Showing papers on "Dominating set published in 1983"
TL;DR: The connected 3-critical graphs of even order are shown to have a 1-factor and some stringent restrictions on their degree sequences and diameters are obtained.
176 citations
TL;DR: Some upper bounds on the size of a minimum set of lines which when removed from G increases the domination number and if T is a tree with at least three points then @a(T - v) > @a (T) if and only if @n is in every minimum dominating set of T.
158 citations
07 Nov 1983
TL;DR: A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set.
Abstract: This paper describes a general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs. The strategy depends on decomposing a planar graph into subgraphs of a form we call k- outerplanar. For fixed k, the problems of interest are solvable optimally in linear time on k-outerplanar graphs by dynamic programming. For general planar graphs, if the problem is a maximization problem, such as maximum independent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least (k-1)/k optimal. If the problem is a minimization problem, such as minimum vertex cover, it gives for each k a linear time algorithm that produces a solution whose size is at most (k + 1)/k optimal. Taking k = c log log n or k = c log n, where n is the number of nodes and c is some constant, we get polynomial time approximation schemes, i.e. algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approximation schemes includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k-outerplanar graphs also enlarges the class of planar graphs for which the problems are known to be solvable.
92 citations
TL;DR: A linear time algorithm for finding a minimum dominating set in a series-parallel graph and its solution to the well-known NP-complete problem of determining whether G has a dominating set D in G satisfying ¦D¦ ≤ k.
78 citations
Journal Article•
73 citations
Proceedings Article•
01 Jan 1983
TL;DR: In this paper, the authors present a general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs by decomposing a planar graph into subgraphs of a form called k-outerplanar.
Abstract: ABSTRACf This paper describes a general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs. The strategy depends on decom posing a planar graph into subgraphs of a form we call k outerplanar. For fixed k, the problems of interest are solv able optimally in linear time on k -outerplanar graphs by dynamic programming. For general planar graphs, if the problem is a maximization problem, such as maximum in dependent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least (k -1)/k optimal. If the problem is a minimization problem, such as minimum vertex cover, it gives for each k a linear time algorithm that produces a solution whose size is at most (k+l)/k optimal. Taking k-cloglogn or k-clogn, where n is the number of nodes and c is some constant, we get po lynomial time approximation schemes, i.e. algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approxi mation schemes includes maximum independent set, max imum tile salvage, partition into triangles, maximum H matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k -outerplanar graphs also enlarges the class of planar graphs for which the problems are known to be solvable.
48 citations
23 citations
Journal Article•
10 citations
TL;DR: In this paper, the authors characterize the polytope of the dominants in this class of graphs, using a polynomial algorithm to find a minimum weight dominating set, which is then used to characterize the dominating set.
Abstract: A graph G is defined to be domishold (Benzaken and Hammer (1978)) if there exist real positive numbers associated to their vertices so that a set of vertices is dominating if and only if the sum of the corresponding weights exceeds a certain threshold b. In this paper, we characterize the polytope of the dominants in this class of graphs, using a polynomial algorithm to find a minimum weight dominating set.
8 citations
TL;DR: It is shown that every connected graph has a spanning tree such that a MDS for the tree is also a M DS for the graph and the “dual” of the set covering problem is proved to have the property that a star packing is maximum if and only if the graph contains no augmenting packing.
Abstract: The minimun dominating set (MDS] problem is to locate a minimum number of facilities at nodes or a graph so that every other node is connected by an edge to at least one facility. For a general graph, the MDS problem is equivalent to the set covering problem and is NP-complete; however, for a tree graph, polynomial time algorithms exist. This paper presents some results that extend the minimum dominating set problem in an effort to lead to new solution procedures. First, it is shown that every connected graph has a spanning tree such that a MDS for the tree is also a MDS for the graph. Second, the “dual” of the set covering problem, called the star-packing problem, is proved to have the property that a star packing is maximum if and only if the graph contains no augmenting packing.