The problem of monitoring an electric power system by placing as few phase measurement units (PMUs) in the system as possible is closely related to the well-known domination problem in graphs. The power domination number γp(G) is the minimum cardinality of a power dominating set of G. In this paper, we investigate the power domination problem in Mycielskian and generalized Mycielskian of graphs...

We calculate the amplitudes of J = 3−− meson production in diffractive DIS within the kt-factorization approach, with a particular attention paid to the ρ3(1690) meson. We find that at all Q2 the ρ3(1690) production cross section is 2–5 times smaller than the ρ(1700) production cross section, which is assumed to be a pure D-wave state. Studying σL and σT separately, we observe domination of ρ3 ...

A graph is k-domination-critical if γ(G)=k, and for any edge e not in G, γ(G+e) = k-1. In this paper we show that the diameter of a domination k-critical graph with k≥2 is at most 2k-2. We also show that for every k≥2, there is a k-domination-critical graph having diameter 3 2 k 1 −     . We also show that the diameter of a 4-domination-critical graph is at most 5.

Let denote the Cartesian product of graphs and A total dominating set of with no isolated vertex is a set of vertices of such that every vertex is adjacent to a vertex in The total domination number of is the minimum cardinality of a total dominating set. In this paper, we give a new lower bound of total domination number of using parameters total domination, packing and -domination numbers of ...

A caterpillar is a tree with the property that after deleting all its vertices of degree 1 a simple path is obtained. The signed 2-domination number γ s (G) and the signed total 2-domination number γ st(G) of a graph G are variants of the signed domination number γs(G) and the signed total domination number γst(G). Their values for caterpillars are studied.

‎In this paper‎, ‎we investigate domination number‎, ‎$gamma$‎, ‎as well‎ ‎as signed domination number‎, ‎$gamma_{_S}$‎, ‎of all cubic Cayley‎ ‎graphs of cyclic and quaternion groups‎. ‎In addition‎, ‎we show that‎ ‎the domination and signed domination numbers of cubic graphs depend‎ on each other‎.

A set S of vertices of a graphG = (V,E) is a dominating set if every vertex of V (G)\S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Velammal ...

We study the parameterized complexity of domination-type problems. (σ, ρ)-domination is a general and unifying framework introduced by Telle: a set D of vertices of a graph G is (σ, ρ)dominating if for any v ∈ D, |N(v) ∩ D| ∈ σ and for any v / ∈ D, |N(v) ∩ D| ∈ ρ. We mainly show that for any σ and ρ the problem of (σ, ρ)-domination is W[2] when parameterized by the size of the dominating set. T...

In11, Kulli and Janakiram initiate the concept of maximal domination in graphs. In this paper, we obtained some bounds characterizations. Also, estimate value number graph products such as join graphs, corona product, cartesian product strong product.

Let Y be a subset of real numbers. A Y dominating function of a graph G = (V, E) is a function f : V → Y such that u∈NG[v] f(u) ≥ 1 for all vertices v ∈ V , where NG[v] = {v} ∪ {u|(u, v) ∈ E}. Let f(S) = u∈S f(u) for any subset S of V and let f(V ) be the weight of f . The Y -domination problem is to find a Y -dominating function of minimum weight for a graph. In this paper, we study the variat...