8 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
Gallai-type theorems and domination parameters
AbstractLet γ(G) denote the minimum cardinality of a dominating set of a graph G = (V,E). A longstanding upper bound for γ(G) is attributed to Berge: For any graph G with n vertices and maximum degree Δ(G), γ(G) ⩽ n − Δ(G). We characterise connected bipartite graphs which achieve this upper bound. For an arbitrary graph G we furnish two conditions which are necessary if γ(G) + Δ(G) = n and are sufficient to achieve n − 1 ⩽ γ(G) + Δ(G) ⩽ n.We further investigate graphs which satisfy similar equations for the independent domination number, i(G), and the irredundance number ir(G). After showing that i(G) ⩽ n − Δ(G) for all graphs, we characterise bipartite graphs which achieve equality.Lastly, we show for the upper irredundance number, IR(G): For a graph G with n vertices and minimum degree δ(G), IR(G) ⩽ n - δ(G). Characterisations are given for classes of graphs which achieve this upper bound for the upper irredundance, upper domination and independence numbers of a graph
Abstract Maximum Sizes of Graphs with Given Domination Parameters
We find the maximum number of edges for a graph of given order and value of parameter for several domination parameters. In particular, we consider the total domination and independent domination numbers
On total restrained domination in graphs
summary:In this paper we initiate the study of total restrained domination in graphs. Let be a graph. A total restrained dominating set is a set where every vertex in is adjacent to a vertex in as well as to another vertex in , and every vertex in is adjacent to another vertex in . The total restrained domination number of , denoted by , is the smallest cardinality of a total restrained dominating set of . First, some exact values and sharp bounds for are given in Section 2. Then the Nordhaus-Gaddum-type results for total restrained domination number are established in Section 3. Finally, we show that the decision problem for is NP-complete even for bipartite and chordal graphs in Section 4