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A connected dominating set (CDS, for short) for graph G is a dominating set which induces a connected subgraph of G. In this paper, we consider the problem of finding a minimum CDS for unit disk graphs, which have recently attracted considerable attention as a model of virtual backbone in multi-hop wireless networks. This optimization problem is known to be NP-hard and many approximation algorithms have been proposed in the literature. This paper proves a lower bound on the performance ratio of a greedy algorithm of Guha and Khuller which was originally proposed for general graphs and is known to be a representative in which the lookahead plays a crucial role in selecting a vertex to be contained in the CDS. More specifically, we show that for any fixed and integer , there is a unit disk graph with bounded degree consisting of at least vertices for which the output of the greedy algorithm is no better than times of an optimum solution to the same graph. [ABSTRACT FROM AUTHOR]

In this paper, we consider the discrete unit square cover (DUSC) problem as follows: given a set 𝒫 of n points and a set 𝒮 of m axis-aligned unit squares in ℝ 2 , the objective is (i) to check whether the union of the squares in 𝒮 covers all the points in 𝒫 , and (ii) if the answer is yes, then select a minimum cardinality subset 𝒮 ∗ ⊆ 𝒮 such that each point in 𝒫 is covered by at least one square in 𝒮 ∗ . For the DUSC problem: (i) we propose a (2 + 4 k − 2) -approximation algorithm, where k (> 2) is an integer parameter that defines a trade-off between the running time and the approximation factor of the algorithm. The running time of our proposed algorithm is O (k m k n + n log n). Our solution of the DUSC problem is based on a simple (1 + 2 k − 2) -approximation algorithm for the subproblem strip square cover (SSC) problem, where all the points in 𝒫 are lying within a horizontal strip of unit height. (ii) we also propose a 2-approximation algorithm, which runs in O (m 4 n) time. The 2-approximation algorithm is based on an algorithm for the subproblem SSC problem. The algorithm for the subproblem is developed using plane sweep and graph search traversal techniques. We also extend this algorithm to get 2-approximation result for the weighted DUSC problem where the squares are assigned weights, and the aim is to choose a subset 𝒮 ∗ ⊆ 𝒮 such that each point in 𝒫 is covered by at least one square in 𝒮 ∗ and the sum of the weights of squares in 𝒮 ∗ is minimized. [ABSTRACT FROM AUTHOR]

Measuring the importance (or centrality) of vertices in a network is a significant topic in complex network analysis, which has significant applications in diverse domains, for example, disease control, spread of rumors, viral marketing and so on. Existing studies mainly focus on social networks with only positive (or friendship) relations, while signed networks with also negative (or enemy) relations are seldom studied. Various signed networks commonly exist in real world, e.g. a network indicating friendship/enmity, love/hate or trust/mistrust relationships. In this paper, we propose a new centrality method named CS_TOTR to give a ranking of vertices in directed signed networks. To design this new method, we use the “status theory” for signed networks, and also adopt the vertex ranking algorithm for a tournament and the topological sorting algorithm for a general directed graph. We apply this new centrality method on the famous Sampson Monastery dataset and obtain a convincing result which shows its validity. [ABSTRACT FROM AUTHOR]

In the past decades, graphs that are determined by their spectrum have received much more and more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. The main aim of this study is to characterize two classes of multicone graphs which are determined by their adjacency, Laplacian and signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let denote a complete graph on vertices. In the paper, we show that multicone graphs and are determined by both their adjacency spectra and their Laplacian spectra, where and denote the Local Higman-Sims graph and the Local graph, respectively. In addition, we prove that these multicone graphs are determined by their signless Laplacian spectra. [ABSTRACT FROM AUTHOR]

In this paper, we examine the main advances registered in the last ten years in Pattern Recognition methodologies based on graph matching and related techniques, analyzing more than 180 papers; the aim is to provide a systematic framework presenting the recent history and the current developments. This is made by introducing a categorization of graph-based techniques and reporting, for each class, the main contributions and the most outstanding research results. [ABSTRACT FROM AUTHOR]

Interval routing is a space efficient method to realize a distributed routing function. In this paper we show that every circular-arc graph allows a shortest path strict 2-interval routing scheme, i.e., by introducing a global order on the vertices and assigning at most two (strict) intervals in this order to the ends of every edge allows to depict a routing function that implies exclusively shortest paths. Since circular-arc graphs do not allow shortest path 1-interval routing schemes in general, the result implies that the class of circular-arc graphs has strict compactness 2, which was a hitherto open question. Additionally, we show that the constructed 2-interval routing scheme is a 1-interval routing scheme with at most one additional interval assigned at each vertex and we outline an algorithm to calculate the routing scheme for circular-arc graphs in 풪( n2) time, where n is the number of vertices. [ABSTRACT FROM AUTHOR]

