Search

A circulant graph C n ( a 1 , … , a k ) is a graph with n vertices { v 0 , … , v n − 1 } such that each vertex v i is adjacent to vertices v ( i + a j ) m o d n , for j = 1 , … , k . In this paper we investigate the vertex colouring problem on circulant graphs. We approach the problem in a purely combinatorial way based on an array representation and propose an exact O ( k 3 log 2 n + n ) algorithm for a subclass of 3-chromatic C n ( a 1 , … , a k ) ’s with k ≥ 2 , which are characterized in the paper. [ABSTRACT FROM AUTHOR]

We study the problem of finding in a graph an inclusionwise maximal independent set of minimum cardinality, known as minimum maximal independent set or independent dominating set problem. This is one of the hardest problems in algorithmic graph theory. In particular, restricted to the class of so called SAT-graphs, this problem coincides with satisfiability , the central problem of theoretical computer science. The class of SAT-graphs, and many other important graph classes, such as graphs of bounded vertex degree or line graphs, can be characterized by finitely many forbidden induced subgraphs. We call such classes finitely defined. The paper [R. Boliac and V. Lozin, Independent domination in finitely defined classes of graphs, Theoretical Computer Science, 301 (2003) 271–284] reveals various conditions under which the independent dominating set problem remains NP-hard in a finitely defined class. In the present paper, we identify a number of finitely defined classes where the problem admits polynomial-time solutions. In particular, we prove that the problem is solvable in polynomial time in the class of P 2 + P 3 -free graphs by correcting a mistake of the first two authors of the paper in their earlier solution. This result is in a sharp contrast with the fact that in the class of P 3 + P 3 -free graphs the problem is known to be NP-hard, since this class contains all SAT-graphs. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the problem of connected edge searching of weighted trees. Barrière et al. claim in [L. Barrière, P. Flocchini, P. Fraigniaud, N. Santoro, Capture of an intruder by mobile agents, in: SPAA’02: Proceedings of the Fourteenth Annual ACM Symposium on Parallel Algorithms and Architectures, ACM, New York, NY, USA, 2002, pp. 200–209] that there exists a polynomial-time algorithm for finding an optimal search strategy, that is, a strategy that minimizes the number of used searchers. However, due to some flaws in their algorithm, the problem turns out to be open. It is proven in this paper that the considered problem is strongly NP-complete even for node-weighted trees (the weight of each edge is 1) with one vertex of degree greater than 2. It is also shown that there exists a polynomial-time algorithm for finding an optimal connected search strategy for a given bounded degree tree with arbitrary weights on the edges and on the vertices. This is an FPT algorithm with respect to the maximum degree of a tree. [Copyright &y& Elsevier]

Abstract Several graph theoretic watermark methods have been proposed to encode numbers as graph structures in software watermarking environments. In this paper we propose an efficient and easily implementable codec system for encoding watermark numbers as reducible permutation flow-graphs and, thus, we extend the class of graphs used in such a watermarking environment. More precisely, we present an algorithm for encoding a watermark number w as a self-inverting permutation π ∗ , an algorithm for encoding the self-inverting permutation π ∗ into a reducible permutation graph F [ π ∗ ] whose structure resembles the structure of real program graphs, as well as decoding algorithms which extract the permutation π ∗ from the reducible permutation graph F [ π ∗ ] and the number w from π ∗. Both the encoding and the decoding process take, in the worst-case, time and space linear in the length of the binary representation of w. The two main components of our proposed codec system, i.e., the self-inverting permutation π ∗ and the reducible permutation graph F [ π ∗ ] , incorporate the binary representation of the watermark w in their structure and possess important structural properties, which make our system resilient to attacks; to this end, we experimentally evaluated our system under edge modification attacks on the graph F [ π ∗ ] and the results suggest that we can detect such attacks with high probability. [ABSTRACT FROM AUTHOR]

