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A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph. In this paper, we provide the first characterization of claw-free equimatchable graphs by identifying the equimatchable claw-free graph families. This characterization implies an efficient recognition algorithm. [ABSTRACT FROM AUTHOR]

A graph G = ( V , E ) is a word-representable graph if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if ( x , y ) ∈ E for each x ≠ y . In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a semi-transitive orientation defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of G is the minimum k such that G is a representable by a word, where each letter occurs k times; such a k exists for any word-representable graph. We show that the representation number of a word-representable graph on n vertices is at most 2 n , while there exist graphs for which it is n / 2 . [ABSTRACT FROM AUTHOR]

In this paper we present improved results on the problem of counting triangles in edge streamed graphs. For graphs with m edges and at least T triangles, we show that an extra look over the stream yields a two-pass streaming algorithm that uses O ( m ε 2.5 T polylog ( m ) ) space and outputs a ( 1 + ε ) approximation of the number of triangles in the graph. This improves upon the two-pass streaming tester of Braverman, Ostrovsky and Vilenchik, ICALP 2013, which distinguishes between triangle-free graphs and graphs with at least T triangle using O ( m T 1 / 3 ) space. Also, in terms of dependence on T , we show that more passes would not lead to a better space bound. In other words, we prove there is no constant pass streaming algorithm that distinguishes between triangle-free graphs from graphs with at least T triangles using O ( m T 1 / 2 + ρ ) space for any constant ρ ≥ 0 . [ABSTRACT FROM AUTHOR]

Just as ordinary hypergraphs are a generalization of graphs, directed hypergraphs (DH) are a natural generalization of digraphs. A DH consists of a set of vertices V and a set of hyperarcs H , where a hyperarc is a pair < S , v > , S non empty subset of V and v ∈ V . DHs have a variety of applications: they have been used to represent functional dependency in databases, Horn formulae in propositional logic, and–or graphs, context free grammars etc. In the paper, after providing a brief historical introduction on the notion of DH and some relevant applications, various problems regarding DHs are surveyed and analyzed. In particular we consider the complexity of the reachability problem (together with its application in the related satisfiability problem for Horn CNF formulae) and the computation of transitive closure and transitive reduction of directed hypergraphs (together with its application to the computation of minimum coverings for a set of functional dependencies). Finally a short introduction to the problem of computing shortest hyperpaths in directed hypergraphs is provided. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the evacuation problem in a network which consists of a directed graph with capacities and transit times on its arcs. This problem can be solved by the algorithm of Hoppe and Tardos [B. Hoppe, É. Tardos, The quickest transshipment problem, Math. Oper. Res. 25(1) (2000) 36–62] in polynomial time. However their running time is high-order polynomial, and hence is not practical in general. Thus, it is necessary to devise a faster algorithm for a tractable and practically useful subclass of this problem. In this paper, we consider a network with a sink such that (i) for each vertex the sum of the transit times of arcs on any path from to takes the same value, and (ii) for each vertex the minimum - cut is determined by the arcs incident to whose tails are reachable from . This class of networks is a generalization of grid networks studied in the paper [N. Kamiyama, N. Katoh, A. Takizawa, An efficient algorithm for evacuation problem in dynamic network flows with uniform arc capacity, IEICE Trans. Infrom. Syst. E89-D (8) (2006) 2372–2379]. We propose an efficient algorithm for this network problem. [Copyright &y& Elsevier]

The scientific literature teems with clique-centric clustering strategies. In this paper we analyze one such method, the paraclique algorithm. Paraclique has found practical utility in a variety of application domains, and has been successfully employed to reduce the effects of noise. Nevertheless, its formal analysis and worst-case guarantees have remained elusive. We address this issue by deriving a series of lower bounds on paraclique densities. [ABSTRACT FROM AUTHOR]

In this paper, we study the competition graphs of oriented complete bipartite graphs. We characterize graphs that can be represented as the competition graphs of oriented complete bipartite graphs. We also present the graphs having the maximum number of edges and the graphs having the minimum number of edges among such graphs. [ABSTRACT FROM AUTHOR]

