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...

The core of a market in indivisible goods can be defined in terms of strong domination or weak domination. The core defined by strong domination is always non-empty, but may contain points which are unstable in a dynamic sense. However, it is shown that there are always stable points in the core, and a characterization is obtained. The core defined by weak domination is always non-empty when th...

Let k be a positive integer. A vertex subset D of a graph G = (V,E) is a perfect k-dominating set of G if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γkp(G). In this paper, we generalize perfect domination to perfect k-domination, where many bounds of γkp(G) are obtained. We ...

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and –1 such that the closed neighbourhood of every vertex contains more +1’s than –1’s. This concept is closely related to combinatorial discrepancy theory as shown by Füredi and Mubayi [J. Combin. Theory, Ser. B 76 (1999) 223–239]. The signed domination number of G is the minimum of the sum...

A set D ⊆ V (G) is a dominating set of G if every vertex not in D is adjacent to at least one vertex in D. A dominating set of G of minimum cardinality is called a γ(G)-set. For each vertex v ∈ V (G), we define the domination value of v to be the number of γ(G)-sets to which v belongs. In this paper, we study some basic properties of the domination value function, thus initiating a local study ...

In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph G, called the sub-k-domination number and denoted subk(G). We show that subk(G) is a computationally efficient sharp lower bound on the k-domination number of G, and improves on several known lower bounds. We also characterize the sub-k-domination numbers of several families of graphs, provide...