Determination of the Optimally Efficient Sets in Special Classes of Graphs

A dominating set is said to be an efficient dominating set if, for every vertex v is an element of V, vertical bar N[v] boolean AND S vertical bar = 1 [1]. A set S is called open irredundant if, for every vertex u is an element of S, there exists a vertex v is an element of V - S for which N(v)boolean AND S = {u}, in which case we say that u efficiently dominates v. There exists a polynomial time algorithm for finding an optimally efficient set in an arbitrary graph. We determine directly the optimally efficient sets in confidentially connected graphs and unbreakable graphs. Also, we determine directly the open irredundant set, closed neighborhood packing set, the influence of a set in confidentially connected graphs and unbreakable graphs.