Diagnosability plays an important role in measuring the reliability of interconnection networks. Conditional faulty set is a special faulty set that does not contain all of neighbors of any vertex in a network. The conditional diagnosability is a metric that can give the maximum cardinality of the conditional faulty sets that the system is guaranteed to identify. This paper shows that the conditional diagnosability of the bubble-sort star graph B S n under the MM model is 6 n − 15 for n ≥ 6 and under the PMC model is 8 n − 21 for n ≥ 5 . [ABSTRACT FROM AUTHOR]

The average eccentricity e c c ( G ) of a graph G is the mean value of eccentricities of all vertices of G . In this paper, we continue the work of Ilić (2012) and resolve a conjecture, obtained by the system AutoGraphiX, about the upper bound on the sum of the average eccentricity and the domination number among connected graphs on n vertices. [ABSTRACT FROM AUTHOR]

An r -dynamic k -coloring of a graph G is a proper vertex k -coloring of G such that the neighbors of any vertex v receive at least min { r , deg ( v ) } different colors. The r -dynamic chromatic number of G , χ r ( G ) , is defined as the smallest k such that G admits an r -dynamic k -coloring. In this paper, first we introduce an upper bound for χ r ( G ) in terms of r , chromatic number, maximum degree and minimum degree. Then, motivated by a conjecture of Montgomery (2001) stating that for a d -regular graph G , χ 2 ( G ) − χ ( G ) ≤ 2 , we prove two upper bounds for χ 2 − χ on regular graphs. Our first upper bound ⌈ 5.437 log d + 2.721 ⌉ improves a result of Alishahi (2011). Also, our second upper bound shows that Montgomery’s conjecture is implied by the existence of a χ ( G ) -coloring for any regular graph G , such that any two vertices whose neighbors are unicolored in this coloring, have no common neighbor. [ABSTRACT FROM AUTHOR]

An orthogonal ray graph is an intersection graph of horizontal rays (closed half-lines) and vertical rays in the plane, which is introduced in connection with the defect-tolerant design of nano-circuits. An orthogonal ray graph is a 3-directional orthogonal ray graph if every vertical ray has the same direction. A 3-directional orthogonal ray graph is a 2-directional orthogonal ray graph if every horizontal ray has the same direction. The characterizations and the complexity of the recognition problem have been open for orthogonal ray graphs and 3-directional orthogonal ray graphs, while various characterizations with a quadratic-time recognition algorithm have been known for 2-directional orthogonal ray graphs. In this paper, we show several characterizations with a linear-time recognition algorithm for orthogonal ray trees by using the 2-directional orthogonal ray trees. We also show that a tree is a 3-directional orthogonal ray graph if and only if it is a 2-directional orthogonal ray graph. Moreover, we show some necessary conditions for orthogonal ray graphs and 3-directional orthogonal ray graphs. [ABSTRACT FROM AUTHOR]

In this paper, we design an algorithm that, given a directed graph G and the Cartesian-product decomposition of its underlying undirected graph G ̃ , produces the directed Cartesian-product decomposition of G in linear time. This is the first time that the linear complexity is achieved for this problem, which has two major consequences. Firstly, it shows that the directed and undirected versions of the Cartesian-product decomposition of graphs are linear-time equivalent problems. And secondly, as there already exists a linear-time algorithm for solving the undirected version of the problem, combined together, it provides the first linear-time algorithm for computing the directed Cartesian-product decomposition of a directed graph. [ABSTRACT FROM AUTHOR]

A cograph is a graph which does not contain any induced path on four vertices. In this paper, we characterize those cographs that are intersection graphs of paths on a grid in the following two cases: (i) the paths on the grid all have at most one bend and the intersections concern edges ( → B 1 -EPG); (ii) the paths on the grid are not bended and the intersections concern vertices ( → B 0 -VPG). In both cases, we give a characterization by a family of forbidden induced subgraphs. We further present linear-time algorithms to recognize B 1 -EPG cographs and B 0 -VPG cographs using their cotree. [ABSTRACT FROM AUTHOR]

