On parameters related to strong and weak domination in graphs

AbstractLet G be a graph. Then μ(G)⩽|V(G)|−δ(G) where μ(G) denotes the weak or independent weak domination number of G and μ(G)⩽|V(G)|−Δ(G) where μ(G) denotes the strong or independent strong domination number of G. We give necessary and sufficient conditions for equality to hold in each case and also describe specific classes of graphs for which equality holds. Finally, we show that the problems of computing iw and ist are NP-hard, even for bipartite graphs

A NOTE ON A RELATION BETWEEN THE WEAK AND STRONG DOMINATION NUMBERS OF A GRAPH

Abstract. In a graph G = (V, E) a vertex is said to dominate itself and all its neighbors. , respectively). The weak (strong, respectively) domination number of G, denoted by γw(G) (γs(G), respectively), is the minimum cardinality of a weak (strong, respectively) dominating set of G. In this note we show that if G is a connected graph of order n ≥ 3, then γw(G) + tγs(G) ≤ n, where t = 3/(Δ + 1) if G is an arbitrary graph, t = 3/5 if G is a block graph, and t = 2/3 if G is a claw free graph