Tree Domination Number Of Middle And Splitting Graphs

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 Middle Graph M(G) of G is defined as follows. The vertex set of M(G) is V(G)ÈE(G). Two vertices x. y in the vertex set of M(G) are adjacent in M(G) if one of the following holds. (i) x, y are in E(G) and x, y are adjacent in G. (ii) xÎV(G), yÎE(G) and y is incident at x in G. Let G be a graph with vertex set V(G) and let V′(G) be a copy of V(G). The splitting graph S(G) of G is the graph, whose vertex set is V(G) È V′(G) and edge set is {uv, u′v and uv′: uvÎE(G)}. In this paper we study the concept of tree domination in middle and splitting graphs.