Abstract Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yannakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with problems on non-oriented edge-colored graphs. The aim is to find a route between two given vertices and . This route can be a walk, a trail or a path. Each time a vertex is crossed by a walk there is an associated non-negative reload cost , where and denote, respectively, the colors of successive edges in this walk. The goal is to find a route whose total reload cost is minimum. Polynomial algorithms and proofs of NP-hardness are given for particular cases: when the triangle inequality is satisfied or not, when reload costs are symmetric (i.e., ) or asymmetric. We also investigate bounded degree graphs and planar graphs. We conclude the paper with the traveling salesman problem with reload costs. [Copyright &y& Elsevier]

Abstract: The frequent connected subgraph mining problem, i.e., the problem of listing all connected graphs that are subgraph isomorphic to at least a certain number of transaction graphs of a database, cannot be solved in output polynomial time in the general case. If, however, the transaction graphs are restricted to forests then the problem becomes tractable. In this paper we generalize the positive result on forests to graphs of bounded tree-width. In particular, we show that for this class of transaction graphs, frequent connected subgraphs can be listed in incremental polynomial time. Since subgraph isomorphism remains NP-complete for bounded tree-width graphs, the positive complexity result of this paper shows that efficient frequent pattern mining is possible even for computationally hard pattern matching operators. [Copyright &y& Elsevier]

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]

Many vertex-partitioning problems can be expressed within a general framework introduced by Telle and Proskurowski. They showed that optimization problems in this framework can be solved in polynomial time on classes of graphs with bounded tree-width. In this paper, we consider a very similar framework, in relationship with more general classes of graphs: we propose a polynomial time algorithm on classes of graphs with bounded clique-width for all the optimization problems in our framework. These classes of graphs are more general than the classes of graphs with bounded tree-width in the sense that classes of graphs with bounded tree-width have also bounded clique-width (but not necessarily the inverse).Our framework includes problems such as independent (dominating) set, p-dominating set, induced bounded degree subgraph, induced p-regular subgraph, perfect matching cut, graph k-coloring and graph list-k-coloring with cardinality constraints (fixed k). This paper thus provides a second (distinct) framework within which the optimization problems can be solved in polynomial time on classes of graphs with bounded clique-width, after a first framework (called MS1) due to the work of Courcelle, Makowsky and Rotics (for which they obtained a linear time algorithm). [Copyright &y& Elsevier]

A class of timed discrete event systems can be modeled by using Timed-Event Graphs, a class of timed Petri nets that can have its firing dynamic described by using an algebra called “Max-plus algebra”. For this kind of systems it may be desirable to enforce some timing constraints in steady state. In this paper, this problem is called a “max-plus regulation problem”. In this context we show a necessary condition for solving these regulation problems and in addition that this condition is sufficient for a large class of problems. The obtained controller is a simple linear static state feedback and can be computed using efficient pseudo-polynomial algorithms. Simulation results will illustrate the applicability of the proposed methodology. [ABSTRACT FROM AUTHOR]

The method of augmenting graphs is a general approach to solve the Maximum Independent Set problem. Our objective is to employ this approach to develop polynomial-time algorithms for some so-called Maximum Set problems, i.e. problems which can be stated as follows. Given a (simple) graph G , find a maximum vertex subset S of G such that the subgraph induced by S satisfies a given property Π . Such problems were shown to be NP-hard in general if the properties considered are non-trivial and hereditary Lewis and Yannakakis (1980) and Yannakakis (1978). In this paper, using the augmenting graph technique, we describe a graph class, in which some problems can be solved in polynomial time. We also prove the NP-hardness of some Maximum Set problems where the considered properties are not hereditary. [ABSTRACT FROM AUTHOR]

Although, the Hamiltonicity of solid grid graphs are polynomial-time decidable, the complexity of the longest cycle problem in these graphs is still open. In this paper, by presenting a linear-time constant-factor approximation algorithm, we show that the longest cycle problem in solid grid graphs is in APX. More precisely, our algorithm finds a cycle of length at least 2 n 3 + 1 in 2-connected n -node solid grid graphs. [ABSTRACT FROM AUTHOR]

