Showing papers on "Dominating set published in 1997"
12 Mar 1997
TL;DR: Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APX-complete even for cubic graphs, which means that unless P=NP these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three.
Abstract: Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APX-complete even for cubic graphs. This means that unless P=NP these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three.
179 citations
TL;DR: The minimum dominating set problem is used to show that the second problem is NP-hard, implying that there is a polynomial algorithm for finding a spanning tree of G with as many different colors as possible.
Abstract: Given a graph G = (V,E) and a (not necessarily proper) edge-coloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible. We show that the first problem is equivalent to finding a common independent set of maximum cardinality in two matroids, implying that there is a polynomial algorithm. We use the minimum dominating set problem to show that the second problem is NP-hard.
77 citations
TL;DR: This work characterize graphs G for which @c(H)=@c"w"c (H) for every connected induced subgraph H of G, and provides a constructive characterization of trees T in which every vertex belongs to some weakly connected dominating set of cardinality @c" w"c(T).
75 citations
TL;DR: A polynomial time approximation algorithm which computes a k-clustering for graphs having a dominating diametral path and the intractability of graph clustering and the hardness of approximating minimum graph clusterings is developed.
43 citations
TL;DR: A new approach to the design of collective communication operations in wormhole-routed mesh networks is described, which extends the concept of dominating sets in graph theory by accounting for the relative distance-insensitivity of the wormhole switching strategy and by taking advantage of a multiport communication architecture.
Abstract: A new approach to the design of collective communication operations in wormhole-routed mesh networks is described. The approach extends the concept of dominating sets in graph theory by accounting for the relative distance-insensitivity of the wormhole switching strategy and by taking advantage of a multiport communication architecture, which allows each node to simultaneously transmit messages on different outgoing channels. Collective communication operations are defined in terms of sets of extended dominating nodes (EDNs). The nodes in a set of EDNs can deliver (receive) messages to (from) a different, larger set of nodes in a single message-passing step under dimension-ordered wormhole routing and without channel contention among messages. The EDN model can be applied to different collective operations in 2D and 3D mesh networks. The authors focus on EDN-based broadcast and global combine operations. Performance evaluation results are presented that confirm the advantage of this approach over other methods.
40 citations
TL;DR: In this article, it was shown that for almost all connected graphs with minimum degree at least 2 and q edges, the domination number is bounded by (q + 1)/3.
Abstract: A dominating set for a graph G = (V,E) is a subset of vertices V′ ⊆ V such that for all v E V − V′ there exists some u E V′ for which {v, u} E E. The domination number of G is the size of its smallest dominating set(s). We show that for almost all connected graphs with minimum degree at least 2 and q edges, the domination number is bounded by (q + 1)/3. From this we derive exact lower bounds for the number of edges of a connected graph with minimum degree at least 2 and a given domination number. We also generalize the bound to k-restricted domination numbers; these measure how many vertices are necessary to dominate a graph if an arbitrary set of k vertices must be incluced in the dominating set. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 139–152, 1997
38 citations
TL;DR: This paper shows how to obtain efficient algorithms to compute a minimum cardinality total dominating set on a variety of graph classes, among them permutation graphs, dually chordal graphs and k -polygon graphs.
29 citations
TL;DR: It is shown that the VC-dimension for set systems induced by stars is computable in polynomial time, and the extremal graphs G with the minimum number of edges such that VC P ( G ) ⩾ k .
27 citations
07 Jul 1997
TL;DR: There is an O(n 2 · (¯m+1) time algorithm to compute the maximum cardinality of an independent set for AT-free graphs, and it is observed that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT- free graphs.
Abstract: An asteroidal triple 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 asteroidal triple. We show that there is an O(n 2 · (¯m+1)) time algorithm to compute the maximum cardinality of an independent set for AT-free graphs, where n is the number of vertices and ¯m is the number of non edges of the input graph. Furthermore we obtain O(n 2 · (¯m+1)) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problem 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.
25 citations
TL;DR: This work characterises connected bipartite graphs which achieve this upper bound for the upper irredundance, upper domination and independence numbers of a graph and investigates graphs which satisfy similar equations for the independent domination number, i ( G ), and the irredundy number ir( G).
24 citations
TL;DR: Both PSPACE-hardness results and polynomial time approximations are presented for most of the problems considered, including minimum vertex coloring, maximum independent set, minimum clique cover, minimum dominating set and minimum independent dominating set.
TL;DR: It is shown by means of an example that the existing relations for the unconstrained case are no longer true when locational constraints are imposed, and a biobjective problem is described, for which a finite dominating set is constructed.
