Showing papers on "Dominating set published in 1987"
TL;DR: This work suggests a general approach for constructing linear-time algorithms in the case where the graph G is defined by certain rules of composition and the desired subgraph H satisfies a property that is “regular” with respect to theserules of composition.
236 citations
TL;DR: Sharp bounds on the cardinality of locating-dominating sets for arbitrary graphs on p vertices and for trees on p trees are given, and a linear algorithm for finding a minimum cardinality locating-Dominating set in an acyclic graph is presented.
Abstract: Locating-dominating sets are of interest in safeguard applications of graphical models of facilities. A subset S of the vertex set V of a graph G is a dominating set if each vertex u ϵ V - S is adjacent to at least one vertex in S. For each v in V - S let S(v) denote the set of vertices in S which are adjacent to v. A dominating set S is defined to be “locating” if for any two vertices v and w in V - S one has S(v) ≠ S(w). Sharp bounds on the cardinality of locating-dominating sets for arbitrary graphs on p vertices and for trees on p vertices are given, and a linear (that is O(P)) algorithm for finding a minimum cardinality locating-dominating set in an acyclic graph is presented.
211 citations
TL;DR: It is shown that the problems steiner tree, dominating set and connected dominating set are NP-complete for chordal bipartite graphs.
127 citations
TL;DR: The shared memory model of parallel computers is used to obtain fast algorithms for finding maximum weighted clique, a maximum weighted independent set, a minimum clique cover, and a minimum weighted dominating set of an interval graph.
47 citations
TL;DR: This paper extends the normal paradigm of dynamic programming to allow a polynomial number of optimal solutions to be computed for each subproblem to yield aPolynomial time algorithm for the dominating set problem on k-trees, where k is fixed.
Abstract: Dynamic programming has long been established as an important technique for demonstrating the existence of polynomial time algorithms for various discrete optimization problems. In this paper we extend the normal paradigm of dynamic programming to allow a polynomial number of optimal solutions to be computed for each subproblem. This technique yields a polynomial time algorithm for the dominating set problem on k-trees, where k is fixed. It is also shown that the dominating set problem is NP-complete for k-trees where k is arbitrary.
45 citations
TL;DR: The line dominating sets are studied and Nordhaus-Gaddum type results are obtained for the line domination number and the line domatic number.
Abstract: A setS of lines is a line dominating set if every line not inS is adjacent to some line ofS. The line domination number of a graph is the cardinality of a minimum line dominating set. In this paper we study the line dominating sets and obtain bounds for the line domination number. Also, Nordhaus-Gaddum type results are obtained for the line domination number and the line domatic number.
34 citations
01 Jan 1987
TL;DR: An emerging theory/methodology for constructing linear-time graph algorithms is illustrated by providing such algorithms for finding the maximum-cut and the maximum cardinality of a minimal dominating set for a generalized-series-parallel graph.
Abstract: This paper extends in several ways the notable work of Takamizawa, Nishizeki and Saito in 1982 [16], which in turn was inspired by that of Watanabe, Ae and Nakamura in 1979 [17]. We illustrate an emerging theory/methodology for constructing linear-time graph algorithms by providing such algorithms for finding the maximum-cut and the maximum cardinality of a minimal dominating set for a generalized-series-parallel graph.
31 citations
Journal Article•
8 citations
TL;DR: The problem is seen to be equivalent to a standard problem in combinatories —that of determining the maximum independent set of a given undirected graph, which is NP-complete with probable exponential computer-time requirements, and therefore is intractable except for the smallest of graphs.