Showing papers on "Dominating set published in 2007"

Location of Alternative-Fuel Stations Using the Flow-Refueling Location Model and Dispersion of Candidate Sites on Arcs

Abstract: The Flow Refueling Location Model (FRLM) is a flow-intercepting model that locates p stations on a network to maximize the refueling of origin–destination flows. Because of the limited driving range of vehicles, network vertices do not constitute a finite dominating set. This paper extends the FRLM by adding candidate sites along arcs using three methods. The first identifies arc segments where a single facility could refuel a path that would otherwise require two facilities at vertices to refuel it. The other methods use the Added-Node Dispersion Problem (ANDP) to disperse candidate sites along arcs by minimax and maximin methods. While none of the methods generate a finite dominating set, results show that adding ANDP sites produces better solutions than mid-path segments or vertices only.

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

Abstract: Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving $\mathcal{NP}$-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by $2k$, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by $335k$. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless $\mathcal{P} = \mathcal{NP}$, planar vertex cover does not have a problem kernel of size smaller than $4k/3$, and planar independent set and planar dominating set do not have kernels of size smaller than $2k$. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to $67 k$, improving significantly the $335 k$ previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363-384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.

Locally Excluding a Minor

Abstract: We introduce the concept of locally excluded minors. Graph classes locally excluding a minor are a common generalisation of the concept of excluded minor classes and of graph classes with bounded local tree-width. We show that first-order model-checking is fixed-parameter tractable on any class of graphs locally excluding a minor. This strictly generalises analogous results by Flum and Grohe on excluded minor classes and Frick and Grohe on classes with bounded local tree-width. As an important consequence of the proof we obtain fixed-parameter algorithms for problems such as dominating or independent set on graph classes excluding a minor, where now the parameter is the size of the dominating set and the excluded minor. We also study graph classes with excluded minors, where the minor may grow slowly with the size of the graphs and show that again, first-order model-checking is fixed-parameter tractable on any such class of graphs.

Algorithms for Graphs Embeddable with Few Crossings per Edge

Abstract: We consider graphs that can be embedded on a surface of bounded genus such that each edge has a bounded number of crossings. We prove that many optimization problems, including maximum independent set, minimum vertex cover, minimum dominating set and many others, admit polynomial time approximation schemes when restricted to such graphs. This extends previous results by Baker and Eppstein to a much broader class of graphs. We also prove that for the considered class of graphs, there are balanced separators of size O(√n) where n is a number of vertices in the graph. On the negative side, we prove that it is intractable to recognize the graphs embeddable in the plane with at most one crossing per edge.

Linear problem kernels for NP-hard problems on planar graphs

Abstract: We develop a generic framework for deriving linear-size problem kernels for NP-hard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for CONNECTED VERTEX COVER, MINIMUM EDGE DOMINATING SET, MAXIMUM TRIANGLE PACKING, AND EFFICIENT DOMINATING Set on planar graphs. On the route to these results, we present effective, problem-specific data reduction rules that are useful in any approach attacking the computational intractability of these problems.

Efficient Exact Algorithms through Enumerating Maximal Independent Sets and Other Techniques

Abstract: We give substantially improved exact exponential-time algorithms for a number of NP-hard problems These algorithms are obtained using a variety of techniques These techniques include: obtaining exact algorithms by enumerating maximal independent sets in a graph, obtaining exact algorithms from parameterized algorithms and a variant of the usual branch-and-bound technique which we call the "colored" branch-and-bound technique These techniques are simple in that they avoid detailed case analyses and yield algorithms that can be easily implemented We show the power of these techniques by applying them to several NP-hard problems and obtaining new improved upper bounds on the running time The specific problems that we tackle are: (1) the Odd Cycle Transversal problem in general undirected graphs, (2) the Feedback Vertex Set problem in directed graphs of maximum degree 4, (3) Feedback Arc Set problem in tournaments, (4) the 4-Hitting Set problem and (5) the Minimum Maximal Matching and the Edge Dominating Set problems The algorithms that we present for these problems are the best known and are a substantial improvement over previous best results For example, for the Minimum Maximal Matching we give an O*(14425n) algorithm improving the previous best result of O*(14422m) [35] For the Odd Cycle Transversal problem, we give an O*(162n) algorithm which improves the previous time bound of O*(17724n) [3]

