Showing papers on "Dominating set published in 2013"
08 Jul 2013
TL;DR: Two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c tw | V | O ( 1 ) time algorithms, also for weighted and counting versions of connectivity problems are presented.
Abstract: It is well known that many local graph problems, like Vertex Cover?and Dominating Set, can be solved in time 2 O ( tw ) | V | O ( 1 ) for graphs G = ( V , E ) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw O ( tw ) | V | O ( 1 ) . Recently, Cygan et al.?(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time c tw | V | O ( 1 ) for a small constant c, e.g., for Hamiltonian Cycle?and Steiner Tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in the runtime in terms of the weights).We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c tw | V | O ( 1 ) time algorithms, also for weighted and counting versions. For example, in this time we can solve Traveling Salesman? or count the number of Hamiltonian cycles. The rank based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying "small" sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the Matrix Tree Theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.
250 citations
TL;DR: A survey of selected recent results on independent domination in graphs is offered and it is shown that not every vertex in S is adjacent to a vertex in S .
196 citations
TL;DR: It is proved that $\gamma_g(G) \le 7n/11$ when $G$ is an isolate-free $n$-vertex forest and that $G) is a forest of nontrivial caterpillars for any isolate-based graph.
Abstract: In the domination game on a graph $G$, two players called Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices dominated; the game ends when the chosen set becomes a dominating set of $G$. Dominator aims to minimize the size of the resulting dominating set, while Staller aims to maximize it. When both players play optimally, the size of the dominating set produced is the game domination number of $G$, denoted by $\gamma_g(G)$ when Dominator plays first and by $\gamma_g^\prime(G)$ when Staller plays first. We prove that $\gamma_g(G) \le 7n/11$ when $G$ is an isolate-free $n$-vertex forest and that $\gamma_g(G) \le \left\lceil7n/10\right\rceil$ for any isolate-free $n$-vertex graph. In both cases we conjecture that $\gamma_g(G) \le 3n/5$ and prove it when $G$ is a forest of nontrivial caterpillars. We also resolve conjectures of Bresar, Klavžar, and Rall by showing that always $\gamma_g^\prime(G)\le\gamma_g(G)+1$, that for $k\ge2$ there a...
124 citations
TL;DR: The state-of-the-art with respect to finding minimum dominating set approximations in distributed systems, where each node locally executes a protocol on its own, communicating with its neighbors in order to achieve a solution with good global properties is summarized.
Abstract: A dominating set is a subset of the nodes of a graph such that all nodes are in the set or adjacent to a node in the set. A minimum dominating set approximation is a dominating set that is not much larger than a dominating set with the fewest possible number of nodes. This article summarizes the state-of-the-art with respect to finding minimum dominating set approximations in distributed systems, where each node locally executes a protocol on its own, communicating with its neighbors in order to achieve a solution with good global properties. Moreover, we present a number of recent results for specific families of graphs in detail. A unit disk graph is given by an embedding of the nodes in the Euclidean plane, where two nodes are joined by an edge exactly if they are in distance at most one. For this family of graphs, we prove an asymptotically tight lower bound on the trade-off between time complexity and approximation ratio of deterministic algorithms. Next, we consider graphs of small arboricity, whose edge sets can be decomposed into a small number of forests. We give two algorithms, a randomized one excelling in its approximation ratio and a uniform deterministic one which is faster and simpler. Finally, we show that in planar graphs, which can be drawn in the Euclidean plane without intersecting edges, a constant approximation factor can be ensured within a constant number of communication rounds.
63 citations
TL;DR: In this article, the authors present open problems and conjectures on visibility graphs of points, segments, and polygons along with necessary backgrounds for understanding them and present a survey article.
Abstract: In this survey article, we present open problems and conjectures on visibility graphs of points, segments, and polygons along with necessary backgrounds for understanding them.
58 citations
TL;DR: If G is an n-vertex maximal outerplanar graph, then @c(G)@?(n+t)/4, where t is the number of vertices of degree 2 in G, is the minimum cardinality of a dominating set of G.
55 citations
TL;DR: Algorithms for enumerating all minimal dominating sets are given, where the running time of each algorithm is within a polynomial factor of the proved upper bound for the graph class in question.
