05C69
• 05C05

Acknowledgments

We are grateful to Professor Gary Chartrand for suggesting the concepts of regular and irregular domination functions to us and kindly providing useful information on this topic. Furthermore, we thank the referee whose valuable suggestions resulted in an improved paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).