On the Domination Number of Some Families of Special Graphs

Abstract

A domination in graphs is part of graph theory which has many applications. Its application includes the morphological analysis, computer network communication, social network theory, CCTV installation, and many others. A set $D$ of vertices of a simple graph $G$, that is a graph without loops and multiple edges, is called a dominating set if every vertex $u\in V(G)-D$ is adjacent to some vertex $v\in D$. The domination number of a graph  $G$, denoted by $\gamma(G)$, is the order of a smallest  dominating set of $G$. A dominating set $D$ with $|D|=\gamma(G)$ is called a minimum dominating set, see Haynes and Henning \cite{Hay1} . This research aims to find the domination number of some families of special graphs, namely Spider Web graph $Wb_{n}$, Helmet graph $H_{n,m}$, Parachute graph $Pc_{n}$, and any regular graph. The results shows that the resulting domination numbers meet the lower bound of an obtained lower bound $\gamma(G)$ of any graphs.