54 citations
01 Jan 2013
TL;DR: This paper presents two metaheuristic algorithms - a hybrid genetic algorithm and a hybrid ant colony optimization algorithm - for the problem of computing minimum weight dominating set and compares their results with that of a greedy heuristic as well as the only other meta heuristic proposed so far in the literature.
Abstract: Minimum weight dominating set (MWDS) finds many uses in solving problems as varied as clustering in wireless networks, multi-document summarization in information retrieval and so on. It is proven to be NP-hard, even for unit disk graphs. Many centralized and distributed, greedy and approximation algorithms have been proposed for the MWDS problem. However, all the approximation algorithms are limited to unit disk graphs which are primarily used to model wireless networks. This assumption fails when applied to other domains. In this paper, we present two metaheuristic algorithms - a hybrid genetic algorithm and a hybrid ant colony optimization algorithm - for the problem of computing minimum weight dominating set. We compare our results with that of a greedy heuristic as well as the only other metaheuristic algorithm proposed so far in the literature and show that our algorithms are far better than these algorithms.
54 citations
Book•
09 May 2013
TL;DR: In this paper, the inverse domination number is defined as the minimum cardinality of a dominating set whose complement contains a minimum dominating set, which implies that every graph with minimum degree at least one has two disjoint dominating sets.
Abstract: The concept of dominating sets introduced by Ore and Berge, is currently receiving much attention in the literature of graph theory. Several types of domination parameters have been studied by imposing several conditions on dominating sets. Ore observed that the complement of every minimal dominating set of a graph with minimum degree at least one is also a dominating set. This implies that every graph with minimum degree at least one has two disjoint dominating sets. Recently several authors initiated the study of the cardinalities of pairs of disjoint dominating sets in graphs. The inverse domination number is the minimum cardinality of a dominating set whose complement contains a minimum dominating set. Motivated by the inverse domination number, there are studies which deals about two disjoint domination number of a graph.
53 citations
TL;DR: This paper shows that inclusion/exclusion as a branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules, and obtains the currently fastest exact exponential-time algorithms for a number of domination problems in graphs.
Abstract: Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times. In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number. This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set, and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k-Set Splitting and its generalisation k-Not-All-Equal Satisfiability.
53 citations
TL;DR: Approximation of the problem by moderately exponential time algorithms is studied and it is shown that it can be approximated within ratio [email protected], for any @e>0, in a time smaller than the one of exact computation and exponentially decreasing with @e.
TL;DR: It is shown that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks.
Abstract: In the study of deterministic distributed algorithms, it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient.Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks.As a corollary of our result, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem. By prior work, there is a deterministic local algorithm that achieves the approximation factor of 4--1/⌊Δ/2⌋ in graphs of maximum degree Δ. This approximation ratio is known to be optimal in the port-numbering model—our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier.Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α
05 Jun 2013
TL;DR: In this paper, a polynomial-time reduction from the Bipartite Dominating Set problem to metric dimension on maximum degree three graphs was shown. But this reduction requires the assumption that FPT≠W[1] fails.
Abstract: The NP-hard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u, w} if the distance (length of a shortest path) between v and u is different from the distance of v and w. We give a polynomial-time computable reduction from the Bipartite Dominating Set problem to Metric Dimension on maximum degree three graphs such that there is a one-to-one correspondence between the solution sets of both problems. There are two main consequences of this: First, it proves that Metric Dimension on maximum degree three graphs is W[2]-hard with respect to the parameter k. This answers an open question concerning the parameterized complexity of Metric Dimension posed by Lokshtanov [Dagstuhl seminar, 2009] and also by Diaz et al. [ESA'12]. Additionally, it implies that a trivial nO(k)-time algorithm cannot be improved to an no(k)-time algorithm, unless the assumption FPT≠W[1] fails. Second, as Bipartite Dominating Set is inapproximable within o(log n), it follows that Metric Dimension on maximum degree three graphs is also inapproximable by a factor of o(log n), unless NP=P. This strengthens the result of Hauptmann et al. [JDA'12] who proved APX-hardness on bounded-degree graphs.
