The Study of Domination Subdivision Number for Cartesian Product of Path, Complete Graph and Grid Graph

Let G=(V,E ) be a simple graph on a vertex set V. In a Graph G, A set D⊆V is a dominating set of G if every vertex in V-D is adjacent to some vertex in D. A dominating set D of G is minimal if for any vertex v∈D, D-{v} is not a dominating set of G. The domination number of a graph G, denoted by γ(G), is the minimum size of a dominating set of vertices in G. The domination subdivision number of a graph G is the minimum number of edges that must subdivided in order to increase the domination number of a graph and it is denoted by sdγ(G). A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v∈V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Total domination subdivision number denoted by sdγt is the minimum number of edges that must be subdivided to increase the total domination number. In this paper the domination subdivision number for some known graphs are investigated. In this paper the domination subdivision number for some known graphs are investigated.