4 research outputs found
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
Bounds on several versions of restrained domination number
We investigate several versions of restraineddomination numbers and present new bounds on these parameters. We generalize theconcept of restrained domination and improve some well-known bounds in the literature.In particular, for a graph of order and minimum degree , we prove thatthe restrained double domination number of is at most . In addition,for a connected cubic graph of order we show thatthe total restrained domination number of is at least andthe restrained double domination number of is at least
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