Abstract: The topological containment problem, TC( ), has been shown to be polynomial-time solvable for any fixed pattern graph , but practical algorithms have been developed only for a few specific pattern graphs. Among these are the wheels with four, five, and six spokes. This paper examines the topological containment problem where the pattern graph is a wheel with seven spokes, and gives a result that describes graphs with no -subdivision, showing how they can be built up, using certain operations, from smaller ‘pieces’ that meet certain conditions. We also discuss algorithmic aspects of the problem. [Copyright &y& Elsevier]

Abstract: The longest path problem asks for a path with the largest number of vertices in a given graph. In contrast to the Hamiltonian path problem, until recently polynomial algorithms for the longest path problem were known only for small graph classes, such as trees. Recently, a polynomial algorithm for this problem on interval graphs has been presented in Ioannidou et al. (2011) [19] with running time on a graph with vertices, thus answering the open question posed in Uehara and Uno (2004) [32]. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect; for instance, several problems– e.g. coloring –are NP-hard on circular-arc graphs, although they can be efficiently solved on interval graphs. In this paper, we prove that for every path of a circular-arc graph , we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph of admits a path on the same vertices as . This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of . As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs. In addition, by exploiting deeper the structure of circular-arc graphs, we manage to get rid of the linear time overhead of the reduction, and thus this algorithm turns out to have the same running time as the one on interval graphs. Our algorithm, which significantly simplifies the approach of Ioannidou et al. (2011) [19], computes in the same time an -approximation of the (exponentially large in worst case) number of different vertex sets that provide a longest path; in the case where is an interval graph, we compute the exact number. Moreover, in contrast to Ioannidou et al. (2011) [19], this algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight. [Copyright &y& Elsevier]

Abstract: We consider the problem of approximating the -chromatic number of a graph. In 2005, Corteel et al. proved that there is no constant for which this problem can be approximated within a factor of in polynomial time, unless . An existence of a constant-factor approximation algorithm for the -chromatic number in general graphs was formulated as an open problem. In this paper we settle this question in a negative way proving that there is no constant , for which the problem can be approximated within a factor , unless . [Copyright &y& Elsevier]

Abstract: In this paper we consider single-machine scheduling problems with position-dependent processing times, i.e., jobs whose processing times are an increasing or decreasing function of their positions in a processing sequence. In addition, the jobs are related by parallel chains and a series–parallel graph precedence constraints, respectively. It is shown that for the problems of minimization of the makespan polynomial algorithms exist. [Copyright &y& Elsevier]

Abstract: The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with edge-disjoint spanning trees. Edmonds generalizes this theorem to matroids with disjoint bases. For any graph that may not have -edge-disjoint spanning trees, the problem of determining what edges should be added to so that the resulting graph has edge-disjoint spanning trees has been studied by Haas (2002) and Liu et al. (2009) , among others. This paper aims to determine, for a matroid that has disjoint bases, the set of elements in such that for any , also has disjoint bases. Using the matroid strength defined by Catlin et al. (1992) , we present a characterization of in terms of the strength of . Consequently, this yields a characterization of edge sets in a graph with at least edge-disjoint spanning trees such that , also has edge-disjoint spanning trees. Polynomial algorithms are also discussed for identifying the set in a matroid , or the edge subset for a connected graph . [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: Let be a graph and be an assignment of thresholds to the vertices of . A subset of vertices is said to be a dynamic monopoly corresponding to if the vertices of can be partitioned into subsets such that and for any , each vertex in has at least neighbors in . Dynamic monopolies are in fact modeling the irreversible spread of influence in social networks. In this paper we first obtain a lower bound for the smallest size of any dynamic monopoly in terms of the average threshold and the order of graph. Also we obtain an upper bound in terms of the minimum vertex cover of graphs. Then we derive the upper bound for the smallest size of any dynamic monopoly when the graph contains at least one odd vertex, where the threshold of any vertex is set as (i.e. strict majority threshold). This bound improves the best known bound for strict majority threshold. We show that the latter bound can be achieved by a polynomial time algorithm. We also show that is an upper bound for the size of strict majority dynamic monopoly, where stands for the matching number of . Finally, we obtain a basic upper bound for the smallest size of any dynamic monopoly, in terms of the average threshold and vertex degrees. Using this bound we derive some other upper bounds. [Copyright &y& Elsevier]

