Showing papers on "Dominating set published in 1999"
TL;DR: The study of a variation of standard domination, namely restrained domination, is initiated and it is shown that the decision problem for γ r ( G ) is NP-complete even for bipartite and chordal graphs.
140 citations
26 Jul 1999
TL;DR: Improved performance ratios for the Independent Set problem in weighted general graphs, weighted bounded-degree graphs, and in sparse graphs are reported.
Abstract: The focus of this study is to clarify the approximability of the important versions of the maximum independent set problem, and to apply, where possible, the technique to related hereditary subgraph and subset problem. We report improved performance ratios for the Independent Set problem in weighted general graphs, weighted bounded-degree graphs, and in sparse graphs. Other problems with better than previously reported ratios include Weighted Set Packing, Longest Subsequence, Maximum Independent Sequence, and Independent Set in hypergraphs.
127 citations
TL;DR: A constructive proof of the existence of dominating pairs in connected AT-free graphs is given and the resulting simple algorithm, based on the well-known lexicographic breadth-first search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previously known for this problem has complexity O(|V|3) for input graph G=(V,E).
Abstract: An independent set of three vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighborhood of the third. A graph is asteroidal triple-free (AT-free) if it contains no asteroidal triple. The motivation for this investigation is provided, in part, by the fact that AT-free graphs offer a common generalization of interval, permutation, trapezoid, and cocomparability graphs.
Previously, the authors have given an existential proof of the fact that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected AT-free graphs. The resulting simple algorithm, based on the well-known lexicographic breadth-first search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previously known for this problem has complexity O(|V|3) for input graph G=(V,E). In addition, we indicate how our algorithm can be extended to find, in time linear in the size of the input, all dominating pairs in a connected AT-free graph with diameter greater than 3. A remarkable feature of the extended algorithm is that, even though there may be O(|V|2) dominating pairs, the algorithm can compute and represent them in linear time.
103 citations
TL;DR: It is shown that there is an O(n4) time algorithm to compute the maximum weight of an independent set for AT-free graphs, and that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT- free graphs.
Abstract: An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an AT. We show that there is an O(n4) time algorithm to compute the maximum weight of an independent set for AT-free graphs. Furthermore, we obtain O(n4) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problems on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally, we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs.
93 citations
TL;DR: For each integer n the authors determine the smallest order of a connected graph with minus domination number equal to n and show that if T is a tree of order n⩾4, then γ(T)−γ− (T)⩽(n−4)/5 and this bound is sharp.
53 citations
TL;DR: New NP-completeness results are obtained for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem inPlanar cubic graphs.
51 citations
TL;DR: In this article, the vertex subsets of a graph G which consist of the vertices contained in all, or in no, respectively, minimum dominating sets of G were studied. And these sets were characterized for trees.
Abstract: In this article we begin the study of the vertex subsets of a graph G which consist of the vertices contained in all, or in no, respectively, minimum dominating sets of G We characterize these sets for trees, and also obtain results on the vertices contained in all minimum independent dominating sets of trees © 1999 John Wiley & Sons, Inc J Graph Theory 31: 163-177, 1999
40 citations
TL;DR: In this paper, it was shown that if a connected graph G of order n has minimum degree at least 2 and is not one of eight exceptional graphs, then γ r (G ) ⩽ ( n − 1)/2.
36 citations
16 Dec 1999
TL;DR: This paper shows that the minimum independent dominating set problem on circle graphs is NP-complete and that for any Ɛ, 0 ≤ ƹ < 1, there does not exist an nƐ-approximation algorithm for the minimumIndependent dominating set problems on n-vertex circle graphs, unless P = NP.
