Showing papers on "Dominating set published in 2011"

Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs

Abstract: We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph $H$ as a minor to graphs excluding $H$ as a topological subgraph. We prove that for a fixed $H$, every graph excluding $H$ as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter $|H|$. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a "typical" application of the structure theorem, we show that on graphs excluding $H$ as a topological subgraph, Partial Dominating Set (find $k$ vertices whose closed neighborhood has maximum size) can be solved in time $f(H,k)\cdot n^{O(1)}$ time. More significantly, we show that on graphs excluding $H$ as a topological subgraph, Graph Isomorphism can be solved in time $n^{f(H)}$. This result unifies and generalizes two previously known important polynomial-time solvable cases of Graph Isomorphism: bounded-degree graphs and $H$-minor free graphs. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition.

Face recognition using local graph structure (LGS)

Abstract: In this paper, a novel algorithm for face recognition based on Local Graph Structure (LGS) has been proposed The features of local graph structures are extracted from the texture in a local graph neighborhood then it's forwarded to the classifier for recognition The idea of LGS comes from dominating set points for a graph of the image The experiments results on ORL face database images demonstrated the effectiveness of the proposed method The advantages of LGS, very simple, fast and can be easily applied in many fields, such as biometrics, pattern recognition, and robotics as preprocessing

Enumeration of minimal dominating sets and variants

Abstract: In this paper, we are interested in the enumeration of minimal dominating sets in graphs A polynomial delay algorithm with polynomial space in split graphs is presented We then introduce a notion of maximal extension (a set of edges added to the graph) that keeps invariant the set of minimal dominating sets, and show that graphs with extensions as split graphs are exactly the ones having chordal graphs as extensions We finish by relating the enumeration of some variants of dominating sets to the enumeration of minimal transversals in hypergraphs

A faster algorithm for dominating set analyzed by the potential method

Abstract: Measure and Conquer is a recently developed technique to analyze worst-case complexity of backtracking algorithms. The traditional measure and conquer analysis concentrates on one branching at once by using only small number of variables. In this paper, we extend the measure and conquer analysis and introduce a new analyzing technique named "potential method" to deal with consecutive branchings together. In potential method, the optimization problem becomes sparse; therefore, we can use large number of variables. We applied this technique to the minimum dominating set problem and obtained the current fastest algorithm that runs in O(1.4864n) time and polynomial space. We also combined this algorithm with a precalculation by dynamic programming and obtained O(1.4689n) time and space algorithm. These results show the power of the potential method and possibilities of future applications to other problems.

On Problems as Hard as CNFSAT

Abstract: The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-Sat. In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2^n), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o(2^n), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every epsilon<1, there is a (large) integer k such that that k-CNF-Sat cannot be computed in time 2^{epsilon n}. In this paper, we show that, for every epsilon < 1, the problems Hitting Set, Set Splitting, and NAE-Sat cannot be computed in time O(2^{epsilon n}) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for Set Cover, and prove that, under this assumption, the fastest known algorithms for Steinter Tree, Connected Vertex Cover, Set Partitioning, and the pseudo-polynomial time algorithm for Subset Sum cannot be significantly improved. Finally, we justify our assumption about the hardness of Set Cover by showing that the parity of the number of set covers cannot be computed in time O(2^{epsilon n}) for any epsilon<1 unless SETH fails.

Computing the domination number of grid graphs

Abstract: Let $\gamma_{m,n}$ denote the size of a minimum dominating set in the $m \times n$ grid graph. For the square grid graph, exact values for $\gamma_{n,n}$ have earlier been published for $n \leq 19$. By using a dynamic programming algorithm, the values of $\gamma_{m,n}$ for $m,n \leq 29$ are here obtained. Minimum dominating sets for square grid graphs up to size $29 \times 29$ are depicted.