Perfect Codes for Metrics Induced by Circulant Graphs

Abstract: An algebraic methodology for defining new metrics over two-dimensional signal spaces is presented in this work. We have mainly considered quadrature amplitude modulation (QAM) constellations which have previously been modeled by quotient rings of Gaussian integers. The metric over these constellations, based on the distance concept in circulant graphs, is one of the main contributions of this work. A detailed analysis of some degree-four circulant graphs has allowed us to detail the weight distribution for these signal spaces. A new family of perfect codes over Gaussian integers will be defined and characterized by providing a solution to the perfect t-dominating set problem over the circulant graphs presented. Finally, we will show how this new metric can be extended to other signal sets by considering hexagonal constellations and circulant graphs of degree six.

Approximation Algorithms and Hardness for Domination with Propagation

Abstract: The power dominating set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-size set of nodes Sthat power dominates all the nodes, where a node vis power dominated if (1) vis in Sor vhas a neighbor in S, or (2) vhas a neighbor wsuch that wand all of its neighbors except vare power dominated. Note that rule (1) is the same as for the dominating set problem, and that rule (2) is a type of propagation rule that applies iteratively. We use nto denote the number of nodes. We show a hardness of approximation threshold of $2^{\log^{1-\epsilon}{n}}$ in contrast to the logarithmic hardness for dominating set. This is the first result separating these two problem. We give an $O(\sqrt{n})$ approximation algorithm for planar graphs, and show that our methods cannot improve on this approximation guarantee. We introduce an extension of PDS called i¾?-round PDS; for i¾?= 1 this is the dominating set problem, and for i¾? i¾? ni¾? 1 this is the PDS problem. Our hardness threshold for PDS also holds for i¾?-round PDS for all i¾? i¾? 4. We give a PTAS for the i¾?-round PDS problem on planar graphs, for $\ell=O(\frac{\log{n}}{\log{\log{n}}})$. We study variants of the greedy algorithm, which is known to work well on covering problems, and show that the approximation guarantees can be i¾?(n), even on planar graphs. Finally, we initiate the study of PDS on directed graphs, and show the same hardness threshold of $2^{\log^{1-\epsilon}{n}}$ for directed acyclic graphs.

Partial vs. Complete Domination: t-Dominating Set

Abstract: We examine the parameterized complexity of t -Dominating Set , the problem of finding a set of at most knodes that dominate at least tnodes of a graph G= (V,E). The classic NP-complete problem Dominating Set , which can be seen to be t -Dominating Set with the restriction that t= n, has long been known to be W[2]-complete when parameterized in k. Whereas this implies W[2]-hardness for t -Dominating Set and the parameter k, we are able to prove fixed-parameter tractability for t -Dominating Set and the parameter t. More precisely, we obtain a quintic problem kernel and a randomized $O((4+\varepsilon)^t\textit{poly}(n))$ algorithm. The algorithm is based on the divide-and-color method introduced to the community earlier this year, rather intuitive and can be derandomized using a standard framework.

A dominating and absorbent set in a wireless ad-hoc network with different transmission ranges

Abstract: Unlike a cellular or wired network, there is no base station or network infrastructure in a wireless ad-hoc network, in which nodes communicate with each other via peer communications. In order to make routing and flooding efficient in such an infrastructureless network, Connected Dominating Set (CDS) as a virtual backbone has been extensively studied. Most of the existing studies on the CDS problem have focused on unit disk graphs, where every node in a network has the same transmission range. However, nodes may have different powers due to difference in functionalities, power control, topology control, and so on. In this case, it is desirable to model such a network as a disk graph where each node has different transmission range. In this paper, we define Minimum Strongly Connected Dominating and Absorbent Set (MSCDAS) in a disk graph, which is the counterpart of minimum CDS in unit disk graph. We propose a constant approximation algorithm when the ratio of the maximum to the minimum in transmission range is bounded. We also present two heuristics and compare the performances of the proposed schemes through simulation.