A subset S of vertices of a graph G is called a k -path vertex cover if every path of order k in G contains at least one vertex from S . The cardinality of a minimum k -path vertex cover is called the k -path vertex cover number of a graph G , denoted by ψ k ( G ) . It is known that the minimum k -path vertex cover problem ( k -PVCP) is NP-hard for all k ≥ 2 . In this paper we consider the weighted version of a k -PVCP ( k -WPVCP), in which vertices are given weights, and the problem is to find a minimum weight set in G such that the graph obtained by deleting this set from G has no P k . This problem was briefly introduced by Tu and Zhou (2011) but has not been studied to much extent. We give solutions and efficient algorithms for the k -WPVCP for some special classes of graphs. In particular, for complete graphs and cycles, and most importantly an algorithm that computes the k -path vertex cover number of a tree with time complexity O ( k ⋅ | V ( G ) | ) is presented. [ABSTRACT FROM AUTHOR]

Abstract: Let be a graph with vertex set . A moplex of is both a clique and a module whose neighborhood is a minimal separator in or empty. A moplex ordering of is an ordered partition of for some integer into moplexes which are defined in the successive transitory elimination graphs, i.e., for , is a moplex of the graph induced by and induces a clique. In this paper we prove the terminal vertex by an execution of the lexicographical depth-first search (LexDFS for short) algorithm on belongs to a moplex whose vertices are numbered consecutively and further that the LexDFS algorithm on defines a moplex ordering of , which is similar to the result about the maximum cardinality search (MCS for short) algorithm on chordal graphs [J.R.S. Blair, B.W. Peyton, An introduction to chordal graphs and clique trees, IMA Volumes in Mathematics and its Applications, 56 (1993) pp. 1–30] and the result about the lexicographical breadth-first search (LexBFS for short) algorithm on general graphs [A. Berry, J.-P. Bordat, Separability generalizes Dirac’s theorem, Discrete Appl. Math., 84 (1998) 43–53]. As a corollary, we can obtain a simple algorithm on a chordal graph to generate all minimal separators and all maximal cliques. [Copyright &y& Elsevier]

Abstract: A graph is a square root of a graph if two vertices are adjacent in if and only if they are at distance one or two in . Computing a square root of a given graph is NP-hard, even when the input graph is restricted to be chordal. In this paper, we show that computing a square root can be done in linear time for a well-known subclass of chordal graphs, the class of trivially perfect graphs. This result is obtained by developing a structural characterization of graphs that have a split square root. We also develop linear time algorithms for determining whether a threshold graph given either by a degree sequence or by a separating structure has a square root. [Copyright &y& Elsevier]

Abstract: A sigraph is a graph in which each edge carries a value called its sign, denoted specially as . Given a sigraph , a new sigraph , called the common-edge sigraph of is that sigraph whose vertex-set is the set of pairs of adjacent edges in and two vertices of are adjacent if the corresponding pairs of adjacent edges of have exactly one edge in common, and the sign of the edge is the sign of the common edge. If all the edges of the sigraph carry + sign then is actually a graph and the corresponding common-edge sigraph is termed as the common-edge graph. In this paper, we characterize common-edge graph and common-edge sigraph and write an algorithm to obtain a corresponding common-edge root graph and common-edge root sigraph from a given common-edge graph and common-edge sigraph respectively. [Copyright &y& Elsevier]

