Abstract:

Let G = (V, E) be a simple graph with p vertices and q edges. A subset S of V(G) is called a strong (weak) efficient dominating set of G if for every vV(G),│Ns[v]∩S│=1(│Nw[v]∩S│=1).Ns(v) ={u V(G) uv E(G), deg(u) ≥ deg(v)}.The minimum cardinality of a strong (weak) efficient dominating set G is called strong (weak) efficient domination number of G and is denoted by γse (γwe ). A graph G is strong efficient if there exists a strong efficient dominating set of G. In this paper, the authors introduced a new parameter called the number of strong efficient dominating sets of a graph G denoted by # γse (G) and studied some Nordhaus- Gaddum type relations on strong efficient domination number of a graph and its derived graph. The relation between the number of strong efficient dominating sets of a graph and its derived graph is also studied

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