Abstract: In this paper we address the problem of finding Simpson points in planar models with locational constraints when distances are measured by polyhedral gauges. Making use of the results we have stated in a previous paper, we show here the existence of a finite set of points in the plane, independent of the weights associated with the users, that contains at least a Simpson point.
Connection between Simpson points as a result of a voting process and Weber points as the outcome of a planning process are explored. It is shown by means of an example that the existing relations for the unconstrained case are no longer true when locational constraints are imposed. In order to reconcile both voting and planning processes, a biobjective problem is described, for which we construct a finite dominating set.
TL;DR: The proof will yield a polynomial-time algorithm which will return an independent set of cardinality at least (7??4)/26 for any such graph, and provides a slight improvement over the best previously known result for triangle-free 5-regular graphs.
TL;DR: A generalized version of two alternative p -Center problems, in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range, is considered.
TL;DR: This paper proposes an on-line algorithm of performance ratio l.SJii + cl and shows that fi c2 is a lower bound for the performance ratio that an on theline dominating set algorithm can possible achieve, where ct and c2 are some positive constants.
TL;DR: It is shown that the dominating setpolytope of G can be described from two linear systems related to G1 and G2, which gives a way to characterize this polytope for classes of graphs that can be recursively decomposed.
TL;DR: This paper addresses the problem of finding a Simpson set for planar models with locational constraints when the metric in use is induced by a round norm by proving the existence of a finite dominating set of locations.
Abstract: The Simpson set is one of the most popular solution concepts in location models with voting. In this paper we address the problem of finding such a set for planar models with locational constraints when the metric in use is induced by a round norm.
After formulating the problem as a mathematical program, we first propose a general algorithm that enables the evaluation of the corresponding objective function. Then, we show how to solve the problem by proving the existence of a finite dominating set of locations, the best of which can be found by inspection using the evaluation algorithm proposed.
11 Aug 1997
TL;DR: This work presents efficient parallel algorithms for finding a minimum weighted connected dominating set, a Minimum weighted Steiner tree for a distance-hereditary graph which take O(log n) time using O(n+m) processors on a CRCW PRAM.
Abstract: We present efficient parallel algorithms for finding a minimum weighted connected dominating set, a minimum weighted Steiner tree for a distance-hereditary graph which take O(log n) time using O(n+m) processors on a CRCW PRAM, where n and m are the number of vertices and edges of a given graph, respectively. We also find a maximum weighted clique of a distance-hereditary graph in O(log/sup 2/ n) time using O(n+m) processors on a CREW PRAM.
TL;DR: In this article, the authors define a universal total dominating function (TDF) which is defined as a function f ∈ V → [0, 1] such that the sum of f values over all vertices adjacent to v is at least one.
Abstract: A total dominating function (TDF) of a graph G=(V, E) is a function f ∶ V → [0,1] such that for each v ∈ V, the sum of f values over all neighbours of ν (i.e., all vertices adjacent to v) is at least one. Integer-valued TDFs are precisely the characteristic functions of total dominating sets of G. A minimal TDF (MTDF) is one such that decreasing any value of it makes it non-TDF. An MTDF f is called universal if convex combinations of f and any other MTDF are minimal. We give a sufficient condition for an MTDF to be universal which generalizes previous results. Also we define a splitting operation on a graph G as follows: take any vertex ν in G and a vertex ω not in G and join ω with all the neighbours of v. A graph G has a universal MTDF if and only if the graph obtained by splitting G has a universal MTDF. A corollary is that graphs obtained by the operation from paths, cycles, complete graphs, wheels, and caterpillar graphs have a universal MTDF.
11 Jul 1997
TL;DR: It is shown that for almost all instances the greedy algorithm has a performance ratio of 2+o(1) and yields a 1+o (1) approximation of the r-dominating set problem.
Abstract: In this paper we analyse the performance of the greedy algorithm for r-independent set on random graphs. We show that for almost all instances
The greedy algorithm has a performance ratio of 2+o(1).
The greedy algorithm yields a 1+o(1) approximation of the r-dominating set problem.
The k-center problem can be solved optimally.
TL;DR: Relations between these two parameters are proved and lower and upper bounds for α d ( G) are given, which is the minimum value of α 2 ( S) where S is any total dominating set of G .
TL;DR: This paper gives linear-time algorithms for finding two minimum (connected) dominating sets with minimum intersection for interval graphs.
TL;DR: In this paper, it was shown that if a graph G is connected and contains no induced subgraph isomorphic to P6 or Ht (the graph obtained by subdividing each edge of K1,t, t ≥ 3, by exactly one vertex), then G has a dominating set which induces a connected graph with clique covering number at most t − 1.