Abstract: Given a positive integer and an undirected edge-weighted connected simple graph with at least edges of positive weight, we wish to partition the graph into edge-disjoint connected components of approximately the same size. We focus on the max-min ratio of the partition, which is the weight of the maximum component divided by that of the minimum component. It has been shown that for some instances, the max-min ratio is at least two. In this paper, for any graph with no edge weight larger than one half of the average weight, we provide a linear-time algorithm for delivering a partition with max-min ratio at most two. Furthermore, by an extreme example, we show that the above restriction on edge weights is the loosest possible. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce an generic framework for the mining of sequences under various constraints. More precisely, we study the enumeration of all partitions of a word into multisets of subsequences. We show that using additional predicates, this generator can be used for frequent subsequences and substrings mining. We define the transition graph whose vertices are multisets of words and arcs are transitions between multisets. We show that is a directed acyclic graph and it admits a covering tree. We use to propose a generic algorithm that enumerates all multisets that satisfies a set of predicates, without redundancy. [Copyright &y& Elsevier]

Abstract: Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept, which we call coarseness, for measuring how blended the elements of are. For , let be the bichromatic discrepancy of X. We say that a partition of S is convex if the convex hulls of its members are pairwise disjoint. The discrepancy of a convex partition Π of S is the minimum over the elements of Π. The coarseness of S is the discrepancy of the convex partition of S with maximum discrepancy. We study the coarseness of bicolored point sets, and relate it to well blended point sets. In particular, we show combinatorial results on the coarseness of general configurations and give efficient algorithms for computing the coarseness of two specific cases, namely when the set of points is in convex position and when the measure is restricted to convex partitions with two elements. [Copyright &y& Elsevier]

Abstract: In this paper, we use the primeval decomposition tree to compute the minimal separators of some graphs and to describe a linear-time algorithm that lists the minimal separators of extended -laden graphs, extending an algorithm for -sparse graphs given by Nikolopoulos and Palios [S.D. Nikolopoulos, L. Palios, Minimal separators in -sparse graphs, Discrete Math. 306 (3) (2006) 381–392]. We also give bounds on the number and total size of all minimal separators of extended -laden graphs and some of their subclasses, such as -tidy and -lite graphs. Moreover, we show that these bounds are tight for all subclasses considered. [Copyright &y& Elsevier]

Abstract: In this paper, we study the class of distance-hereditary comparability graphs, that is, those graphs which admit a transitive orientation and are completely decomposable with respect to the split decomposition. We provide a characterization based on a restricted number of forbidden subgraphs. We also provide further characterizations and one of them, based on the split decomposition, is used to devise a recognizing algorithm working in linear time. Finally, we show how to build distance-hereditary comparability graphs. [Copyright &y& Elsevier]

Abstract: Classification of weather maps at various isobaric levels as a methodological tool is used in several problems related to meteorology, climatology, atmospheric pollution and to other fields for many years. Initially the classification was performed manually. The criteria used by the person performing the classification are features of isobars or isopleths of geopotential height, depending on the type of maps to be classified. Although manual classifications integrate the perceptual experience and other unquantifiable qualities of the meteorology specialists involved, these are typically subjective and time consuming. Furthermore, during the last years different approaches of automated methods for atmospheric circulation classification have been proposed, which present automated and so-called objective classifications. In this paper a new method of atmospheric circulation classification of isobaric maps is presented. The method is based on graph theory. It starts with an intelligent prototype selection using an over-partitioning mode of fuzzy c-means (FCM) algorithm, proceeds to a graph formulation for the entire dataset and produces the clusters based on the contemporary dominant sets clustering method. Graph theory is a novel mathematical approach, allowing a more efficient representation of spatially correlated data, compared to the classical Euclidian space representation approaches, used in conventional classification methods. The method has been applied to the classification of 850hPa atmospheric circulation over the Eastern Mediterranean. The evaluation of the automated methods is performed by statistical indexes; results indicate that the classification is adequately comparable with other state-of-the-art automated map classification methods, for a variable number of clusters. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the minimum flow problem on network flows in which the lower arc capacities vary with time. We will show that this problem for set {0,1,…, T} of time points can be solved by at most n minimum flow computations, by combining of preflow-pull algorithm and reoptimization techniques (no matter how many values of T are given). Running time of the presented algorithm is O(n 2 m). [Copyright &y& Elsevier]

