Dominating set games

Abstract

In this paper, we study cooperative cost games arising from domination problems on graphs. We introduce three games to model the cost allocation problem and we derive a necessary and sufficient condition for the balancedness of all three games.

Introduction

In this paper, we consider cooperative cost games that arise from domination problems on graphs. A domination problem consists of a given graph G=(V,E), a positive integer $\text{k∈}\text{N}$, and a non-negative function $\text{w:V→}\text{R}{}_{\mathrm{+}}$ that assigns a fixed cost to each vertex. A k-dominating set is a set D⊂V such that the distance between any vertex in V and at least one vertex in D is at most k. A k-domination problem is the problem of finding a so-called minimum weighted k-dominating set of G, ie. a k-dominating set that minimizes the total cost of its vertices.

Domination problems are widely studied in graph theory. In [12] domination problems are investigated where the underlying graph is a tree. Some results of [12] are extended to larger classes of graphs in [5]. In [8] an overview of literature on domination problems is given.

An illustration of a domination problem is the following example. Consider a number of regions in which certain facilities are going to be placed. There is a fixed cost for the placement of a facility in a certain region. The problem is to select the regions in which to place facilities at minimum costs, such that each region is served by a facility in it or by a facility in a neighbouring region. The problem of placing the facilities at minimum costs can be regarded as a domination problem. Let G=(V,E) be the graph where regions correspond to vertices and edges represent pairs of regions that are neighbours. The fixed cost can be described by a map $\text{w:V→}\text{R}{}_{\mathrm{+}}$. The problem of placing the facilities at minimum cost is equivalent to finding a minimum-weighted 1-dominating set on G.

A natural question that now arises is how to allocate the total costs of placing the facilities among the participating regions. In this paper, we use cooperative game theory to study this problem. We introduce three cooperative cost games that model the cost allocation problem.

The three dominating set games have in common is that the cost of the grand coalition N equals the minimum-weighted k-domination number. However, the cost of coalitions may take different values in each of the three games. This is due to the fact that the coalitions have different possibilities of placing the facilities in each of the three games. In the relaxed dominating set game coalitions are allowed to use every vertex and every edge of the graph. These games belong to the class of combinatorial optimization games introduced in [4].

In some situations it makes sense to assume that the coalitions will place facilities in their own region. For instance, because regions outside the coalition do not allow the placement of a facility in their region. Therefore, we introduce the intermediate dominating set game. In this game coalitions are only allowed to place facilities in their own regions. However, coalitions are still allowed to use every edge of the graph. Consider for example the situation where the facilities are libraries. Then coalitions will place the libraries in their own regions, but inhabitants of one region may freely travel through regions outside the coalition in order to reach a library.

Now consider a situation where the facilities are power stations. Then coalitions will place the power stations in their own regions. However, cooperation between non-adjacent members of a coalition can be obstructed by regions outside the coalition, if these regions do not allow transportation of electricity through their region. For this reason we define the rigid dominating set game. In this game coalitions are only allowed to use vertices and edges of the induced subgraph corresponding to the coalition.

In spite of the differences between these games, we will obtain a common necessary and sufficient condition for non-emptiness of the cores of all three dominating set games. In particular, if one of the dominating set games possesses a core element, then the other two dominating set games possess core elements as well. We also derive relations between the cores of the dominating set games. Furthermore, we present a class of graphs for which the corresponding dominating set games have a non-empty core for all cost functions $\text{w:V→}\text{R}{}_{\mathrm{+}}$ and all $\text{k∈}\text{N}$.

Other game theoretical approaches to location problems include facility location games (cf. [9], [13]) and minimum spanning forest games [7]. In [9], [13] a general class of facility location problems is studied. They mainly focus on tree graphs, and for these graphs non-emptiness of the core is established. In [7] no restrictions are made on the proximity of the facilities. If the underlying graph is a tree, non-emptiness of the core is shown.

This paper is organized as follows. In Section 2 we recall notions from cooperative game theory, graph theory and matrix theory. In Section 3, we introduce three cooperative cost games that model the cost allocation problem arising from domination problems on graphs. In Section 4, we focus on the core of the dominating set games.