Abstract: In the context of designing a scalable overlay network to support decentralized topic-based pub/sub communication, the Minimum Topic-Connected Overlay problem (Min-TCO in short) has been investigated: given a set of topics and a collection of users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. It is known that Min-TCO is -hard and approximable within in polynomial time. In this paper, we further investigate the problem and some of its special instances. We give various hardness results for instances where the number of topics in which a user is interested in is bounded by a constant, and also for the instances where the number of users interested in a common topic is a constant. For the latter case, we present a first constant approximation algorithm. We also present some polynomial-time algorithms for very restricted instances of Min-TCO. [Copyright &y& Elsevier]

Abstract: Clique-width is a relatively new parameterization of graphs, philosophically similar to treewidth. Clique-width is more encompassing in the sense that a graph of bounded treewidth is also of bounded clique-width (but not the converse). For graphs of bounded clique-width, given the clique-width decomposition, every optimization, enumeration or evaluation problem that can be defined by a monadic second-order logic formula using quantifiers on vertices, but not on edges, can be solved in polynomial time. This is reminiscent of the situation for graphs of bounded treewidth, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded clique-width are a larger class than graphs of bounded treewidth, on which we can resolve fewer, but still many, optimization problems efficiently. One of the major open questions regarding clique-width is whether graphs of clique-width at most , for fixed , can be recognized in polynomial time. In this paper, we present the first polynomial-time algorithm () to recognize graphs of clique-width at most 3. [Copyright &y& Elsevier]

Abstract: In this paper we merge recent developments on exact algorithms for finding an ordering of vertices of a given graph that minimizes bandwidth (the Bandwidth problem) and for finding an embedding of a given graph into a line that minimizes distortion (the Distortion problem). For both problems we develop algorithms that work in time and polynomial space. For Bandwidth, this improves algorithm by Feige and Kilian from 2000, for Distortion this is the first polynomial space exact algorithm that works in time we are aware of. As a byproduct, we enhance the —time and —space algorithm for Distortion by Fomin et al. to an algorithm working in —time and space. [Copyright &y& Elsevier]

Abstract: In this paper we show that the problem of identifying an edge of a graph such that there exists an optimal vertex cover of containing exactly one of the vertices and is NP-hard. Such an edge is called a weak edge. We then develop a polynomial time approximation algorithm for the vertex cover problem with performance guarantee , where is an upper bound on a measure related to a weak edge of a graph. A related problem of identifying an edge such that there exists an optimal vertex cover containing both vertices and is also shown to be NP-hard. Further, we discuss a new relaxation of the vertex cover problem which is used in our approximation algorithm to obtain smaller values of . We also obtain linear programming representations of the vertex cover problem on special graphs. [Copyright &y& Elsevier]

Abstract: This paper is aimed at investigating some computational aspects of different isoperimetric problems on weighted trees. In this regard, we consider different connectivity parameters called minimum normalized cuts/isoperimetric numbers defined through taking the minimum of the maximum or the mean of the normalized outgoing flows from a set of subdomains of vertices, where these subdomains constitute a partition/subpartition. We show that the decision problem for the case of taking -partitions and the maximum (called the max normalized cut problem NCP), and the other two decision problems for the mean version (referred to as IPP and NCP) are -complete problems for weighted trees. On the other hand, we show that the decision problem for the case of taking -subpartitions and the maximum (called the max isoperimetric problem IPP) can be solved in linear time for any weighted tree and any . On the basis of this fact, we provide polynomial time -approximation algorithms for all different versions of the th isoperimetric numbers considered. Moreover, for when the number of partitions/subpartitions, , is a fixed constant, we prove, as an extension of a result of Mohar (1989) for the case (usually referred to as the Cheeger constant), that the max and mean isoperimetric numbers of weighted trees, and their max minimum normalized cut can be computed in polynomial time. We also prove some hardness results for the case of simple unweighted graphs and trees. [Copyright &y& Elsevier]