Power domination in cylinders, tori, and generalized Petersen graphs

Abstract: A set S of vertices is defined to be a power dominating set (PDS) of a graph G if every vertex and every edge in G can be monitored by the set S according to a set of rules for power system monitoring. The minimum cardinality of a PDS of G is its power domination number. In this article, we find upper bounds for the power domination number of some families of Cartesian products of graphs: the cylinders Pn□Cm for integers n ≥ 2, m ≥ 3, and the tori Cn□Cm for integers n,m ≥ 3. We apply similar techniques to present upper bounds for the power domination number of generalized Petersen graphs P(m,k). We prove those upper bounds provide the exact values of the power domination numbers if the integers m,n, and k satisfy some given relations. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 58(1), 43–49 2011 © 2011 Wiley Periodicals, Inc.

Self-Stabilizing Small k-Dominating Sets

Abstract: A self-stabilizing algorithm, after transient faults hit the system and place it in some arbitrary global state, recovers in finite time without external (eg, human) intervention In this paper, we propose a distributed asynchronous silent self-stabilizing algorithm for finding a minimal k-dominating set of at most n/(k+1) processes in an arbitrary identified network of size n We propose a transformer that allows our algorithm work under an unfair daemon (the weakest scheduling assumption) The complexity of our solution is in O(n) rounds and O(D n²) steps using O(log n + k log n) bits per process where D is the diameter of the network

Minimum d-blockers and d-transversals in graphs

Abstract: We consider a set V of elements and an optimization problem on V: the search for a maximum (or minimum) cardinality subset of V verifying a given property ?. A d-transversal is a subset of V which intersects any optimum solution in at least d elements while a d-blocker is a subset of V whose removal deteriorates the value of an optimum solution by at least d. We present some general characteristics of these problems, we review some situations which have been studied (matchings, s---t paths and s---t cuts in graphs) and we study d-transversals and d-blockers of stable sets or vertex covers in bipartite and in split graphs.

Exact Exponential-Time Algorithms for Domination Problems in Graphs

Abstract: This PhD thesis studies exact exponential-time algorithms for domination problems in graphs. Domination problems in graphs are a special kind of subset problems in graphs. A subset problem in a graph is a problem where one is given a graph G=(V,E), and one is asked whether there exist some subset S of a set of items U in the graph (mostly U is either the vertices V or the edges E) that satisfies certain properties. Domination problems in graphs are subset problems in which there is a domination criterion based on a neighbourhood relation in the graph that decides which elements of U are dominated by a given subset S, and where one of the properties that a solution subset S must satisfy is that S must dominate its complement U\S. The most well-known graph domination problem is the Dominating Set problem where the set U is the set of vertices V of G, a vertex subset S dominates all vertices in G that have a neighbour in S, and one is asked to compute the smallest vertex subset S of V that dominates all vertices in V\S. Other examples of domination problems in graphs are Independent Set, Edge Dominating Set, Total Dominating Set, Red-Blue Dominating Set, Partial Dominating Set, and #Dominating Set. We study exact exponential-time algorithms for these problems. These are algorithms that, when executed, use a number of operations that is exponential in a complexity parameter of the input in the worst case. That is, these algorithms use exponential time. Exact exponential-time algorithms then return an optimal solution to the problem. This in contrast to other fields of algorithm design where one sometimes trades running time for other properties of the algorithm or the returned solution. In this thesis, we also study parameterised algorithms for domination problems in graph on graph decompositions. These are algorithms whose worst-case running times are polynomial in the size of the graph and exponential in the graph-width parameter associated to the graph decomposition. Our study led to faster exact exponential-time algorithms for many well-known graph domination problems. This includes an O(1.4969^n)-time algorithm for Dominating Set, an O(1.2114^n)-time algorithm for Independent Set, an O(1.3226^n)-time algorithm for Edge Dominating Set, an O(1.4969^n)-time algorithm for Total Dominating Set, an O(1.2252^n)-time algorithm for Red-Blue Dominating Set, an O(1.5014^n)-time algorithm for Partial Dominating Set, and an O(1.5002^n)-time algorithm for #Dominating Set. We also obtained faster algorithms for these and many other graph domination problems on some prominent types of graph decompositions: tree decompositions, branch decompositions, and, for some problems, clique decompositions (also called k-expressions). A series of interesting new insights and techniques arose from this study. We mention the techniques of inclusion/exclusion-based branching and extended inclusion/exclusion-based branching. We also mention our generalisation of the fast subset convolution algorithm, which we translated to the setting of state-based dynamic programming algorithms on graph decompositions. This thesis also contains an accessible introduction to the field of exact exponential-time algorithms.

