Let G be a graph with vertex set V . A set D ⊆ V is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in V \D has a neighbor in V \D. The minimum cardinality of a total restrained dominating set of G is called the total restrained domination number of G, and is denoted by γtr(G). In this paper, we prove that if G is a connected graph of order n ≥ 4...

Let G = (V, E) be a graph. A k-connected restrained dominating set is a set S ⊆ V , where S is a restrained dominating set and G[S] has at most k components. The k-connected restrained domination number of G, denoted by γ k r (G), is the smallest cardi-nality of a k-connected restrained dominating set of G. In this talk, I will give some exact values and sharp bounds for γ k r (G). Then the nec...

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′ ⊆...