Showing papers on "Dominating set published in 1991"
TL;DR: The bondage number b(G) of a graph G is defined to be the cardinality of a smallest set E of edges for which σ(G−E)>σ(G).
160 citations
TL;DR: A graph G is well-covered if every maximal independent set of points in G is also maximum as discussed by the authors, which is equivalent to the property that the greedy algorithm for constructing a maximal independent subset always results in a maximum independent set.
Abstract: A graph G is well-covered (or w-c) if every maximal independent set of points in G is also maximum. Clearly, this is equivalent to the property that the greedy algorithm for constructing a maximal independent set always results in a maximum independent set. Although the problem of independence number is well-known to be NP-complete, it is trivially polynomial for well-covered graphs. The concept of well-coveredness was introduced by the author in [P1] and was first discussed therein with respect to its relationship to a number of other properties involving the independence number. Since then, a number of results about well-covered graphs have been obtained. It is our purpose in this paper to survey these results for the first time. As the reader will see, many of the results we will discuss are quite recent and have not as yet appeared in print.
152 citations
TL;DR: This paper presents an extremely simple O(n) algorithm which simultaneously solves the following three problems on circular-arc graphs: the maximum independent set, the minimum clique cover, and the minimum dominating set problems, whereas the best previous bounds for the latter two problems were O (n2) and O( n3), respectively.
110 citations
TL;DR: The literature is surveyed and some theorems that subsume most previous results and are more general than previous results are proved, aimed primarily at investigators who wish to know whether an FDS exists for a specific problem.
Abstract: A research theme involving location on networks, since its inception, has been the identification of a finite dominating set (FDS), or a finite set of points to which an optimal solution must belong. We attempt to unify and generalize results of this sort. We survey the literature and then prove some theorems that subsume most previous results and that are, at the same time, more general than previous results. The paper is aimed primarily at investigators who wish to know whether an FDS exists for a specific problem.
109 citations
TL;DR: It is shown that, unless P = NP, no polynomial-time approximation algorithm can guarantee to find an independent dominating set with size within a factor of K of the optimal, where K is any fixed constant >1.
71 citations
TL;DR: For certain classes of graphs, a polynomial algorithm is given for finding a dominating clique and Forbidden subgraph conditions sufficient to imply the existence of a dominatingClique are given.
65 citations
TL;DR: The complexity of the minimum cardinality dominating set problem and some of its variations on several families of perfect graphs are reviewed, with emphasis on the dynamic programming approach to the design of algorithms.
64 citations
TL;DR: A simple greedy algorithm for finding small dominating sets in undirected graphs of N nodes and M edges is analyzed and it is shown that dg < N + 1 - v2M + 1, where dg is the cardinality of the dominating set returned by the algorithm.
60 citations
TL;DR: This paper shows that the upper irredundance number and the upper domination number are equal for all chordal graphs, and a class of graphs not containing a set of forbidden subgraphs.
54 citations
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01 Jan 1991
TL;DR: In this paper, the authors introduce the concept of dominating sets in directed graphs and present a model for finding dominating cliques efficiently in strongly chordal graphs and undirected path graphs.
Abstract: Introduction (S.T. Hedetniemi, R.C. Laskar). Theoretical. Chessboard Domination Problems (E.J. Cockayne). On the Queen Domination Problem (C.M. Grinstead, B. Hahne, D. Van Stone). Recent Problems and Results About Kernels in Directed Graphs (C. Berge, P. Duchet). Critical Concepts in Domination (D.P. Summer). The Bondage Number of a Graph (J.F. Fink et al.). Chordal Graphs and Upper Irredundance, Upper Domination and Independence (M.S. Jacobson and K. Peters). Regular Totally Domatically Full Graphs (B. Zelinka). Domatically Critical and Domatically Full Graphs (D. Rall). On Generalised Minimal Domination Parameters for Paths (B. Bollobas, E.J. Cockayne, C.M. Mynhardt). New Models. Dominating Cliques in Graphs (M.B. Cozzens, L.L. Kelleher). Covering all Cliques of Graph (Z. Tuza). Factor Domination in Graphs (R.C. Brigham, R. Dutton). The Least Point Covering and Domination Numbers of a Graph (E. Sampathkumar). Algorithmic. Dominating Sets in Perfect Graphs (D.G. Corneil, L.K. Stewart). Unit Disk Graphs (B.N. Clark, C.J. Colbourn, D.S. Johnson). Permutation Graphs: Connected Domination and Steiner Trees (C.J. Colbourn, L.K. Stewart). The Discipline Number of a Graph (V. Chvatal, W.J. Cook). Best Location of Service Centers in a Tree-Like Network under Budget Constraints (J. McHugh, Y. Perl). Dominating Cycles in Halin Graphs (M. Skowronska, M.M. Syslo). Finding Dominating Cliques Efficiently, in Strongly Chordal Graphs and Undirected Path Graphs (D. Kratsch). On Minimum Dominating Sets with Minimum Intersection (D.L. Grinstead, P.J. Slater). Bibliography. Bibliography on Domination in Graphs and Some Basic Definitions of Domination Parameters (S.T. Hedetniemi, R.C. Laskar).
37 citations
TL;DR: An upper bound on the number of edges a connected graph with a given number of vertices and a given domination number can have is given and the extremal graphs attaining this upper bound are characterized.
TL;DR: This paper presents a linear algorithm for finding two minimum dominating sets with minimum possible intersection in a tree T, and shows that simply determining whether or not there exist two disjoint minimum dominate sets is NP-hard for arbitrary bipartile graphs.
TL;DR: The subcube identification problem is addressed and the problem is modeled as a graph theoretical problem and shown to be NP complete.
TL;DR: A set S ⊂ V of a graph G is a total point cover (t.p.c.) if S is a point cover containing all isolates of G, if any, and the number α t ( G ) is the minimum cardinality of a t.
TL;DR: It is proved that a domatically critical graph G is domatically full if d ( G ) ⩽ 3 and examples to show this does not extend to the cases d (G )>3.
TL;DR: It is shown that the problem of finding minimum abort set is polynomial-time solvable for such wait-for graphs that are edge-disjoint reducible.
TL;DR: This paper investigates graphs on well-ordered sets of type ?