Showing papers on "Dominating set published in 1998"
Book•
01 Jan 1998
TL;DR: Bounds on the domination number domination, independence and irredundance efficiency, redundancy and the duals changing and unchanging domination conditions on the dominating set varieties of domination multiproperty and multiset parameters sums and products of parameters dominating functions frameworks for domination domination complexity and algorithms are presented.
Abstract: Bounds on the domination number domination, independence and irredundance efficiency, redundancy and the duals changing and unchanging domination conditions on the dominating set varieties of domination multiproperty and multiset parameters sums and products of parameters dominating functions frameworks for domination domination complexity and algorithms.
3,265 citations
Book•
01 Jan 1998
TL;DR: A survey of domination-related parameters topics on directed graphs graphs can be found in this article with respect to the domination number bondage, insensitivity, and reinforcement of graph dominating functions.
Abstract: LP-duality, complementarity and generality of graphical subset parameters dominating functions in graphs fractional domination and related parameters majority domination and its generalizations convexity of external domination-related functions of graphs combinatorial problems on chessboards - II domination in cartesian products - Vizing's conjecture algorithms complexity results domination parameters of a graph global domination distance domination in graphs domatic numbers of graphs and their variants - a survey domination-related parameters topics on domination in directed graphs graphs critical with respect to the domination number bondage, insensitivity and reinforcement.
1,289 citations
TL;DR: 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.
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 i
1,026 citations
TL;DR: In this paper, a polynomial time algorithm with a (c+1) H(Δ) +c-1 approximation factor was proposed, where c is the Steiner approximation ratio.
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(Δ)+2 and H(Δ)+2 are presented, where Δ 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(Δ) +c-1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644 ).
639 citations
TL;DR: A survey of the major results on paired domination with an emphasis on bounds for the paired domination number can be found in this paper, where a set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching.
Abstract: A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The minimum cardinality of a paired dominating set of G is the paired domination number of G. This chapter presents a survey of the major results on paired domination with an emphasis on bounds for the paired domination number.
257 citations
TL;DR: This work studies the computational complexity of partitioning the vertices of a graph into generalized dominating sets, parameterized by two sets of nonnegative integers σ and ρ which constrain the neighborhood N(υ) of vertices.
Abstract: We study the computational complexity of partitioning the vertices of a graph into generalized dominating sets. Generalized dominating sets are parameterized by two sets of nonnegative integers σ and ρ which constrain the neighborhood N(υ) of vertices. A set S of vertices of a graph is said to be a (σ, ρ)-set if ∀υ ∈ S : |N(υ) ∩ S| ∈ σ and ∀υ n ∈ S : |N(υ) ∩ S| ∈ ρ. The (k, σ, ρ)-partition problem asks for the existence of a partition V1, V2, ..., Vk of vertices of a given graph G such that Vi, i = 1, 2, ...,k is a (σ, ρ)-set of G. We study the computational complexity of this problem as the parameters σ, ρ and k vary.
110 citations
TL;DR: It is shown that the problem of determining whether G has an efficient edge dominating set is NP-complete when G is restricted to a bipartite graph, and a linear time algorithm is presented to solve the weighted efficient edge domination problem on bipartITE permutation graphs, which form a subclass of bipartites graphs using the technique of dynamic programming.
59 citations
TL;DR: This solution shows that the recognition problem, whether a connected graph G has the property γ ( G ) = β ( G ), is solvable in polynomial time.
41 citations
TL;DR: It is proved that membership in W[2] and thus W,[2]-completeness is proved, which is a natural extension W∗[2], defined in terms of circuit families of depth bounded by a function of the parameter.
35 citations
11 Mar 1998
TL;DR: For appropriately partitioned bounded degree graphs, it is shown that the running time of the algorithm under the P-RAM computational model is of $O(1)$, which is an improvement over the previous best P- RAM complexity for this class of graphs.
Abstract: The parallel construction of maximal independent sets is a useful building block for many algorithms in the computational sciences, including graph coloring and multigrid coarse grid creation on unstructured meshes. We present an efficient asynchronous maximal independent set algorithm for use on parallel computers, for ``well partitioned'''' graphs, that arise from finite element (FE) models. For appropriately partitioned bounded degree graphs, it is shown that the running time of our algorithm under the P-RAM computational model is of $O(1)$, which is an improvement over the previous best P-RAM complexity for this class of graphs. We present numerical experiments on an IBM SP, that confirm our P-RAM complexity model is indicative of the performance one can expect with practical partitions on graphs from FE problems.