TL;DR: In this article, the authors constructed pairwise-incomparable bounds on the projective dimensions of edge ideals using combinatorial properties of the associated graphs and drew heavily from the topic of dominating sets.
01 Nov 2013
TL;DR: This work presents an Energy Constrained minimum Dominating Set based efficient clustering called ECDS and proposes multiple extensions to the distributed algorithm for the energy constrained dominating set that perform well in terms of energy usage, node lifetime, and clustering time in comparison and are very suitable for wireless sensor networks.
Abstract: Using partitioning in sensor networks to create clusters for routing, data management, and for controlling communication has been proven as a way to ensure long range deployment and to deal with sensor network shortcomings such as limited energy and short communication ranges. Choosing a cluster head within each cluster is important because cluster heads use additional energy for their responsibilities and that burden needs to be carefully passed around among nodes in a cluster. Many existing protocols either choose cluster heads randomly or use nodes with the highest remaining energy. We present an Energy Constrained minimum Dominating Set based efficient clustering called ECDS to model the problem of optimally choosing cluster heads with energy constraints. Our proposed randomized distributed algorithm for the constrained dominating set runs in O(lognlog@D) rounds with high probability where @D is the maximum degree of a node in the graph. We provide an approximation ratio for the ECDS algorithm of expected size 8H"@D|OPT| and with high probability a size of O(|OPT|logn) where n is the number of nodes, H is the harmonic function and OPT means the optimal size. We propose multiple extensions to the distributed algorithm for the energy constrained dominating set. We experimentally show that these extensions perform well in terms of energy usage, node lifetime, and clustering time in comparison and, thus, are very suitable for wireless sensor networks.
TL;DR: This paper proposes a new strategy for computing a smaller-size 3-connected m-dominating set in a unit disk graph with any m ≥ 1 and shows the approximation ratio of the algorithm is constant and its running time is polynomial.
Abstract: In this paper, we study the problem of computing quality fault-tolerant virtual backbone in homogeneous wireless network, which is defined as the k-connected m-dominating set problem in a unit disk graph. This problem is NP-hard, and thus many efforts have been made to find a constant factor approximation algorithm for it, but never succeeded so far with arbitrary k ≥ 3 and m ≥ 1 pair. We propose a new strategy for computing a smaller-size 3-connected m-dominating set in a unit disk graph with any m ≥ 1. We show the approximation ratio of our algorithm is constant and its running time is polynomial. We also conduct a simulation to examine the average performance of our algorithm. Our result implies that while there exists a constant factor approximation algorithm for the k-connected m-dominating set problem with arbitrary k ≤ 3 and m ≥ 1 pair, the k-connected m-dominating set problem is still open with k > 3.
TL;DR: The parameterized edge dominating set problem can be solved in O^*(2.3147^k) time and polynomial space and can be reduced to a quadratic kernel with O(k^3) edges.
TL;DR: It is proved that if a circulant graph of large degree has an efficient dominating set, then either its elements are equally spaced, or the graph is the wreath product of a smaller circulants graph with an efficient dominate set and a complete graph.
TL;DR: It is shown that the power domination of a connected cubic graph on $n$ vertices different from $K_{3,3}$ is at most $n/4$ and this bound is tight and the minimum cardinality of a power dominating set of a graph is its power domination number.
Abstract: In this paper, we continue the study of power domination in graphs (see [T. W. Haynes et al., SIAM J. Discrete Math., 15 (2002), pp. 519--529; P. Dorbec et al., SIAM J. Discrete Math., 22 (2008), pp. 554--567; A. Aazami et al., SIAM J. Discrete Math., 23 (2009), pp. 1382--1399]). Power domination in graphs was birthed from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A set of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set following a set of rules (according to Kirschoff laws) for power system monitoring. The minimum cardinality of a power dominating set of a graph is its power domination number. We show that the power domination of a connected cubic graph on $n$ vertices different from $K_{3,3}$ is at most $n/4$ and this bound is tight. More generally, we show that for $k \ge 1$, the $k$-power domination number of a connected $(k+2)$-regular graph on $n$ ...
