The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of O(H (\Delta)) are presented, where \Delta is the maximum degree, and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has at least one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln n. We also consider the ...

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

We describe four versions of a Greedy Randomized Adaptive Search Procedure (GRASP) for finding approximate solutions of general instances of the Steiner Problem in Graphs. Di#erent construction and local search algorithms are presented. Preliminary computational results with one of the versions on a variety of test problems are reported. On the majority of instances from the OR-Library, a set of standard test problems, the GRASP produced optimal solutions. On those that optimal solutions were not found, the GRASP found good quality approximate solutions.

Abstract Let G = (V; E) be a connected graph. A connected dominating set S ae V is a dominating set that induces a connected subgraph of G. The connected domination number of G, denoted fl

In this paper those graphs are studied for which a so-called strong ordering of the vertex set exists. This class of graphs, called strongly orderable graphs, generalizes the strongly chordal graphs and the chordal bipartite graphs in a quite natural way. We consider two characteristic elimination orderings for strongly orderable graphs, one on the vertex set and the second on the edge set, and prove that these graphs can be recognized in O(|V | + |E|)|V | time. Moreover, a special strong ordering of a strongly orderable graph can be produced in the same time bound. We present variations of greedy algorithms that compute a minimum coloring, a maximum clique, a minimum clique partition and a maximum independent set of a strongly orderable graph in linear time if such a special strong ordering is given. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Strongly chordal graphs; Chordal bipartite graphs; Strong ordering; Simple elimination ordering; Bisimplicial edge withou...

HT-graphs have been introduced in [11] and investigated with respect to location problems on graphs. In this paper two new characterizations of these graphs are given and then it is shown that the central vertex, connected r-domination and Steiner trees problems are linear or almost linear time solvable in HT-graphs.

In the last quarter of the century there has been a growing interest in the design and analysis of algorithms in different branches of mathematics, science and engineering. In particular, within the past 15 years there has appeared a wealth of literature on algorithms in graph theory, a significant part of which is related to domination in graphs. Since there are well over one hundred papers on domination or domination-related algorithms, this chapter will necessarily have to focus on a limited collection of these algorithms. The intention of this chapter on domination algorithms for special graph classes is twofold. The major goal is to demonstrate some of the main techniques for designing efficient algorithms for domination problems. For this purpose we consider some particular examples and focus on five graph classes and five well researched variants of the domination problem. Another goal is to present a fairly comprehensive bibliography of algorithms for domination problems. We refer the reader who is unfamiliar with the design and analysis of algorithms to [57].

One way of overcoming the intractable nature of problems in graphs is by considering them on special classes of graphs. In this thesis, we study the properties of two important classes of graphs -- asteroidal triple-free graphs and distancehereditary graphs, and use their properties to design efficient polynomial-time algorithms for a variety of problems that are NP -complete on general graphs. Three independent vertices in a graph are said to constitute an asteroidal triple (AT) if there is a path between every two of them that avoids the neighbourhood of the third. Graphs which do not have an AT are said to be AT-free. These graphs strictly contain the well-known class of cocomparability graphs, but are not necessarily perfect. We present O(n 3 ) algorithms for the problems of finding a minimum cardinality connected dominating set and a Steiner set on these graphs. These are the first known algorithms for these problems on any non-trivial class of non-perfect graphs -- in addition...

Abstract. The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(�) + 2 and H(�) + 2 are presented, where � is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (cn + 1) ln n where cn ln k is the approximation factor for the node weighted Steiner tree problem (currently cn = 1.6103). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c + 1)H(�) + c − 1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644).