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 $G=(V,E)$ be a graph. A total restrained dominating set is a set $S\subseteq V$ where every vertex in $V-S$ is adjacent to a vertex in $S$ as well as to another vertex in $V-S$, and every vertex in $S$ is adjacent to another vertex in $S$. The total restrained domination number of $G$, denoted by $\gamma _r^t(G)$, is the smallest cardinality of a total restrained dominating set of $G$. First, some exact values and sharp bounds for $\gamma _r^t(G)$ 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 $\gamma _r^t(G)$ is NP-complete even for bipartite and chordal graphs in Section 4