Showing papers on "Dominating set published in 1979"

An algorithmic approach to network location problems.

Abstract: Problems of finding p-centers and dominating sets of radius r in networks are discussed in this paper. Let n be the number of vertices and |El be the number of edges of a network. With the assumption that the distance-matrix of the network is available, we design an 0(I|EI n *lg n) algorithm for finding an absolute 1-center of a vertex-weighted network and an O(|E| * n + n2 * lg n) algorithm for finding an absolute 1 -center of a vertex-unweighted network (the problem of finding a vertex 1-center of a network is trivial). We show that the problem of finding a (vertex or absolute) p-center (for 1< p < n) of a (vertex-weighted or vertex-unweighted) network, and the problem of finding a dominating set of radius r are NP-hard even in the case where the network has a simple structure (e.g., a planar graph of maximum vertex degree 3). However, we describe an algorithm of complexity 0((tEI| * n2p l/(p _ 1)!) lg n) (respectively, 0(IEIP * n2p l/(p _ 1)!)) for finding an absolute p-center in a vertex-weighted (respectively, vertex-unweighted) network. We proceed by discussing the problems of finding p-centers and dominating sets of networks whose underlying graphs are trees. When the network is a vertex-weighted tree, we obtain the following algorithms: An 0(n * lg n) algorithm for finding the (vertex or absolute) 1-center; an 0(n) algorithm for finding a (vertex or absolute) dominating set of radius r; an 0(n2 * lg n) algorithm for finding a (vertex or absolute) p-center for any 1 < p < n. Some generalizations of these problems are discussed. When the network is a vertex-unweighted tree, 0(n) algorithms for finding the (vertex or absolute) 1-center and an abWolute 2-center are already known; we extend these results by giving an 0(n * lg,-2n) algorithm for finding an absolute p-center (where 3 p < n) and an 0(n * lgP 1n) algorithm for finding a vertex p-center (where 2 p < n). In part II we treat the p-median problem.

On some subclasses of well‐covered graphs

Abstract: A set of points in a graph is independent if no two points in the set are adjacent. A graph is well covered if every maximal independent set is a maximum independent set or, equivalently, if every independent set is contained in a maximum independent set. The well-covered graphs are classified by the Wn property: For a positive integer n, a graph G belongs to class Wn if ≥ n and any n disjoint independent sets are contained in n disjoint maximum independent sets. Constructions are presented that show how to build infinite families of Wn graphs containing arbitrarily large independent sets. A characterization of Wn graphs in terms of well-covered subgraphs is given, as well as bounds for the size of a maximum independent set and the minimum and maximum degrees of points in Wn graphs.

Minimum dominating cycles in 2-trees

Abstract: We consider the class of 2-trees and present a linear time algorithm for finding minimum dominating cycles of such graphs. We stress the use of a particular representation of these graphs called a recursive representation, and some linear operations on directed trees associated with these graphs.