Abstract: In practice, it is important to construct node-disjoint paths in networks, because they can be used to increase the transmission rate and enhance the transmission reliability. The hyper-star networks were introduced to be a competitive model for both the hypercubes and the star graphs. In this paper, one-to-many node-disjoint paths are constructed between a fixed node and other nodes of such that each of these paths has length at most 4 more than the shortest path to that node. Moreover, their maximum length is not greater than the diameter+2. [Copyright &y& Elsevier]

Abstract: A graph is called an interval graph if there exists a set of intervals corresponding to the vertex set of and two vertices are adjacent to each other if and only if the two corresponding intervals are intersecting with each other. In this paper, we apply the reliability-based backup 2-center modal proposed by Wang, Wu and Chao, in which each server may fail with a given probability, to interval graphs and present an time algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the concept of -power domination which is a common generalization of domination and power domination. We extend several known results for power domination to -power domination. Concerning the complexity of the -power domination problem, we first show that deciding whether a graph admits a -power dominating set of size at most is NP-complete for chordal graphs and for bipartite graphs. We then give a linear algorithm for the problem on trees. Finally, we propose sharp upper bounds for the power domination number of connected graphs and of connected claw-free -regular graphs. [Copyright &y& Elsevier]

Abstract: In this paper, we study connected plane graphs with link component number equal to the nullity and call them near-extremal graphs. We first study near-extremal graphs with minimum degree at least 3 and prove that a connected plane graph with minimum degree at least 3 is a near-extremal graph if and only if is isomorphic to , the complete graph with 4 vertices. The result is obtained by studying general graphs using the knowledge of bicycle space and the Tutte polynomial. Then a simple algorithm is given to judge whether a connected plane graph is a near-extremal graph or not. Finally we study the construction of near-extremal graphs and prove that all near-extremal graphs can be constructed from a loop and by two graph operations. [Copyright &y& Elsevier]

Abstract: A ring star in a graph is a subgraph that can be decomposed into a cycle (or ring) and a set of edges with exactly one vertex in the cycle. In the minimum ring-star problem (mrsp) the cost of a ring star is given by the sum of the costs of its edges, which vary, depending on whether the edge is part of the ring or not. The goal is to find a ring-star spanning subgraph minimizing the sum of all ring and assignment costs. In this paper we show that the mrsp can be reduced to a minimum (constrained) Steiner arborescence problem on a layered graph. This reduction is used to introduce a new integer programming formulation for the mrsp. We prove that the dual bound generated by the linear relaxation of this formulation always dominates the one provided by an early model from the literature. Based on our new formulation, we developed a branch-and-cut algorithm for the mrsp. On the primal side, we devised a grasp heuristic to generate good upper bounds for the problem. Computational tests with these algorithms were conducted on a benchmark of public domain. In these experiments both our exact and heuristics algorithms had excellent performances, noticeably in dealing with instances whose optimal solution has few vertices in the ring. In addition, we also investigate the minimum spanning caterpillar problem (mscp) which has the same input as the mrsp and admits feasible solutions that can be viewed as ring stars with paths in the place of rings. We present an easy reduction of the mscp to the mrsp, which makes it possible to solve to optimality instances of the former problem too. Experiments carried out with the mscp revealed that our branch-and-cut algorithm is capable to solve to optimality instances with up to 200 vertices in reasonable time. [Copyright &y& Elsevier]

Abstract: An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of , denoted , is the minimum number of colors required for acyclic vertex coloring of graph . For a family of graphs, the acyclic chromatic number of , denoted by , is defined as the maximum over all the graphs . In this paper we show that where is the family of graphs of maximum degree 5 and give a linear time algorithm to achieve this bound. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the -maxian problem on interval graphs where each interval has a positive weight. First we propose a linear time algorithm which is similar to the algorithm of Bespamyatnikh for the unweighted 1-median problem on interval graphs to solve the weighted 1-maxian problem. For a 2-maxian problem, we show that two intervals with the minimum right endpoint and the maximum left endpoint, respectively, are an optimal solution. It can be easily extended to the -maxian problem (). [ABSTRACT FROM AUTHOR]

