A dissociation set in a graph G = (V, E) is a vertex subset D such that the subgraph G[D] induced on D has vertex degree at most 1. A 3-path vertex cover in a graph is a vertex subset C such that every path of three v...
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A dissociation set in a graph G = (V, E) is a vertex subset D such that the subgraph G[D] induced on D has vertex degree at most 1. A 3-path vertex cover in a graph is a vertex subset C such that every path of three vertices contains at least one vertex from C. A vertex set D is a dissociation set if and only if V \ D is a 3-path vertex cover. There are many applications for dissociation sets and 3-path vertex covers. However, it is NP-hard to compute a dissociation set of maximum size or a 3-path vertex cover of minimum size in graphs. Several exact algorithms have been proposed for these two problems and they can be solved in O*(1.4658(n)) time in n-vertex graphs. In this paper, we reveal some interesting structural properties of the two problems, which allow us to solve them in O*(1.4656(n)) time and polynomial space or O*(1.3659(n)) time and exponential space. (C) 2016 Elsevier B.V. All rights reserved.
A set D subset of V of a graph G = (V, E) is called an open neighborhood locating-dominating set (OLD-set) if (i)N-G(v) boolean AND D not equal empty set for all v is an element of V, and (ii)N-G(u) boolean AND D not ...
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A set D subset of V of a graph G = (V, E) is called an open neighborhood locating-dominating set (OLD-set) if (i)N-G(v) boolean AND D not equal empty set for all v is an element of V, and (ii)N-G(u) boolean AND D not equal N-G(v) boolean AND D for every pair of distinct vertices u, v is an element of V. Given a graph G = (V, E), the MIN OLD-SET problem is to find an OLD-set of minimum cardinality. Given a graph G = (V, E) and a positive integer k, the DECIDE OLD-SET problem is to decide whether G has an OLD-set of cardinality at most k. The DECIDE OLD-SET problem is known to NP-complete for general graphs. In this paper we extend the NP-completeness result of the DECIDE OLD-SET problem by showing that it remains NP-complete for bipartite graphs, planar graphs, split graphs and doubly chordal graphs. We prove that the DECIDE OLD-SET problem can be solved in linear time for bounded tree-width graphs. We, then, propose a linear time algorithm for the MIN OLD SET problem in trees. We also propose a (2 + 3 ln Delta)-approximation algorithm for the MIN OLD-SET problem and show that the MIN OLD-SET problem cannot be approximated within 1/2(1 - epsilon) ln vertical bar V vertical bar for any epsilon > 0 unless NP subset of DTIME(vertical bar V vertical bar(0(log log vertical bar V vertical bar))) Finally, we prove that the MIN OLD-SET problem is APX-complete for bipartite graphs of maximum degree 3. (C) 2015 Elsevier B.V. All rights reserved.
An alliance in a graph is a set of vertices that is either safe under attacks from the neighborhood (defensive), capable of attacking its neighbors (offensive), or simultaneously defensive and offensive (powerful). An...
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An alliance in a graph is a set of vertices that is either safe under attacks from the neighborhood (defensive), capable of attacking its neighbors (offensive), or simultaneously defensive and offensive (powerful). An alliance is global if all nonmembers are adjacent to some members of the alliance. The concept of alliances was introduced by Kristiansen et al. (2004). After that many results concerning the complexity for finding a minimum alliance were shown. It was shown that the problem of finding a minimum alliance of any variant is NP-hard in general. For some variants, it was shown that the problem becomes polynomial time solvable when restricted to trees or series-parallel graphs. In this paper, we show that the problems for all variants are efficiently solvable for much larger graph classes. We present a polynomial-time algorithm for graphs of bounded clique-width. We also show that the problem is fixed-parameter tractable when parameterized by the vertex cover number. (C) 2017 Elsevier B.V. All rights reserved.
In this paper, we study the approximability of the Maximum Labeled Path problem: given a vertex-labeled directed acyclic graph D, find a path in D that collects a maximum number of distinct labels. For any epsilon >...
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In this paper, we study the approximability of the Maximum Labeled Path problem: given a vertex-labeled directed acyclic graph D, find a path in D that collects a maximum number of distinct labels. For any epsilon > 0, we provide a polynomial time approximation algorithm that computes a solution of value at least OPT1-epsilon and a self-reduction showing that any constant ratio approximation algorithm for this problem can be converted into a PTAS. This last result, combined with the APX-hardness of the problem, shows that the problem cannot be approximated within any constant ratio unless P = NP.
In this brief, we study first- and second-order consensus algorithms for the scale-free small-world Koch network, where vertices are subject to white noise. We focus on three cases of consensus schemes: 1) first-order...
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In this brief, we study first- and second-order consensus algorithms for the scale-free small-world Koch network, where vertices are subject to white noise. We focus on three cases of consensus schemes: 1) first-order leaderless algorithm;2) first-order algorithm with a single leader;and 3) second-order leaderless algorithm. We are concerned with the coherence of the Koch network in the H-2 norm, which captures the level of agreement of vertices in face of stochastic disturbances. Based on the particular network construction, we derive explicit expressions of the coherence for all the three consensus algorithms, as well as their dependence on the network size. Particularly, for the first-order leader-follower model, we show that coherence relies on the shortest-path distance between the leader and the largest-degree vertices, as well as the degree of the leader. The asymptotic behaviors for coherence of the three consensus algorithms in Koch network behave differently from those associated with other networks lacking scale-free small-world features, indicating significant influences of the scale-free small-world topology on the performance of the consensus algorithms in noisy environments.
For an edge-weighted graph , in which the vertices are partitioned into k clusters , a spanning tree T of G is a clustered spanning tree if T can be cut into k subtrees by removing edges such that each subtree is a sp...
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For an edge-weighted graph , in which the vertices are partitioned into k clusters , a spanning tree T of G is a clustered spanning tree if T can be cut into k subtrees by removing edges such that each subtree is a spanning tree for one cluster. In this paper, we show the inapproximability of finding a clustered spanning tree with minimum routing cost, where the routing cost is the total distance summed over all pairs of vertices. We present a 2-approximation for the case that the input is a complete weighted graph whose edge weights obey the triangle inequality. We also study a variant in which the objective function is the total distance summed over all pairs of vertices of different clusters. We show that the problem is polynomial-time solvable when the number of clusters k is 2 and NP-hard for . Finally, we propose a polynomial-time 2-approximation algorithm for the case of three clusters.
Reconstruction of family trees, or pedigree reconstruction, for a group of individuals is a fundamental problem in genetics. The problem is known to be NP-hard even for datasets known to only contain siblings. Some re...
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Reconstruction of family trees, or pedigree reconstruction, for a group of individuals is a fundamental problem in genetics. The problem is known to be NP-hard even for datasets known to only contain siblings. Some recent methods have been developed to accurately and efficiently reconstruct pedigrees. These methods, however, still consider relatively simple pedigrees, for example, they are not able to handle half-sibling situations where a pair of individuals only share one parent. In this work, we propose an efficient method, IPED2, based on our previous work, which specifically targets reconstruction of complicated pedigrees that include half-siblings. We note that the presence of half-siblings makes the reconstruction problem significantly more challenging which is why previous methods exclude the possibility of half-siblings. We proposed a novel model as well as an efficient graph algorithm and experiments show that our algorithm achieves relatively accurate reconstruction. To our knowledge, this is the first method that is able to handle pedigree reconstruction from genotype data when half-sibling exists in any generation of the pedigree.
We present parameterized algorithms for the k-path problem, the p-packing of q-sets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only...
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We present parameterized algorithms for the k-path problem, the p-packing of q-sets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only in the parameter (k, p, q) and using polynomial space. The constant bases of the exponentials are significantly smaller than in previous works;for example, for the k-path problem the improvement is from 2 to 1.66. We also show how to detect if a d-regular graph admits an edge coloring with d colors in time within a polynomial factor of 2((d-1)n/2). Our techniques generalize an algebraic approach studied in various recent works. (c) 2017 Elsevier Inc. All rights reserved.
Untargeted metabolomics makes it possible to identify compounds that undergo significant changes in concentration in different experimental conditions. The resulting metabolomic profile characterizes the perturbation ...
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Untargeted metabolomics makes it possible to identify compounds that undergo significant changes in concentration in different experimental conditions. The resulting metabolomic profile characterizes the perturbation concerned, but does not explain the underlying biochemical mechanisms. Bioinformatics methods make it possible to interpret results in light of the whole metabolism. This knowledge is modelled into a network, which can be mined using algorithms that originate in graph theory. These algorithms can extract sub-networks related to the compounds identified. Several attempts have been made to adapt them to obtain more biologically meaningful results. However, there is still no consensus on this kind of analysis of metabolic networks. This review presents the main graph approaches used to interpret metabolomic data using metabolic networks. Their advantages and drawbacks are discussed, and the impacts of their parameters are emphasized. We also provide some guidelines for relevant sub-network extraction and also suggest a range of applications for most methods.
A determinant property of the structure of a biological network is the distribution of local connectivity patterns, i.e., network motifs. In this work, a method for creating directed, unweighted networks while promoti...
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A determinant property of the structure of a biological network is the distribution of local connectivity patterns, i.e., network motifs. In this work, a method for creating directed, unweighted networks while promoting a certain combination of motifs is presented. This motif-based network algorithm starts with an empty graph and randomly connects the nodes by advancing or discouraging the formation of chosen motifs. The in-or out-degree distribution of the generated networks can be explicitly chosen. The algorithm is shown to perform well in producing networks with high occurrences of the targeted motifs, both ones consisting of three nodes as well as ones consisting of four nodes. Moreover, the algorithm can also be tuned to bring about global network characteristics found in many natural networks, such as small-worldness and modularity.
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