A set S of vertices is defined to be a power dominating set (PDS) of a graph G if every vertex and every edge in G can be monitored by the set S according to a set of rules for power system monitoring. The minimum cardinality of a PDS of G is its power domination number. In this article, we find upper bounds for the power domination number of some families of Cartesian products of graphs: the c...

Let γt(G) denote the total domination number of the graph G. G is said to be total domination edge critical, or simply γt-critical, if γt(G+ e) < γt(G) for each edge e ∈ E(G). In this paper we study a family H of 4-critical graphs with diameter three, in which every vertex is a diametrical vertex, and every diametrical pair dominates the graph. We also generalize the self-complementary graphs, ...

We use the domination number of a parametrized random digraph family called proportional-edge proximity catch digraphs (PCDs) for testing multivariate spatial point patterns. This digraph family is based on relative positions of data points from various classes. We extend the results on the distribution of the domination number of proportional-edge PCDs, and use the domination number as a stati...

The bondage number b(G) of a graph G is the smallest number of edges of G whose removal results in a graph having the domination number larger than that of G. In a sense, the bondage number b(G) measures integrity and reliability of the domination number γ(G) with respect to the edge removal, which corresponds, e.g., to link failures in communication networks. We show that, for a graph G having...

A Roman dominating function on a graph G = (V,E) is a function f : V −→ {0, 1, 2} satisfying the condition that every vertex v for which f(v) = 0 is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function is the value w(f) = ∑ v∈V f(v). The Roman domination number of a graph G, denoted by γR(G), equals the minimum weight of a Roman dominating function on ...

For the terminology and notations not defined here, we adopt those in Bondy and Murty [1] and Xu [2] and consider simple graphs only. Let G = (V,E) be a graph with vertex set V = V (G) and edge set E = E(G). For any vertex v ∈ V , NG(v) denotes the open neighborhood of v in G and NG[v] = NG(v) ∪ {v} the closed one. dG(v) = |NG(v)| is called the degree of v in G, ∆ and δ denote the maximum degre...