Weighted Steiner Connected Dominating Set and its Application to Multicast Routing in Wireless MANETs

Abstract: In this paper, we first propose three centralized learning automata-based heuristic algorithms for approximating a near optimal solution to the minimum weight Steiner connected dominating set (WSCDS) problem. Finding the Steiner connected dominating set of the network graph is a promising approach for multicast routing in wireless ad-hoc networks. Therefore, we present a distributed implementation of the last approximation algorithm proposed in this paper (Algorithm III) for multicast routing in wireless mobile ad-hoc networks. The proposed WSCDS algorithms are compared with the well-known existing algorithms and the obtained results show that Algorithm III outperforms the others both in terms of the dominating set size and running time. Our simulation experiments also show the superiority of the proposed multicast routing algorithm over the best previous methods in terms of the packet delivery ratio, multicast route lifetime, and end-to-end delay.

On the Power Dominating Sets of Hypercubes

Abstract: The performance of electrical networks is monitored by expensive Phasor Measurement Units (PMUs). It is economically beneficial to determine the optimal placement and the minimum number of PMUs required to effectively monitor an entire network. This problem has a graph theory model involving power dominating sets in a graph. A set S of vertices in a graph is called a power dominating set if every vertex and every edge in the graph is “observed” by S according to a set of observation rules. The power domination number of a graph is the minimum cardinality of a power dominating set of the graph. In this paper, the power domination number is determined for hypercubes Qn with n = 2 k , where k is any positive integer.

Domination in the Corona and Join of Graphs

Abstract: In this paper we characterized the dominating sets, total dominating sets, and secure total dominating sets in the corona of two graphs. The secure total dominating sets in the join of two graphs were also investigated. As direct consequences, the domination, total domination, and secure total domination numbers of these graphs were obtained.

On connected domination in unit ball graphs

Abstract: Given a simple undirected graph, the minimum connected dominating set problem is to find a minimum cardinality subset of vertices D inducing a connected subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual routing backbone in wireless networks. This paper first proposes a constant-ratio approximation algorithm for the minimum connected dominating set problem in unit ball graphs and then introduces and studies the edge-weighted bottleneck connected dominating set problem, which seeks a minimum edge weight in the graph such that the corresponding bottleneck subgraph has a connected dominating set of size k. In wireless network applications this problem can be used to determine an optimal transmission range for a network with a predefined size of the virtual backbone. We show that the problem is hard to approximate within a factor better than 2 in graphs whose edge weights satisfy the triangle inequality and provide a 3-approximation algorithm for such graphs. We also show that for fixed k the problem is polynomially solvable in unit disk and unit ball graphs.

The Hub Number of Sierpiński-Like Graphs

Abstract: A set Q⊆V is a hub set of a graph G=(V,E) if, for every pair of vertices u,v∈V∖Q, there exists a path from u to v such that all intermediate vertices are in Q. The hub number of G is the minimum size of a hub set in G. This paper derives the hub numbers of Sierpinski-like graphs including: Sierpinski graphs, extended Sierpinski graphs, and Sierpinski gasket graphs. Meanwhile, the corresponding minimum hub sets are also obtained.

A characterization of locating-total domination edge critical graphs

Abstract: For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number t(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V (G) n D, NG(u)\ D 6 NG(v)\ D. The locating-total domination number t L(G)