Paired-domination in generalized claw-free graphs

Abstract: In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998) 199–206). A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by $$\gamma_{\rm pr}(G)$$ , is the minimum cardinality of a paired-dominating set of G. If G does not contain a graph F as an induced subgraph, then G is said to be F-free. Haynes and Slater (Networks 32 (1998) 199–206) showed that if G is a connected graph of order $$n \ge 3$$ , then $$\gamma_{\rm pr}(G) \le n-1$$ and this bound is sharp for graphs of arbitrarily large order. Every graph is $$K_{1,a+2}$$ -free for some integer a ≥ 0. We show that for every integer a ≥ 0, if G is a connected $$K_{1,a+2}$$ -free graph of order n ≥ 2, then $$\gamma_{\rm pr}(G) \le 2(an + 1)/(2a+1)$$ with infinitely many extremal graphs.

Perfect domination in rectangular grid graphs

Abstract: A dominating set S in a graph G is said to be perfect if every vertex of G not in S is adjacent to just one vertex of S. Given a vertex subset S of a side Pm of an m×n grid graph G, the perfect dominating sets S in G with S = S ∩ V (Pm) can be determined via an exhaustive algorithm Θ of running time O(2m+n). Extending Θ to infinite grid graphs of width m − 1, periodicity makes the binary decision tree of Θ prunable into a finite threaded tree, a closed walk of which yields all such sets S. The graphs induced by the complements of such sets S can be codified by arrays of ordered pairs of positive integers via Θ, for the growth and determination of which a speedier algorithm exists. A recent characterization of grid graphs having total perfect codes S (with just 1-cubes as induced components), due to Klostermeyer and Goldwasser, is given in terms of Θ, which allows to show that these sets S are restrictions of only one total perfect code S1 in the integer lattice graph Λ of R. Moreover, the complement Λ− S1 yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in Λ are in 1-1 correspondence with the doubly infinite {0, 1}-sequences.

Independent dominating sets and hamiltonian cycles

Abstract: A graph is uniquely hamiltonian if it contains exactly one hamiltonian cycle. In this note we prove that there are no r-regular uniquely hamiltonian graphs when r > 22. This improves upon earlier results of Thomassen. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 233–244, 2007

Upper total domination versus upper paired-domination

Abstract: Click on the link to view the abstract. Keywords: Bounds, upper total domination, upper paired-domination Quaestiones Mathematicae 30 (2007), 1–12

A New Upper Bound on the Total Domination Number of a Graph

Abstract: A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. Let $G$ be a connected graph of order $n$ with minimum degree at least two and with maximum degree at least three. We define a vertex as large if it has degree more than $2$ and we let ${\cal L}$ be the set of all large vertices of $G$. Let $P$ be any component of $G - {\cal L}$; it is a path. If $|P| \equiv 0 \, ( {\rm mod} \, 4)$ and either the two ends of $P$ are adjacent in $G$ to the same large vertex or the two ends of $P$ are adjacent to different, but adjacent, large vertices in $G$, we call $P$ a $0$-path. If $|P| \ge 5$ and $|P| \equiv 1 \, ( {\rm mod} \, 4)$ with the two ends of $P$ adjacent in $G$ to the same large vertex, we call $P$ a $1$-path. If $|P| \equiv 3 \, ( {\rm mod} \, 4)$, we call $P$ a $3$-path. For $i \in \{0,1,3\}$, we denote the number of $i$-paths in $G$ by $p_i$. We show that the total domination number of $G$ is at most $(n + p_0 + p_1 + p_3)/2$. This result generalizes a result shown in several manuscripts (see, for example, J. Graph Theory 46 (2004), 207–210) which states that if $G$ is a graph of order $n$ with minimum degree at least three, then the total domination of $G$ is at most $n/2$. It also generalizes a result by Lam and Wei stating that if $G$ is a graph of order $n$ with minimum degree at least two and with no degree-$2$ vertex adjacent to two other degree-$2$ vertices, then the total domination of $G$ is at most $n/2$.

Restricted domination parameters in graphs

Abstract: In a graph G, a vertex dominates itself and its neighbors. A subset S ⊂eqV(G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γ m , and of an m-tuple dominating set is mtupledom. For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number r k (G,f_π) is the smallest integer r such that given any subset K of (at most) k vertices of G, there exists a π set containing K of (at most) cardinality r. We show that for 1< k < n where n is the order of G: (a) if G has minimum degree m, then r k (G,γ m ) < (mn+k)/(m+1); (b) if G has minimum degree 3, then r k (G,γ) < (3n+5k)/8; and (c) if G is connected with minimum degree at least 2, then r k (G,ddom) < 3n/4 + 2k/7. These bounds are sharp.