Abstract: A graph G = (V, E) is called a circle graph if there is a one-to-one correspondence between vertices in V and a set C of chords in a circle such that two vertices in V are adjacent if and only if the corresponding chords in C intersect. A subset V′ of V is a dominating set of G if for all u ∈ V either u ∈ V′ or u has a neighbor in V′. In addition, if no two vertices in V′ are adjacent, then V′ is called an independent dominating set; if G[V′] is connected, then V′ is called a connected dominating set. Keil (Discrete Applied Mathematics, 42 (1993), 51-63) shows that the minimum dominating set problem and the minimum connected dominating set problem are both NP-complete even for circle graphs. He leaves open the complexity of the minimum independent dominating set problem. In this paper we show that the minimum independent dominating set problem on circle graphs is NP-complete. Furthermore we show that for any Ɛ, 0 ≤ Ɛ < 1, there does not exist an nƐ-approximation algorithm for the minimum independent dominating set problem on n-vertex circle graphs, unless P = NP. Several other related domination problems on circle graphs are also shown to be as hard to approximate.
20 citations
TL;DR: It is shown that every graph G has a dominating target with at most an (G) vertices and every dominating target D of G is a dominating set of G for every set S such that G[D∪S] is connected.
20 citations
TL;DR: In this paper, a dominating set D of a graph G = (V,E) is defined as a cototal dominating set if every vertex v ∈ V−D is not an isolated vertex in the induced subgraph V − D>.
Abstract: A dominating set D of a graph G = (V,E) is a cototal dominating (c.t.d.) set if every vertex v ∈ V−D is not an isolated vertex in the induced subgraph V − D>. The cototal domination number γcl (G) of G is the minimum cardinality of a c.t.d. set. The cototal domatic number dCt (G) of G is defined like domatic number d(G). In this paper, we establish some results concerning these parameters.
TL;DR: A finite dominating set is identified for this problem of locating an undesirable facility in a bounded polygonal region, using Euclidean distances, under an objective function that generalizes the maximin and maxisum criteria, and includes other criteria such as the linear combinations of these criterions.
17 Jun 1999
TL;DR: It is proved that claw-free graphs, containing an induced dominating path, have a Hamiltonian path, and that two-connected claw- free graphs,containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, haveA Hamiltonian cycle.
Abstract: We prove that claw-free graphs, containing an induced dominating path, have a Hamiltonian path, and that two-connected claw-free graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Hamiltonian cycle. As a consequence, we obtain linear time algorithms for both problems if the input is restricted to (claw,net)-free graphs. These graphs enjoy those interesting structural properties.
TL;DR: This paper is the first attempt to demonstrate the fruitfulness of this contention taking the notion of domination in graphs as a basis for generalizing results in graph theory.
17 Jun 1999
TL;DR: An O(n2.376) algorithm for recognizing claw-free AT-free graphs and a linear-time algorithm for computing the set of all central vertices of a claw- free AT- free graph are presented.
Abstract: We present an O(n2376) algorithm for recognizing claw-free AT-free graphs and a linear-time algorithm for computing the set of all central vertices of a claw-free AT-free graph In addition, we give efficient algorithms that solve the problems INDEPENDENT SET, DOMINATING SET, and COLORING We argue that all running times achieved are optimal unless better algorithms for a number of famous graph problems such as triangle recognition and bipartite matching have been found Our algorithms exploit the structure of 2LexBFS schemes
Journal Article•
TL;DR: In this article, the domination number of the cardinal product of path graphs was determined and some bounds on the domination numbers were given for P6 and Pn.F or P7 and P8.
Abstract: Here we determine the domination numbers of the cardinal product of path graphs P6 ◊ Pn .F orP7 ◊ Pn and P8 ◊ Pn we give some bounds.
TL;DR: This work reports results of its investigation into the nature of connected separable graphs having unique minimum psd-sets, and characterize block-cactus graphs (with at least two blocks) having this property.
TL;DR: The strong domination number γ st (G ) is defined as the minimum cardinality of a strong dominating set and was introduced by Sampathkumar and Pushpa Latha in 1996 as mentioned in this paper.
