Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. We define the restrained bondage number br(G) of a nonempty graph G to be the minimum cardinality among all sets of edges E′ ⊆...

For any graph G, let V (G) and E(G) denote the vertex set and the edge set of G respectively. The Boolean function graph B(G, L(G),NINC) of G is a graph with vertex set V (G) ∪ E(G) and two vertices in B(G, L(G),NINC) are adjacent if and only if they correspond to two adjacent vertices of G, two adjacent edges of G or to a vertex and an edge not incident to it in G. For brevity, this graph is d...

For any graph G, let V (G) and E(G) denote the vertex set and the edge set of G respectively. The Boolean function graph B(G, L(G),NINC) of G is a graph with vertex set V (G) ∪ E(G) and two vertices in B(G, L(G),NINC) are adjacent if and only if they correspond to two adjacent vertices of G, two adjacent edges of G or to a vertex and an edge not incident to it in G. For brevity, this graph is d...

A longest sequence $(v_1,\ldots,v_k)$ of vertices a graph $G$ is Grundy total dominating if for all $i$, $N(v_i) \setminus \bigcup_{j=1}^{i-1}N(v_j)\not=\emptyset$. The length $k$ the called domination number and denoted $\gamma_{gr}^{t}(G)$. In this paper, studied on four standard products. For direct product we show that $\gamma_{gr}^t(G\times H) \geq \gamma_{gr}^t(G)\gamma_{gr}^t(H)$, conjec...

A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = k n = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the conception of k-domina...