TL;DR: It is shown that if G is an n-vertex maximal outerplanar graph with n>=3 having k vertices of degree 2, then G has a dominating set of size at most @?n+k4@?
TL;DR: The relationships between the annihilation number and the total domination number of a graph are investigated, and it is shown that @c"t(T)@?a(T)+1, and the extremal trees achieving equality in this bound are characterized.
16 Dec 2013
TL;DR: This work reduces (in polynomial time) the enumeration of minimal dominating sets in interval and permutation graphs to the enumerations of paths in directed acyclic graphs(DAGs), improving considerably upon previously known results on interval graphs.
Abstract: We reduce (in polynomial time) the enumeration of minimal dominating sets in interval and permutation graphs to the enumeration of paths in directed acyclic graphs(DAGs). As a consequence, we can enumerate with linear delay, after a polynomial time pre-processing, minimal dominating sets in interval and permutation graphs. We can also count them in polynomial time. This improves considerably upon previously known results on interval graphs, and up to our knowledge no output polynomial time algorithm for the enumeration of minimal dominating sets and their counting were known for permutation graphs.
TL;DR: This work gives a polynomial time algorithm for finding an efficient dominating set in a class of graphs properly containing the class of hereditary efficiently dominatable graphs by reducing the problem to the maximum weight independent set problem in claw-free graphs.
Abstract: An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the graph's vertex set. We introduce graphs that are hereditary efficiently dominatable in that sense that every induced subgraph of the graph contains an efficient dominating set. We prove a decomposition theorem for (bull, fork, C4)-free graphs, based on which we characterize, in terms of forbidden induced subgraphs, the class of hereditary efficiently dominatable graphs. We also give a decomposition theorem for hereditary efficiently dominatable graphs and examine some algorithmic aspects of such graphs. In particular, we give a polynomial time algorithm for finding an efficient dominating set (if one exists) in a class of graphs properly containing the class of hereditary efficiently dominatable graphs by reducing the problem to the maximum weight independent set problem in claw-free graphs.
26 Aug 2013
TL;DR: The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be \(\mathbb{NP}\)-complete even for very restricted graph classes.
Abstract: Let G be a finite undirected graph. A vertex dominates itself and its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be \(\mathbb{NP}\)-complete even for very restricted graph classes.
05 Sep 2013
TL;DR: This work presents tight approximation results for the max min vertex cover problem, namely a polynomial approximation algorithm which guarantees an n − 1/2 approximation ratio, while showing that unless P = NP, the problem is inapproximable within ratio n e − (1/2) for any strictly positive e.
Abstract: We address the max min vertex cover problem, which is the maximization version of the well studied min independent dominating set problem, known to be NP-hard and highly inapproximable in polynomial time. We present tight approximation results for this problem on general graphs, namely a polynomial approximation algorithm which guarantees an n − 1/2 approximation ratio, while showing that unless P = NP, the problem is inapproximable within ratio n e − (1/2) for any strictly positive e. We also analyze the problem on various restricted classes of graph, on which we show polynomiality or constant-approximability of the problem. Finally, we show that the problem is fixed-parameter tractable with respect to the size of an optimal solution, to treewidth and to the size of a maximum matching.
TL;DR: This paper investigates the hypergraphs satisfying @t(H)=@c(H), and proves that their recognition problem is NP-hard already on the class of linear hyper graphs of rank 3, while on unrestricted problem instances it lies inside the complexity class @Q"2^p.
TL;DR: This work disproves a conjecture by Skupien that every tree of order n has at most 2^n^/^2 minimal dominating sets and provides an algorithm for listing all minimal dominate sets of a tree in time O(1.4656^n).
TL;DR: A linear-time labeling algorithm for the mixed domination problem in cacti is presented and an incomplete proof of the NP-completeness of the mixed dominance problem in split graphs is fixed.
TL;DR: The value of the domination number for some circulant graphs is obtained and a corresponding dominating set is also determined and a necessary and sufficient condition for a subgroup to be an efficient dominating set in circulants.
27 Feb 2013
TL;DR: In this article, the first linear kernels for dominating set and connected dominating set problems on graphs excluding a fixed graph H as a topological minor were given for both types of problems.
Abstract: We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor.