The power domination problem is to find a minimum placement of phase measurement units (PMUs) for observing the whole electric power system, which is closely related to the classical domination problem in graphs. For a graph G = (V , E), the power domination number of G is the minimum cardinality of a set S ⊆ V such that PMUs placed on every vertex of S results in all of V being observed. A ver...

A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of f is w(f) = ∑ v∈V f(v). The Roman domination number is the minimum weight of an RDF in G. It is known that for every graph G, the Roman domination number of G is bounded above...

For a graph G a subset D of the vertex set of G is a k-dominating set if every vertex not in D has at least k neighbors in D. The k-domination number γk(G) is the minimum cardinality among the k-dominating sets of G. Note that the 1-domination number γ1(G) is the usual domination number γ(G). Fink and Jacobson showed in 1985 that the inequality γk(G) ≥ γ(G) + k − 2 is valid for every connected ...

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [S. Arumugam, C. Sivagnanam, Neighborhood total domination in graphs, Opuscula Math. 31 (2011) 519–531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph G is a dominating set S in G with the property that the subgraph induced by the open neighborhood of t...

Let G = (V,E) be a graph without isolated vertices. A set S ⊆ V is a paired-domination set if every vertex in V − S is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang an...

Let γ(G) and ι(G) be the domination and independent domination numbers of a graph G, respectively. In this paper, we define the Price of Independence of a graph G as the ratio ι(G) γ(G) . Firstly, we bound the Price of Independence by values depending on the number of vertices. Secondly, we consider the question of computing the Price of Independence of a given graph. Unfortunately, the decisio...

The recently introduced concept of k-power domination generalizes domination and power domination, the latter concept being used for monitoring an electric power system. The k-power domination problem is to determine a minimum size vertex subset S of a graph G such that after setting X = N [S], and iteratively adding to X vertices x that have a neighbour v in X such that at most k neighbours of...

Input data for communication network design/optimization problems involving multi-hour or uncertain traffic can consist of a large set of traffic matrices. These matrices are explicitly considered in problem formulations for link dimensioning. However, many of these matrices are usually dominated by others so only a relatively small subset of matrices would be sufficient to obtain proper link c...

This paper examines the relationship between coalition-proof Nash equilibria based on different dominance relations. Konishi, Le Breton, and Weber (1999) pointed out that the set of coalition-proof Nash equilibria under weak domination does not necessarily coincide with that under strict domination. We show that, if a game satisfies the conditions of anonymity, monotone externality, and strateg...

A graph G is dot-critical if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. We show that the totally dot-critical graphs essentially include the much-studied domination vertex-critical and edge-critical graphs as special cases. We investigate these properties, and provide a characterization of dot...