Efficient Triple Connected Domination Number of a Graph

The concept of triple connected graphs with real life application was introduced in [16] by considering the existence of a path containing any three vertices of a graph G. In [3], G. Mahadevan et. al., was introduced the concept of triple connected domination number of a graph. A subset S of V of a nontrivial connected graph G is said to be 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 subset S of V of a nontrivial graph G is said to be an efficient dominating set, if every vertex is dominated exactly once. The minimum cardinality taken over all efficient dominating sets is called the efficient domination number and is denoted by e . In this paper we introduce new domination parameter efficient triple connected domination number of a graph with real life application. A subset S of V of a nontrivial connected graph G is said to be an efficient triple connected dominating set, if S is an efficient dominating set and the induced subgraph is triple connected. The minimum cardinality taken over all efficient triple connected dominating sets is called the efficient triple connected domination number and is denoted by etc. We also determine this number for some standard graphs and obtain bounds for general graph. Its relationship with other graph theoretical parameters are also investigated.