Smaller Connected Dominating Sets in Ad Hoc and Sensor Networks based on Coverage by Two-Hop Neighbors

Abstract: In this paper, we focus on the construction of an efficient dominating set in ad hoc and sensor networks A set of nodes is said to be dominating if each node is either itself dominant or neighbor of a dominant node Application of such a set may for example be broadcasting, where the size of the set greatly impacts on energy consumption Obtaining small sets is thus of prime importance As a basis for our work, we use a heuristic given by Dai and Wu for constructing such a set Their approach, in conjunction with the elimination of message overhead by Stojmenovic, has been recently shown to be an excellent compromise with respect to a wide range of metrics In this paper, we present an enhanced definition to obtain smaller sets in the specific case where 2-hop information is considered In our new definition, a node μ is not dominant if there exists in its 2-hop neighborhood a connected set of nodes with higher priorities that covers μ and its 1-hop neighbors This new rule requires the same level of knowledge used by the original heuristic: only neighbors of nodes and neighbors of neighbors must be known to apply it However, it takes advantage of some topological knowledge originally not taken into account, that may be used to deduce communication links between 1 -hop and 2-hop neighbors We provide the proof that the new set is a subset of the one obtained with the original heuristic We also give the proof that our set is always dominating for any graph, and connected for any connected graph Two versions are considered: with topological and positional information, which differ in whether or not nodes are aware of links between their 2-hop neighbors that are not 1-hop neighbors An algorithm for locally applying the concept at each node is described We finally provide experimental data that demonstrates the superiority of our rule in obtaining smaller dominating sets A centralized algorithm is used as a benchmark in the comparisons The overhead of the size of connected dominating set is reduced by about 15% with the topological variant and by about 30% with the positional variant of our new definition

A transition from total domination in graphs to transversals in hypergraphs

Abstract: A set S of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set in G. The transversal number of a hypergraph is the minimum number of vertices meeting every edge. We observe that total domination in graphs can be translated to the problem of finding transversals in hypergraphs. In this paper we survey bounds on the total domination of a graph in terms of the order of the graph, and provide a transition from total domination in graphs to transversals in hypergraphs.

Distributed approximation of capacitated dominating sets

Abstract: We study local, distributed algorithms for the capacitated minimum dominating set (CapMDS) problem, which arises in various distributed network applications. Given a network graph G = (V,E), and a capacity cap(v) ∈ N for each node v ∈ V , the CapMDS problem asks for a subset S ⊆ V of minimal cardinality, such that every network node not in S is covered by at least one neighbor in S, and every node v ∈ S covers at most cap(v) of its neighbors. We prove that in general graphs and even with uniform capacities, the problem is inherently non-local, i.e., every distributed algorithm achieving a non-trivial approximation ratio must have a time complexity that essentially grows linearly with the network diameter. On the other hand, if for some parameter e > 0, capacities can be violated by a factor of 1 + e, CapMDS becomes much more local. Particularly, based on a novel distributed randomized rounding technique, we present a distributed bi-criteria algorithm that achieves an O(log Δ)-approximation in time O(log3n + log(n)/e), where n and Δ denote the number of nodes and the maximal degree in G, respectively. Finally, we prove that in geometric network graphs typically arising in wireless settings, the uniform problem can be approximated within a constant factor in logarithmic time, whereas the non-uniform problem remains entirely non-local.

The total domination and total bondage numbers of extended de Bruijn and Kautz digraphs

Abstract: In this paper we consider the total domination number and the total bondage number for digraphs. The total bondage number, defined as the minimum number of edges whose removal enlarges the total domination number, measures to some extent the robustness of a network where a minimum total dominating set is required. We determine the total domination number and total bondage number of the extended de Burijn digraph and the extended Kautz digraph, proposed by Shibata and Gonda in 1995, which generalize the classical de Bruijn digraph and the Kautz digraph.