Abstract: Given a simple and finite connected graph G, the distance is the length of the shortest -path in G. Cicerone and Di Stefano [Graphs with bounded induced distance, Discrete Applied Mathematics 108 (2001), pp. 3–21] introduced and studied the class of k-distance-hereditary graphs, i.e., graphs where the distance in each connected induced subgraph is at most k times the distance in the whole graph. In this paper we make a step forward in the study of such graphs by providing characterizations for k-distance-hereditary graphs, , in terms of both forbidden subgraphs and cycle-chord conditions. Such results lead to a polynomial-time recognition algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number and the harmonious chromatic number of -tidy graphs and ()-graphs, for every fixed q. These classes include cographs, -sparse and -lite graphs. We also obtain a polynomial time algorithm to determine the Grundy number of ()-graphs. All these coloring problems are known to be NP-hard for general graphs. [Copyright &y& Elsevier]

Abstract: The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time ) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph of G admits a path on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of . As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight. [Copyright &y& Elsevier]

Abstract: Combinatorial maps explicitly encode orientations of edges around vertices, and have been used in many fields. In this paper, we address the problem of searching for patterns in model maps by putting forward the concept of symbol graph. A symbol graph will be constructed and stored for each model map in the preprocessing. Furthermore, an algorithm for submap isomorphism is presented based on symbol sequence searching in the symbol graphs. The computational complexity of this algorithm is quadratic in the worst case if we neglect the preprocessing step. [Copyright &y& Elsevier]

Abstract: Minesweeper is a popular single player game. It has been shown that the Minesweeper consistency problem is NP-complete and the Minesweeper counting problem is #P-complete. In this paper, we present efficient algorithms for solving these problems for Minesweeper graphs with bounded treewidth. Our algorithms turn out to be much better than those based directly on dynamic programming. The algorithms mostly use of algebraic operations on multivariate polynomials, so that one may use existing software to implement them easily. [Copyright &y& Elsevier]

Abstract: We consider the minimum rainbow subgraph problem (MRS): given a graph , whose edges are coloured with colours. Find a subgraph of of minimum order and with edges such that each colour occurs exactly once. For graphs with maximum degree there is a greedy polynomial-time approximation algorithm for the MRS problem with an approximation ratio of . In this paper we present a polynomial-time approximation algorithm with an approximation ratio of for . [ABSTRACT FROM AUTHOR]

Abstract: Given an undirected graph and two specified nonadjacent nodes a and b of V, a cut separator is a subset such that and a and b belong to different connected components of the graph induced by . Given a nonnegative cost vector , cut separator of minimum cost. This new problem is closely related to the vertex separator problem. In this paper, we give a polynomial time algorithm for this problem. We also present four equivalent linear formulations, and we show their tightness. Using these results we obtain an explicit short polyhedral description of the dominant of the cut separator polytope. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce and study cooperative variants of the Traveling Salesperson Problem. In these problems a salesperson has to make deliveries to customers who are willing to help in the process. Customer cooperativeness may be manifested in several modes: they may assist by approaching the salesperson, by reselling the goods they purchased to other customers, or by doing both. Several objectives are of interest: minimizing the total distance traveled by all of the participants, minimizing the maximal distance traveled by a participant and minimizing the total time until all of the deliveries are made. All of the combinations of cooperation modes and objective functions are considered, both in weighted undirected graphs and in Euclidean space. We show that most of the problems have a constant approximation algorithm, many of the others admit a PTAS, and a few are solvable in polynomial time. On the intractability side we provide NP-hardness proofs and inapproximability factors, some of which are tight. [Copyright &y& Elsevier]