Abstract: Motivated by earlier work on dominating cliques, we show that if a graph G is connected and contains no induced subgraph isomorphic to P6 or Ht (the graph obtained by subdividing each edge of K1,t, t ≥ 3, by exactly one vertex), then G has a dominating set which induces a connected graph with clique covering number at most t − 1. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 101–105, 1997
Journal Article•
01 Jan 1997
TL;DR: The problem of minimizing the cardinality d(S) of a dominating set of a Shiftable Interval Graph is formally stated, its strong NP-completeness is proved, and upper and lower bounds for d( S) are discussed.
Abstract: In this paper the problem of computing a minimum dominating set of a Shiftable Interval Graph (SIG) is studied. SIG's can be considered a generalization of interval graphs, in the sense that a SIG identifies a whole family of interval graphs. An optimization problem defined on a SIG consists in determining an interval graph of the family which optimizes the value of the chosen measure. In this paper, in particular, we studied the problem of minimizing the cardinality d(S) of a dominating set. The problem is formally stated, its strong NP-completeness is proved, and upper and lower bounds for d(S) are discussed. Special cases solvable at optimality are characterized. Several algorithms are proposed to solve the problem on arbitrary SIG's, and are tested on 220 randomly chosen problem and on five ad hoc designed examples which emphasize the differences among the proposed algorithms.
Book•
01 Jan 1997
TL;DR: Improved approximations of independent dominating set in bounded degree graphs and a new characterization of P 4-connected graphs are presented.
Abstract: Hypergraphs and decision trees.- Improved approximations of independent dominating set in bounded degree graphs.- A new characterization of P 4-connected graphs.- Node rewriting in hypergraphs.- On ?-partitioning the n-cube.- Embedding complete binary trees in product graphs.- Clique and anticlique partitions of graphs.- Optimal parallel routing in star graphs.- Counting edges in a dag.- Closure properties of context-free Hyperedge Replacement Systems.- Upward drawings of search trees.- More general parallel tree contraction: Register allocation and broadcasting in a tree.- System diagnosis with smallest risk of error.- Efficient algorithms for shortest path queries in planar digraphs.- LexBFS-orderings and powers of graphs.- Efficient Union-Find for planar graphs and other sparse graph classes.- Switchbox routing in VLSI design: Closing the complexity gap.- Detecting diamond necklaces in labeled dags.- Algebraic graph derivations for graphical calculi.- Definability equals recognizability of partial 3-trees.- One, two, three, many, or: Complexity aspects of dynamic network flows with dedicated arcs.- Approximate maxima finding of continuous functions under restricted budget (Extended abstract).- The Optimal Cost Chromatic Partition problem for trees and interval graphs.- Modifying networks to obtain low cost trees.- On the hardness of allocating frequencies for hybrid networks.- Homogeneous sets and domination problems.- Independent spanning trees of product graphs.- Designing distrance-preserving fault-tolerant topologies.- Shortest path algorithms for nearly acyclic directed graphs.- Computing disjoint paths with length constraints.- Generalized edge-rankings of trees.
Journal Article•
01 Jan 1997
TL;DR: In this article, the authors presented a scheme which constructs a long cycle in each component of C, x T. If T satisfies certain degree constraints, then the cycle thus traced is shown to be a dominating set, and in some cases, a vertex cover of that component.
Abstract: Let C, x 7’ denote the Kronecker product of a cycle C, and a tree T. If m is odd, then C,,, x T is connected, otherwise this graph consists of two isomorphic components. This paper presents a scheme which constructs a long cycle in each component of C, x T. If T satisfies certain degree constraints, then the cycle thus traced is shown to be a dominating set, and in some cases, a vertex cover of that component. The procedure builds on (i) results on longest cycles in C,,, x P,, and (ii) a path factor of T. Additional results include characterizations for the existence of a Hamiltonian cycle and for that of a Hamiltonian path in Cm x T.
TL;DR: In this paper, it was shown that an optimal solution to the rectilinear network design problem is contained in a grid graph defined by the set of given points and the barriers.
Abstract: Given a set of points on a Cartesian plane and the coordinate axes, the rectilinear network design problem is to find a network, with arcs parallel to either one of the axes, that minimizes the fixed and the variable costs of interactions between a specified set of pairs of points. We show that, even in the presence of arbitrary barriers, an optimal solution to the problem (when feasible) is contained in a grid graph defined by the set of given points and the barriers. This converts the spatial problem to a combinatorial problem. Finally we show connections between the rectilinear network design problem and a number of well-known problems. Thus this paper unifies the known dominating set results for these problems and extends the results to the case with barriers.