Strong Triple Connected Domination Number of a Graph

The concept of triple connected graphs with real life application was introduced in (14) by considering the existence of a path containing any three vertices of a graph G. In (3), G. Mahadevan et. al., introduced Smarandachely triple connected domination number of a graph. In this paper, we introduce a new domination parameter, called strong triple connected domination number of a graph. A subset S of V of a nontrivial graph G is said to be smarandachely triple connected dominating set, if S is a dominating set and the induced sub graph is triple connected. The minimum cardinality taken over all triple connected dominating sets is called the triple connected domination number and is denoted by tc. A set D ⊆ V(G) is a strong dominating set of G, if for every vertex x ∈ V(G) − D there is a vertex y ∈ D with xy ∈ E(G) and d(x,G) ≤ d(y,G). The strong domination number st(G) is defined as the minimum cardinality of a strong domination set. A subset S of V of a nontrivial graph G is said to be strong triple connected dominating set, if S is a strong dominating set and the induced sub graph is triple connected. The minimum cardinality taken over all strong triple connected dominating sets is called the strong triple connected domination number and is denoted by stc. We determine this number for some standard graphs and obtain bounds for general graph. Its relationship with other graph theoretical parameters are also investigated.