Abstract: Given an -vertex directed graph and a set of requests, we consider assigning a set of edges to each vertex in so that for every request in the union of the edge sets assigned to and contains a path from to . The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinalities of the edge sets assigned to each vertex. This problem has been shown to be NP-hard in general, though it is polynomially solvable for some restricted classes of graphs and restricted request structures, such as bidirectional trees with requests of all pairs of vertices. In this paper, we give an advanced investigation about the difficulty of MCD by focusing on the relationship between its (in)approximability and request structures. We first show that MCD with general has lower and upper bounds on approximation ratio under the assumption . We then assume forms a clique structure, called Subset-Full, which is a natural setting in the context of the application. Interestingly, under this natural setting, MCD becomes 2-approximable, though it has still no polynomial time approximation algorithm whose factor is better than 677/676 unless . Finally, we show that this approximation ratio can be improved to 3/2 for the undirected variant of MCD with Subset-Full. [Copyright &y& Elsevier]

Abstract: The Cops and Robbers game as originally defined independently by Quilliot and by Nowakowski and Winkler in the 1980s has been much studied, but very few results pertain to the algorithmic and complexity aspects of it. In this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is NP-hard and that the parameterized version of the problem is W[2]-hard; the proof extends to the case where the robber moves time faster than the cops. We show that on split graphs, the problem is polynomially solvable if but is NP-hard if . We further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for . Finally, we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops. [Copyright &y& Elsevier]

Abstract: Given an undirected edge-weighted network in which one edge capacity is allowed to vary, we propose in this paper a polynomial algorithm that needs only two 2-cut-trees computations to provide the all pairs maximum 2-route flow values. [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: This paper deals with the problem of additional sensor location in order to recover the state and input observability for structured linear systems. The proposed method is based on a graph-theoretic approach and assumes only the knowledge of the system’s structure. It allows one to provide the minimal number of the required sensors and either their pertinent location or a necessary and sufficient condition which allows one to check if a sensor location is adequate or not. We obtain a sensor placement procedure based on classical and well-known graph theory algorithms, which have polynomial complexity orders. [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: The -sparse Graph Sandwich Problem asks, given two graphs and , whether there exists a graph such that and is -sparse. In this paper we present a polynomial-time algorithm for solving the Graph Sandwich Problem for -sparse graphs. [Copyright &y& Elsevier]

Abstract: This paper introduces a family of modular, self-similar, small-world graphs with clustering zero. Relevant properties of this family are comparable to those of some networks associated with technological systems with a low clustering, like the power grid or some electronic circuits. Moreover, the graphs are outerplanar and it is know that many algorithms that are NP-complete for general graphs perform polynomial in outerplanar graphs. Therefore the graphs constitute a good mathematical model for these systems. [Copyright &y& Elsevier]

Abstract: A set of vertices in a graph with no isolated vertices is a paired-dominating set of if every vertex of is adjacent to at least one vertex in and if the subgraph induced by contains a perfect matching. Let be a permutation graph and be its corresponding permutation. In this paper we present an time algorithm for finding a minimum cardinality paired-dominating set for a permutation graph with vertices and edges. [Copyright &y& Elsevier]

Abstract: Suppose that each vertex of a graph G is either a supply vertex or a demand vertex and is assigned a positive real number, called the supply or the demand. Each demand vertex can receive “power” from at most one supply vertex through edges in G. One thus wishes to partition G into connected components by deleting edges from G so that each component C either has no supply vertex or has exactly one supply vertex whose supply is at least the sum of demands in C, and wishes to maximize the fulfillment, that is, the sum of demands in all components with supply vertices. This maximization problem is known to be NP-hard even for trees having exactly one supply vertex and strongly NP-hard for general graphs. In this paper, we focus on the approximability of the problem. We first show that the problem is MAXSNP-hard and hence there is no polynomial-time approximation scheme (PTAS) for general graphs unless . We then present a fully polynomial-time approximation scheme (FPTAS) for series-parallel graphs having exactly one supply vertex. [Copyright &y& Elsevier]

