Tree Domination Number in Total Graph T(G) of a Graph G.

Abstract

Let G = (V, E) be a connected graph. A subset D of V is called a dominating set of G if N[D] = V. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by g(G). A dominating set D of a graph G is called a tree dominating set (ntr - set) if the induced subgraph áDñ is a tree. The tree domination number γ_tr(G) of G is the minimum cardinality of a tree dominating set. The total graph T(G) of a graph G is a graph such that the vertex set T(G) corresponds to the vertices and edges of G and two vertices are adjacent in T(G) if and only if their corresponding elements are either adjacent or incident in G. In this paper, we have studied some bounds for tree domination number of Total graph T(G) of a graph. Also, we have found the tree domination number of T(G) for some graphs.

In this paper, tree domination number of total graphs of some standard graphs are obtained. Also we have characterized graphs for which γ_tr(T(G)) = 1, 2 or n ‒ 2.