33 citations
TL;DR: In this paper, the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an infinite number of) convex functions is addressed.
Abstract: A number of optimization methods require as a first step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this paper, we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an infinite number of) convex functions, and we show how to obtain a convex dominating set in terms of dominating sets of simpler problems. The applicability of the results obtained is illustrated with the statement of new localization results in the fields of linear regression and location.
14 Dec 1998
TL;DR: The minimum weighted independent dominating set and the minimum weighted efficient dominating set in trapezoid graphs can both be found in O(n log n) time, both of the algorithms require only O( n) space.
Abstract: The weighted independent domination problem in trapezoid graphs was solved in O(n2) time [1]; the weighted efficient domination problem in trapezoid graphs was solved in O(n log log n + m) time [8], where m denotes the number of edges in the complement of the trapezoid graph. In this paper, we show that the minimum weighted independent dominating set and the minimum weighted efficient dominating set in trapezoid graphs can both be found in O(n log n) time. Both of the algorithms require only O(n) space. Since m can be as large as Ω(n2), comparing to previous results, our algorithms clearly give more efficient solutions to the related problems.
TL;DR: In this paper, it was shown that the conjecture holds if Y(H)=#γc(H), and for paths Pm and Pn, a lower bound and an upper bound for γ(PmXPn) are obtained.
Abstract: Let G= (V,E) be a simple graph. A subset D of V is called a dominating set of G if for every vertex κ,eV—D,κ is adjacent to at least one vertex of D. Let γ(G) and γc(G) denote the domination and connected domination number of G, respectively. In 1965,Vizing conjectured that if GXH is the Cartesian product of G and H, then
$$\gamma (G \times H) \geqslant \gamma (G) \cdot \gamma (H).$$
. In this paper, it is showed that the conjecture holds if Y(H)=#γc(H). And for paths Pm and Pn, a lower bound and an upper bound for γ(PmXPn) are obtained.
TL;DR: In this paper, the authors extend the dominating set for the 1-facility centdian problem to the corresponding 2-centdian problems and propose a solution procedure for a network that improves the complexity of the exhaustive search in dominating set.
12 Aug 1998
TL;DR: An optimal O(n) algorithm for finding a dominating set D of an undirected graph G such that every vertex not in D is adjacent to at least one vertex in D.
Abstract: A dominating set D of an undirected graph G is a set of vertices such that every vertex not in D is adjacent to at least one vertex in D. Given a undirected graph G, the minimal cardinality dominating set problem is to find a dominating set of G with minimum number of vertices. The minimal cardinality dominating set problem is NP-hard for general graphs. For permutation graphs, the best-known algorithm ran in O(n log log n) time, where n is the number of vertices. In this paper, we present an optimal O(n) algorithm.
01 Jan 1998
TL;DR: In this paper, it was shown that the conjecture holds if γ(H)≠γσ(H), and for paths Pm and Pn+ a lower bound and an upper bound for γ (Pm ×Pn) are obtained.
Abstract: Let G= (V,E) be a simple graph. A subset D of V is called a dominating set of G iffor every vertex x∈V--D,x is adjacent to at least one vertex of D. Let γ(G) and γ,(G) denotethe domination and connected domination number of G, respectively. In 1965, Vizing conjectured that if G×H is the Cartesian product of G and H, then γ(G × H) ≥ γ(G) . γ(H).In this paper, it is showed that the conjecture holds if γ(H)≠γσ(H). And for paths Pm and Pn+ a lower bound and an upper bound for γ(Pm ×Pn) are obtained.
TL;DR: Gallai-type equalities for the strong P -domination number are proved, which generalize Nieminen's result.
TL;DR: The domatic number of all not domatically full graphs of this class of graphs is determined, which extends a result of Zelinka (1986).
14 Dec 1998
TL;DR: The inapproximability of non NP-hard optimization problems is investigated and it is shown that problems LOG DOMINATING SET, TOURNAMENT DOMINating SET and RICH HYPERGRAPH VERTEX COVER cannot be approximated to a constant ratio in polynomial time unless the corresponding NP- hard versions are also approximable in deterministic subexponential time.