Abstract: The class of fork-free graphs is an extension of claw-free graphs and their subclass of line graphs. The first polynomial-time solution to the maximum weight independent set problem in the class of line graphs, which is equivalent to the maximum matching problem in general graphs, has been proposed by Edmonds in 1965 and then extended to the entire class of claw-free graphs by Minty in 1980. Recently, Alekseev proposed a solution for the larger class of fork-free graphs, but only for the unweighted version of the problem, i.e., finding an independent set of maximum cardinality. In the present paper, we describe the first polynomial-time algorithm to solve the problem for weighted fork-free graphs. [Copyright &y& Elsevier]

Abstract: In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of of the optimum in -time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of of the optimum in -time, where is the minimum k such that there is a spanning tree of the graph with maximum degree k. [Copyright &y& Elsevier]

Abstract: Let M be a finite set of vectors in of cardinality m and the central hyperplane arrangement represented by M. An independent subset of M of cardinality n is called a Camion basis, if it determines a simplex region in the arrangement . In this paper, we first present a new characterization of Camion bases, in the case where M is the column set of the node-edge incidence matrix (without one row) of a given connected digraph. Then, a general characterization of Camion bases as well as a recognition procedure which runs in are given. Finally, an algorithm which finds a Camion basis is presented. For certain classes of matrices, including totally unimodular matrices, it is proven to run in polynomial time and faster than the algorithm due to Fonlupt and Raco. [Copyright &y& Elsevier]

Abstract: Let be an undirected tree, in which each edge is associated with a non-negative cost, and let be a collection of k distinct pairs of vertices. Given a requirement parameter , the partial multicut on a tree problem asks to find a minimum cost set of edges whose removal from T disconnects at least t out of these k pairs. This problem generalizes the well-known multicut on a tree problem, in which we are required to disconnect all given pairs. The main contribution of this paper is an -approximation algorithm for partial multicut on a tree, whose run time is strongly polynomial for any fixed . This result is achieved by introducing problem-specific insight to the general framework of using the Lagrangian relaxation technique in approximation algorithms. Our algorithm utilizes a heuristic for the closely related prize-collecting variant, in which we are not required to disconnect all pairs, but rather incur penalties for failing to do so. We provide a Lagrangian multiplier preserving algorithm for the latter problem, with an approximation factor of 2. Finally, we present a new 2-approximation algorithm for multicut on a tree, based on LP-rounding. [Copyright &y& Elsevier]

Abstract: It has been a challenging open problem whether there is a polynomial time approximation algorithm for the VERTEX COVER problem whose approximation ratio is bounded by a constant less than 2. In this paper, we study the VERTEX COVER problem on graphs with perfect matching (shortly, VC-PM). We show that if the VC-PM problem has a polynomial time approximation algorithm with approximation ratio bounded by a constant less than 2, then so does the VERTEX COVER problem on general graphs. Approximation algorithms for VC-PM are developed, which induce improvements over previously known algorithms on sparse graphs. For example, for graphs of average degree 5, the approximation ratio of our algorithm is 1.414, compared with the previously best ratio 1.615 by Halldórsson and Radhakrishnan. [Copyright &y& Elsevier]

In this paper, we study a chromatic aspect for the class of P6-free graphs. Here, the focus of our interest are graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colorability can be decided in polynomial time and, if so, a proper 3-coloring can be determined also in polynomial time. Note that the 3-colorability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. for triangle- and K1,5-free graphs (Discrete Math. 162 (1–3) (1996) 313–317). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-coloring of a given graph in general. Our approach is based on an encoding of the problem with Boolean formulas making use of the existence of bounded dominating subgraphs. Together with a structural analysis of the non-perfect K4-free members of the graph class in consideration we obtain our main result that 3-colorability can be decided in polynomial time for the class of P6-free graphs. [Copyright &y& Elsevier]

A fork is a graph that is obtained from K1,3 by subdividing one edge. It is known that for K1,3-free graphs the problem of finding the largest independent set can be solved in a polynomial time. In this paper, we prove that this is also true for the wider class of fork-free graphs. [Copyright &y& Elsevier]