Let G = (V , E) be a graph. A subset D ⊆ V is a dominating set if every vertex not in D is adjacent to a vertex in D. A dominating set D is called a total dominating set if every vertex in D is adjacent to a vertex in D. The domination (resp. total domination) number of G is the smallest cardinality of a dominating (resp. total dominating) set of G. The bondage (resp. total bondage) number of a...

In this paper we compute for paths and cycles certain graph domination invariants like locating domination number, differentiating domination number, global alliance number etc., We also do some comparison analysis of certain parameters defined by combining the domination measures and the second smallest eigen value of the Laplacian matrix of all connected graphs of order 4.While discussing app...

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...

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.

Eternal inflation is a term that describes a number of different phenomena which have been classified by Winitzki. According to Winitzki’s classification these phases can be characterized by the topology of the percolating structures in the inflating, “white,” region. In this paper we discuss these phases, the transitions between them, and the way they are seen by a “Census Taker”; a hypothetic...

‎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.

We present an interpretation of the physics of space-times undergoing eternal inflation by repeated nucleation of bubbles. In many cases the physics can be interpreted in terms of the quantum mechanics of a system with a finite number of states. If this interpretation is correct, the conventional picture of these space-times is misleading.

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 ...