TL;DR: It is proved that for all graphs G, Θ ( G) = Γ ( G ) where Γ( G ) is the maximum cardinality of a minimal dominating set of G and where Θ( G) is themaximum cardinality among all perfect neighborhood sets of G.
29 Apr 1999
TL;DR: In this article, it was shown that the problem of finding a minimum dominating set in an n-vertex convex bipartite graph with a given convex-round enumeration is solvable in O(n 2 ) time.
Abstract: A bipartite graph G=(X,Y;E) is convex if there exists a linear enumeration, L, of the vertices of X such that the neighbours of each vertex of Y are consecutive in L. We show that the problem of finding a minimum dominating set, or a minimum independent dominating set, in an n-vertex convex bipartite graph is solvable in time O(n^2). This improves previous O(n^3) algorithms for these problems. Recently, a new class of graphs called convex-round graphs have been introduced by Bang-Jensen, Huang and Yeo. These are the graphs in which vertices can be circularly enumerated so that the neighbours of every vertex are consecutive in the enumeration. As a byproduct, we show that a minimum dominating set, or a minimum independent dominating set, in a convex-round graph can be computed in time O(n^3). Using a reduction to circular arc graphs, we show that a minimum total dominating set in a convex-round graph (with a given convex-round enumeration) can be computed in time O(n).
01 Jan 1999
TL;DR: Point-set dominating sets are defined in this paper as a subset of the vertex set of a graph such that the subgraph induced by the vertices of a vertex set is connected, and the maximum number of classes of a partition of such vertices is the point-set domatic number.
Abstract: A subset $D$ of the vertex set $V(G)$ of a graph $G$ is called point-set dominating, if for each subset $S\subseteq V(G)-D$ there exists a vertex $v\in D$ such that the subgraph of $G$ induced by $S\cup\{v\}$ is connected. The maximum number of classes of a partition of $V(G)$, all of whose classes are point-set dominating sets, is the point-set domatic number $d_p(G)$ of $G$. Its basic properties are studied in the paper.
TL;DR: The domination numbers of Helm graph, web graph and Levi graph are investigated and it is proved that the domination number denoted by is the minimum cardinality of dominating set in G.
Abstract: A set is dominating set of a graph G, if every vertex in V-S is adjacent to at least one vertex in S. The domination number denoted by is defined to be the minimum cardinality of dominating set in G. We investigate the domination numbers of Helm graph, web graph and Levi graph. Also we testing our theoretical results in computer by introduce a matlab procedure to calculate the domination numbers , dominating set S and draw this graphs that illustrated the vertices of domination this graphs. It is proved that:
16 Dec 1999
TL;DR: This paper presents O(1)-approximation algorithms for all three problems -- MDS, MCDS, and MTDS on circle graphs -- for any circle graph with n vertices and m edges, for which these algorithms take O(n2 + nm) time and O( n2) space.
Abstract: A graph G = (V, E) is called a circle graph if there is a one-to-one correspondence between vertices in V and a set C of chords in a circle such that two vertices in V are adjacent if and only if the corresponding chords in C intersect. A subset V′ of V is a dominating set of G if for all u ∈ V either u ∈ V′ or u has a neighbor in V′. In addition, if G[V′] is connected, then V′ is called a connected dominating set; if G[V′] has no isolated vertices, then V′ is called a total dominating set. Keil (Discrete Applied Mathematics, 42 (1993), 51-63) shows that the minimum dominating set problem (MDS), the minimum connected dominating set problem (MCDS) and the minimum total domination problem (MTDS) are all NP-complete even for circle graphs. He mentions designing approximation algorithms for these problems as being open. This paper presents O(1)-approximation algorithms for all three problems -- MDS, MCDS, and MTDS on circle graphs. For any circle graph with n vertices and m edges, these algorithms take O(n2 + nm) time and O(n2) space. These results, along with the result on the hardness of approximating minimum independent dominating set on circle graphs (Damian-Iordache and Pemmaraju, in this proceedings) advance our understanding of domination problems on circle graphs significantly.