The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and m...

Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such partitions. When independent sets are replaced by dominating sets, the parameters corresponding to the chromatic and achromatic numbers are the domatic ...

A subset S of V is called a total dominating set if every vertex in V is adjacent to some vertex in S. The total domination number γt (G) of G is the minimum cardinality taken over all total dominating sets of G. A dominating set is called a connected dominating set if the induced subgraph 〈S〉 is connected. The connected domination number γc(G) of G is the minimum cardinality taken over all min...

A subset D of V is called a dom strong dominating set if for every v V – D, there exists u1, u2 D such that u1v, u2v  E(G) and deg (u1 ) ≥ deg (v). The minimum cardinality of a dom strong dominating set is called dom strong domination number and is denoted by γdsd. In this paper, we introduce the concept of nonsplit dom strong domination number of a graph. A dom strong dominating set D of a ...

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating 2-broadcasts along with the associated parameter, the dominating 2-broadcast number. We prove that computing the dominating 2-broadcast number is a NP-complete problem, but can be achieved in linear time fo...

A subset D of the vertex set V (G) of a graph G is called a dominating set of G if every vertex in V − D is adjacent to a vertex in D. The minimum cardinality of a dominating set is called the domination number and is denoted by γ(G). A dominating set D such that δ(< D >) = 0 is called an isolate dominating set. The minimum cardinality of an isolate dominating set is called the isolate dominati...

A set D ⊆ V (G) of a graph G = (V,E) is a liar’s dominating set if (1) for all v ∈ V (G) |N [v] ∩ D| ≥ 2 and (2) for every pair u, v ∈ V (G) of distinct vertices, |N [u] ∪ N [v] ∩ D| ≥ 3. In this paper, we consider the liar’s domination number of some middle graphs. Every triple dominating set is a liar’s dominating set and every liar’s dominating set must be a double dominating set. So, the li...

A dominating coloring by k colors is a proper k coloring where every color i has a representative vertex xi adjacent to at least one vertex in each of the other classes. The b-chromatic number, b(G), of a graph G is the largest integer k such that G admits a dominating coloring by k colors. A graph G = (V,E) is said b-monotonous if b(H1) ≥ b(H2) for every induced subgraph H1 of G and every subg...

Let G = (V, E) be a graph. Let D be a minimum secure total dominating set of G. If V – D contains a secure total dominating set D' of G, then D' is called an inverse secure total dominating set with respect to D. The inverse secure total domination number γst(G) of G is the minimum cardinality of an inverse secure total dominating set of G. The disjoint secure total domination number γstγst(G) ...