ON THE DOMINATION AND INDEPENDENT SETS OF G^M_{m,n} GRAPH

Abstract

In graph theory, the theory of domination has several applications in various fields of science and technology, which is considered as a turn up field of research. In real life, it is extremely important in fields like network desigh, wireless sensor networks,logistics, mobile computing, telecommunication and others, Problems with facility location, communication or electrical network monitoring can lead to dominance. Undirected graphs is one of the most excellent models in connection with distributed computation and parellel processing. A set $S\subset V$ is said to be a dominating set of a graph $G$ if every vertex in $ V-S $ is adjacent to atleast one vertex in $S$. The domination number $\gamma{(G)}$ of the graph $ G $ is the minimum cardinality of a dominating set of $ G $. An independent dominating set $ S \subset V $ is exists if no edges in the induced subgraph $\langle S \rangle$ and the independent dominating number $\gamma_i(G)$ is the minimum cardinality of an independent dominating set of $ G $. in this paper, some results on dominating sets and independent dominating sets of $\uppercase{G}^{M}_{m,n}$ graph on a finite sebset of natural numbers are presented and the domination numbers are obtained for various values of $m,n$.