Abstract: The contribution of this paper is twofold. It presents a new approach to the matched drawability problem of pairs of planar graphs and it provides four algorithms based on this approach for drawing the pairs , , and . Further, it initiates the study of the matched drawability of triples of planar graphs: it presents an algorithm to compute a matched drawing of a triple of cycles and an algorithm to compute a matched drawing of a caterpillar and two unlabeled level planar graphs. The results extend previous work on the subject and relate to existing literature about simultaneous embeddability and unlabeled level planarity. [Copyright &y& Elsevier]

Abstract: This paper considers a new class of network flows, called dynamic generative network flows in which, the flow commodity is dynamically generated at a source node and dynamically consumed at a sink node and the arc-flow bounds are time dependent. Then the maximum dynamic flow problem in such networks for a pre-specified time horizon T is defined and mathematically formulated in both arc flow and path flow presentations. By exploiting the special structure of the problem, an efficient algorithm is developed to solve the general form of the dynamic problem as a minimum cost static flow problem. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that -connected graphs have either a dominating path or two disjoint paths, wherein the length of the two paths is bounded by the minimum among and a parameter defined on the neighborhood condition of any four independent vertices of the graph. [Copyright &y& Elsevier]

Abstract: A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing. In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing. [Copyright &y& Elsevier]

Abstract: The goal of the Cluster Editing problem is to make the fewest changes to the edge set of an input graph such that the resulting graph is a disjoint union of cliques. This problem is NP-complete but recently, several parameterized algorithms have been proposed. In this paper, we present a number of surprisingly simple search tree algorithms for Weighted Cluster Editing assuming that edge insertion and deletion costs are positive integers. We show that the smallest search tree has size for edit cost , resulting in the currently fastest parameterized algorithm, both for this problem and its unweighted counterpart. We have implemented and compared our algorithms, and achieved promising results. 1 [1] This is an extended version of two articles published in: Proc. of the 6th Asia Pacific Bioinformatics Conference, APBC 2008, in: Series on Advances in Bioinformatics and Computational Biology, vol. 5, Imperial College Press, pp. 211–220; and in: Proc. of the 2nd Conference on Combinatorial Optimization and Applications, COCOA 2008, in: LNCS, vol. 5038, Springer, pp. 289–302. [Copyright &y& Elsevier]

Abstract: The paper is devoted to demonstrating a randomized algorithm for determining a dominating set in a given graph having a maximum degree of five. The algorithm follows the Las Vegas technique. Furthermore, the concept of a 2-separated collection of subsets of vertices in graphs is used. The suggested algorithm is based on a condition of the upper bound of the cardinality of a local dominating set. If the condition is not satisfied, then the algorithm halts with an appropriate message. Otherwise, the algorithm determines the dominating set. The given algorithm is considered a polynomial-time approximation one. [Copyright &y& Elsevier]

Abstract: Let be a graph without isolated vertices. For a positive integer , a set is a -distance paired-dominating set if each vertex in is within distance of a vertex in and the subgraph induced by contains a perfect matching. In this paper, we present two linear time algorithms to find a minimum cardinality -distance paired-dominating set in interval graphs and block graphs, which are two subclasses of chordal graphs. In addition, we present a characterization of trees with unique minimum -distance paired-dominating set. [Copyright &y& Elsevier]

Abstract: Let be a simple graph without isolated vertices. A vertex set is a paired-dominating set if every vertex in has a neighbor in and the induced subgraph has a perfect matching. In this paper, we investigate the approximation hardness of paired-domination in graphs. For weighted paired-domination, an approximation algorithm in general graphs and an exact dynamic programming style algorithm in trees are also given. [Copyright &y& Elsevier]

Abstract: This paper revisits the ‘branchwidth territories’ of Kloks, Kratochvíl and Müller [T. Kloks, J. Kratochvíl, H. Müller, New branchwidth territories, in: 16th Ann. Symp. on Theoretical Aspect of Computer Science, STACS, in: Lecture Notes in Computer Science, vol. 1563, 1999, pp. 173–183] to provide a simpler proof, and a faster algorithm for computing the branchwidth of an interval graph. We also generalize the algorithm to the class of chordal graphs, albeit at the expense of exponential running time. Compliance with the ternary constraint of the branchwidth definition is facilitated by a simple new tool called -troikas: three sets of size at most each are a -troika of set , if any two have union . We give a straightforward algorithm, computing branchwidth for an interval graph on edges, vertices and maximal cliques. We also prove a conjecture of Mazoit [F. Mazoit, A general scheme for deciding the branchwidth, Technical Report RR2004-34, LIP — École Normale Supérieure de Lyon, 2004. http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-34.pdf], by showing that branchwidth can be computed in polynomial time for a chordal graph given with a clique tree having a polynomial number of subtrees. [Copyright &y& Elsevier]

Abstract: This paper presents a modification on the stair-shaped volume of fluid (VOF) advection algorithm called defined donating and accepting region (DDAR) scheme which shows the importance of the more accurate definition of donating and accepting region. The algorithm which is suitable for uniform Cartesian grids uses multi-dimensional method of cell boundary flux integration. The performance of the modified scheme presented here has been compared with the performance of a number of alternative schemes taking into account translation, rotation and shear advection tests. In some aspects, the DDAR scheme is shown to be more accurate than linear constant and flux limited schemes, and in some way comparable with the linear piecewise schemes (PLIC). Without using any geometrical information, DDAR scheme predicts flux correctly using information of cell volume fraction and its face velocities and consequently improves the performance the first order DDR schemes. [Copyright &y& Elsevier]

Abstract: In the majority problem, we are given balls coloured black or white and we are allowed to query whether two balls have the same colour or not. The goal is to find a ball of majority colour in the minimum number of queries. The answer is known to be where is the number of 1’s in the binary representation of . In this paper we study randomized algorithms for determining majority, which are allowed to err with probability at most . We show that any such algorithm must have expected running time at least . Moreover, we provide a randomized algorithm which shows that this result is best possible. These extend a result of De Marco and Pelc [G. De Marco, A. Pelc, Randomized algorithms for determining the majority on graphs, Combin. Probab. Comput. 15 (2006) 823–834]. [Copyright &y& Elsevier]

Abstract: It is a well-known result in the theory of simple games that a game is weighted if and only if it is trade robust. In this paper we propose a variant of trade robustness, that we call invariant-trade robustness, which is enough to determine whether a simple game is weighted. To test whether a simple game is invariant-trade robust we do not need to consider all winning coalitions; a reduced subset of minimal winning coalitions is enough. We make a comparison between the two methods (trade robustness and invariant-trade robustness) to check whether a simple game is weighted. We also provide by means of algorithms a full classification using both methods, for simple games with less than 8 voters according to the maximum level of (invariant-)trade robustness they achieve. [Copyright &y& Elsevier]

Abstract: For a fixed family of graphs, an -packing in a graph is a set of pairwise vertex-disjoint subgraphs of , each isomorphic to an element of . Finding an -packing that maximizes the number of covered edges is a natural generalization of the maximum matching problem, which is just . In this paper we provide new approximation algorithms and hardness results for the -packing problem where . We show that already for the -packing problem is APX-complete, and, in fact, we show that it remains so even for graphs with maximum degree 4. On the positive side, we give an approximation algorithm with approximation ratio at most 2 for every fixed . For we obtain better approximations. For we obtain a simple -approximation, achieving a known ratio that follows from a more involved algorithm of Halldórsson. For , we obtain a -approximation, and for we obtain a -approximation. [Copyright &y& Elsevier]

Abstract: Given an undirected graph with penalties associated with its vertices and costs associated with its edges, a Prize Collecting Steiner (PCS) tree is either an isolated vertex of or else any tree of , be it spanning or not. The weight of a PCS tree equals the sum of the costs for its edges plus the sum of the penalties for the vertices of not spanned by the PCS tree. Accordingly, the Prize Collecting Steiner Problem in Graphs (PCSPG) is to find a PCS tree with the lowest weight. In this paper, after reformulating and re-interpreting a given PCSPG formulation, we use a Lagrangian Non Delayed Relax and Cut (NDRC) algorithm to generate primal and dual bounds to the problem. The algorithm is capable of adequately dealing with the exponentially many candidate inequalities to dualize. It incorporates ingredients such as a new PCSPG reduction test, an effective Lagrangian heuristic and a modification in the NDRC framework that allows duality gaps to be further reduced. The Lagrangian heuristic suggested here dominates their PCSPG counterparts in the literature. The NDRC PCSPG lower bounds, most of the time, nearly matched the corresponding Linear Programming relaxation bounds. [Copyright &y& Elsevier]

Abstract: Recently, it has been shown in a series of works that the representation of graphs by Ordered Binary Decision Diagrams (OBDDs) often leads to good algorithmic behavior. However, the question for which graph classes an OBDD representation is advantageous, has not been investigated, yet. In this paper, the space requirements for the OBDD representation of certain graph classes, specifically cographs, several types of graphs with few s, unit interval graphs, interval graphs and bipartite graphs are investigated. Upper and lower bounds are proven for all these graph classes and it is shown that in most (but not all) cases a representation of the graphs by OBDDs is advantageous with respect to space requirements. [Copyright &y& Elsevier]

Abstract: Let G be an embedded planar digraph. A maximum upward planar subgraph of G is an embedding preserving subgraph that is upward planar and, among those, has the maximum number of edges. This paper presents an extensive study on the problem of computing maximum upward planar subgraphs of embedded planar digraphs: Complexity results, algorithms, and experiments are presented. Namely: (i) we prove that the addressed problem is NP-Hard; (ii) a fast heuristic and an exponential-time exact algorithm are described; (iii) a wide experimental analysis is performed to show the effectiveness of our techniques. [Copyright &y& Elsevier]

Abstract: We present in this paper a codeword for labelled -path graphs. Structural properties of this codeword are investigated, leading to the solution of two important problems: determining the exact number of labelled -path graphs with vertices and locating a hamiltonian path in a given -path graph in time . The corresponding encoding scheme is also presented, providing linear-time algorithms for encoding and decoding. [Copyright &y& Elsevier]

Splitting off a pair of edges su, sv in a graph G means replacing these two edges by a new edge uv. This operation is well known in graph theory. Let G=(V+s,E+F) be a graph which is k-edge-connected in V and suppose that |F| is even. Here F denotes the set of edges incident with s. Lova´sz (Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979) proved that if k&ges;2 then the edges in F can be split off in pairs preserving the k-edge-connectivity in V. This result was recently extended to the case where a bipartition R∪Q=V is given and every split edge must connect R and Q (SIAM J. Discrete Math. 12 (2) (1999) 160). In this paper, we investigate an even more general problem, where two disjoint subsets R,Q⊂V are given and the goal is to split off (the largest possible subset of) the edges of F preserving k-edge-connectivity in V in such a way that every split edge incident with a vertex from R has the other end-vertex in Q. Motivated by connectivity augmentation problems, we introduce another extension, the so-called split completion version of our problem. Here, the smallest set F&ast; of edges incident to s has to be found for which all the edges of F+F&ast; can be split off in the augmented graph G=(V+s, E+F+F&ast;) preserving k-edge-connectivity and in such a way that every split edge incident with a vertex from R has the other end-vertex in Q. We solve each of the above extensions when k is even: we give min–max formulae and polynomial algorithms to find the optima. For the case when k is odd we show how to find a solution to the split completion problem using at most two edges more than the optimum. [Copyright &y& Elsevier]