On the r-domination number of a graph

The (classical) domination number of a graph is the cardinality of a smallest subset of its vertex set with the property that each vertex of the graph is in the subset or adjacent to a vertex in the subset. Since its introduction to the literature during the early 1960s, this graph parameter has been researched extensively and nds application in the generic facility location problem where a smallest number of facilities must be located on the vertices of the graph, at most one facility per vertex, so that there is at least one facility in the closed neighbourhood of each vertex of the graph. The placement constraint in the above application may be relaxed in the sense that multiple facilities may possibly be located at a vertex of the graph and the adjacency criterion may be strengthened in the sense that a graph vertex may possibly be required to be adjacent to multiple facilities. More speci cally, the number of facilities that can possibly be located at the i-th vertex of the graph may be restricted to at most ri ≥ 0 and it may be required that there should be at least si ≥ 0 facilities in the closed neighbourhood of this vertex. If the graph has n vertices, then these restriction and su ciency speci cations give rise to a pair of vectors r = [r1, . . . , rn] and s = [s1, . . . , sn]. The smallest number of facilities that can be located on the vertices of a graph satisfying these generalised placement conditions is called the 〈r, s〉-domination number of the graph. The classical domination number of a graph is therefore its 〈r, s〉-domination number in the special case where r = [1, . . . , 1] and s = [1, . . . , 1]. The exact values of the 〈r, s〉-domination number, or at least upper bounds on the 〈r, s〉domination number, are established analytically in this dissertation for arbitrary graphs and various special graph classes in the general case, in the case where the vector s is a step function and in the balanced case where r = [r, . . . , r] and s = [s, . . . , s]. A linear algorithm is put forward for computing the 〈r, s〉-domination number of a tree, and two exponential-time (but polynomial-space) algorithms are designed for computing the 〈r, s〉domination number of an arbitrary graph. The e ciencies of these algorithms are compared to one another and to that of an integer programming approach toward computing the 〈r, s〉domination number of a graph. iii Stellenbosch University http://scholar.sun.ac.za