On the Domination Number of Some Graph Operations

Abstract

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. This research aims to characterize the domination number of some graph operations, namely joint graphs, coronation of graphs, graph compositions, tensor product of two graphs, and graph amalgamation. The results shows that most of the resulting domination numbers attain the given lower bound of $\gamma(G)$.