Abstract: The inapproximability of non NP-hard optimization problems is investigated. Based on self-reducibility and approximation preserving reductions, it is shown that problems LOG DOMINATING SET, TOURNAMENT DOMINATING SET and RICH HYPERGRAPH VERTEX COVER cannot be approximated to a constant ratio in polynomial time unless the corresponding NP-hard versions are also approximable in deterministic subexponential time. A direct connection is established between non NP-hard problems and a PCP characterization of NP. Reductions from the PCP characterization show that LOG CLIQUE is not approximable in polynomial time and MAX SPARSE SAT does not have a PTAS under the assumption that SAT cannot be solved in deterministic 2O(log n√n) time and that NP ⊈ DTIME(2o(n)).
TL;DR: In this paper, a backtracking algorithm was developed to find the smallest cardinality of an open P-dominating set, i.e., the set where every node v 6 V is adjacent to at least Pl nodes in the graph.
Abstract: --Let G = (V, E) be a graph with node set V, edge set E and IVI = n. A set S C V is an open h-dominating set if every node v~ 6 V is adjacent to at least h nodes in S. Consider a vector P = [Pl,P2 .... ,P,] with p~ positive integers and Pi ~ deg(vi). A set S C V is an open P-dominating set if every vi 6 V is adjacent to at least Pl nodes in S. We develop a backtracking algorithm that finds the open P-domination number of G, which is the smallest cardinality of an open P-dominating set. A comparative computational experiment is presented on arbitrary graphs of different sizes. Keywords--Multiple domination, Backtracking. 1. INTRODUCTION The notion of domination in graphs was introduced formally by K6nig [1] and Ore [2]. Since then a tremendous volume of research work related to the concept of domination has been published. For an extensive bibliography see [3]. Here, we follow in general the notation and terminology given by Harary [4]. Let G = (V, E) be a graph with IV[ = n and [El = m. The
01 Jan 1998
TL;DR: In this paper, the authors considered on-line dominating set problems for general and permutation (simple) graphs and proved that the minimum dominating set problem is NP-hard. But they also showed that it is possible to find a dominating set of minimum cardinality.
Abstract: A dominating set of a graph G = (V, E) is a subset V’ of V such that for each vertex u ∈ V — V’ there is a vertex v ∈ V’ so that (u, v) ∈ E. The minimum dominating set problem is to find a set V’ of minimum cardinality, which is denoted by o(G). It is well known that the minimum dominating set problem is NP-complete [9]. In this paper we consider on-line dominating set problems for general and permutation (simple) graphs.
TL;DR: Let M be a finite subset of vertices of a connected graph G and assume that every vertex v E M has a dominating radius r(v)E N U {0}.
TL;DR: This work proposes polynomial-time algorithms for the problem of computing minimum dominating sets of n intervals on lines in three cases: (1) the lines intersect at a single point, (2) all lines except one are parallel, and (3) one line with t weighted points and the minimum dominating set must maximize the sum of the weights of the points covered.
Abstract: We study the problem of computing minimum dominating sets of n intervals on lines in three cases: (1) the lines intersect at a single point, (2) all lines except one are parallel, and (3) one line with t weighted points on it and the minimum dominating set must maximize the sum of the weights of the points covered. We propose polynomial-time algorithms for the first two problems, which are special cases of the minimum dominating set problem for path graphs which is known to be NP-hard. The third problem requires identifying the structure of minimum dominating sets of intervals on a line so as to be able to select one that maximizes the weight sum of the weighted points covered. Assuming that presorting has been performed, the first problem has an O(n) -time solution, while the second and the third problems are solved by dynamic programming algorithms, requiring O(n log n) and O(n + t) time, respectively.
TL;DR: It is proved that a conjecture of Sampathkumar (1990) that γ 1 ⩽ 3p 5 for any connected graph G of order p ⩾ 2 is proved.
TL;DR: It is shown that the perfect dominating set problem on G can be solved in O(log2 n) time using O(n+m) procesors on a CREW PRAM.
Abstract: In the literature, there are quite a few sequential and parallel algorithms for solving problems on distance-hereditary graphs. With an n-vertex distance-hereditary graph G, we show that the perfect dominating set problem on G can be solved in O(log2 n) time using O(n+m) procesors on a CREW PRAM