Showing papers on "Dominating set published in 1989"
TL;DR: It is shown that if a connected graph G has minimum degree two and is not one of seven exceptional graphs, then γ(G)γ 2/5|V|, which is the minimum cardinality of a subset of V such that every vertex is either in the set or is adjacent to some vertex in theSet.
Abstract: The domination number γ(G) of a graph G = (V, E) is the minimum cardinality of a subset of V such that every vertex is either in the set or is adjacent to some vertex in the set. We show that if a connected graph G has minimum degree two and is not one of seven exceptional graphs, then γ(G)γ 2/5|V|. We also characterize those connected graphs with γ(G)γ 2/5|V|.
157 citations
01 Jan 1989
TL;DR: A set of vertices of a graph G is said to be a global dominating set of G if it dominates G as well as G¯¯¯ as discussed by the authors and the global domination number of G is the minimum cardinality of a globally dominating set.
Abstract: A set of vertices of a graph G is said to be a global dominating set of G if it dominates G as well as G¯¯¯. The global domination number of G is the minimum cardinality of a global dominating set of G and the global domatic number of G is the maximum order of a partition {V1,⋯,Vn} of V such that each Vi is a global dominating set. The author obtains several easy bounds for these parameters. The paper is marred by a vast number of printing errors, particularly omissions of mathematical symbols.
71 citations
TL;DR: This work gives an algorithm for optimally solving a class of maxdominance problems, including the problem of computing a minimum independent dominating set in a permutation graph, the related problem of finding the shortest maximal increasing subsequence, and the issue of enumerating restricted empty rectangles.
Abstract: Given a planar setS ofn points,maxdominance problems consist of computing, for everyp eS, some function of the maxima of the subset ofS that is dominated byp A number of geometric and graph-theoretic problems can be formulated as maxdominance problems, including the problem of computing a minimum independent dominating set in a permutation graph, the related problem of finding the shortest maximal increasing subsequence, the problem of enumerating restricted empty rectangles, and the related problem of computing the largest empty rectangle We give an algorithm for optimally solving a class of maxdominance problems A straightforward application of our algorithm yields improved time bounds for the above-mentioned problems The techniques used in the algorithm are of independent interest, and include a linear-time tree computation that is likely to arise in other contexts
51 citations
TL;DR: In this paper, a linear time algorithm for finding a minimum cardinality total dominating set of a block graph is presented, where every vertex in V is adjacent to a vertex in D. The algorithm is based on the concept of dominating set.
20 citations
17 Aug 1989
TL;DR: It is shown that recognition, transitive orientation, maximum node weighted clique, minimum node coloring, minimum weight dominating set, minimum fill-in and isomorphism for cographs is in NC.
Abstract: Cograph arise in such application areas as examination scheduling and automatic clustering of index terms. It is shown that recognition, transitive orientation, maximum node weighted clique, minimum node coloring, minimum weight dominating set, minimum fill-in and isomorphism for cographs is in NC. The model of computation is CRCW P-RAM.
6 citations
Proceedings Article•
01 Jan 1989
5 citations
18 Oct 1989
TL;DR: Graphs whose blocks are complete subgraphs are said to be block-complete graphs, and polynomial time algorithms to solve several problems, problems which are not believed to be polynomial for general graphs are presented.
Abstract: Graphs whose blocks are complete subgraphs are said to be block-complete graphs. Polynomial time algorithms to solve several problems, problems which are not believed to be polynomial for general graphs, are presented for connected block-complete graphs. These include: finding a minimum vertex cover, finding a minimum dominating set of radius r, and finding a minimum m-centrix radius r augmentation.
1 citations