In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense k-uniform hypergraphs. It has been known that the perfect matching problem for the classes of hypergraphs H with minimum ((k - 1)-wise) vertex degreeδ(H) at least c|V(H)| is NP-complete for $c < \frac{1}{k}$ and trivial for c ≥ ½, leaving the status of the problem with c in the interval $[\frac{1}{k}, \frac{1}{2})$ widely open. In this paper we show, somehow surprisingly, that ½ is not the threshold for tractability of the perfect matching problem, and prove the existence of an ε > 0 such that the perfect matching problem for the class of hypergraphs H with δ(H) ≥ (½ - ε)|V(H)| is solvable in polynomial time. This seems to be the first polynomial time algorithm for the perfect matching problem on hypergraphs for which the existence problem is nontrivial. In addition, we consider parallel complexity of the problem, which could be also of independent interest. [ABSTRACT FROM AUTHOR]

This paper studies the problem of orienting all edges of a weighted graph such that the maximum weighted outdegree of vertices is minimized. This problem, which has applications in the guard arrangement for example, can be shown to be $\mathcal{NP}$-hard generally. In this paper we first give optimal orientation algorithms which run in polynomial time for the following special cases: (i) the input is an unweighted graph, and (ii) the input graph is a tree. Then, by using those algorithms as sub-procedures, we provide a simple, combinatorial, $\min \{\frac{w_{max}}{w_{min}},(2 - \varepsilon)\}$-approximation algorithm for the general case, where wmax and wmin are the maximum and the minimum weights of edges, respectively, and ε is some small positive real number that depends on the input. [ABSTRACT FROM AUTHOR]

This paper deals with a special kind of complex systems which depend on an arbitrary (and usually large) number n of random Boolean variables. The so-called complex stochastic Boolean systems often appear in many different scientific, technical or social areas. Clearly, there are 2n binary states associated to such a complex system. Each one of them is given by a binary string u = (u1,...,un) ∈ {0, 1}n of n bits, which has a certain occurrence probability Pr{u}. The behavior of a complex stochastic Boolean system is determined by the current values of its 2n binary n-tuple probabilities Pr{u} and by the ordering between pairs of them. Hence, the intrinsic order graph provides a useful representation of these systems by displaying (scaling) the 2n binary n-tuples which are ordered in decreasing probability of occurrence. The intrinsic order reduces the complexity of the problem from exponential (2n binary n-tuples) to linear (n Boolean variables). For any fixed binary n-tuple u, this paper presents a new, simple algorithm enabling rapid, elegant determination of all the binary n-tuples v with occurrence probabilities less than or equal to (greater than or equal to) Pr{u}. This algorithm is closely related to the lexicographic (truth-table) order in {0, 1}n, and this is illustrated through the connections (paths) in the intrinsic order graph. [ABSTRACT FROM AUTHOR]

A recent paper posed the question: "Graph Matching: What are we really talking about?". Far from providing a definite answer to that question, in this paper we will try to characterize the role that graphs play within the Pattern Recognition field. To this aim two taxonomies are presented and discussed. The first includes almost all the graph matching algorithms proposed from the lake seventies, and describes the different classes of algorithms. The second taxonomy considers the types of common applications of graph-based techniques in the Pattern Recognition and Machine Vision field. [ABSTRACT FROM AUTHOR]

A recent paper posed the question: "Graph Matching: What are we really talking about?". Far from providing a definite answer to that question, in this paper we will try to characterize the role that graphs play within the Pattern Recognition field. To this aim two taxonomies are presented and discussed. The first includes almost all the graph matching algorithms proposed from the lake seventies, and describes the different classes of algorithms. The second taxonomy considers the types of common applications of graph-based techniques in the Pattern Recognition and Machine Vision field. [ABSTRACT FROM AUTHOR]

A set D ⊆ V is a restrained dominating set of a graph G = (V, E) if every vertex in V\D is adjacent to a vertex in D and a vertex in V\D. Given a graph G and a positive integer k, the restrained domination problem is to check whether G has a restrained dominating set of size at most k. The restrained domination problem is known to be NP-complete even for chordal graphs. In this paper, we propose a linear time algorithm to compute a minimum restrained dominating set of a proper interval graph. We present a polynomial time reduction that proves the NP-completeness of the restrained domination problem for undirected path graphs, chordal bipartite graphs, circle graphs, and planar graphs. [ABSTRACT FROM AUTHOR]

In this paper we study linear codes over the rings Ri = ℤ4 + wℤ4, where w2 = i, i = 1, 2w. We characterize self-dual codes over Ri and propose a construction method for self-dual codes over these rings. We also briefly discuss circulant self-dual codes and Type II codes. Some examples are given. [ABSTRACT FROM AUTHOR]

A star is a graph in which some node is incident with every edge of the graph, i.e., a graph of diameter at most 2. A star forest is a graph in which each connected component is a star. Given a connected graph G in which the edges may be weighted positively. A spanning star forest of G is a subgraph of G which is a star forest spanning the nodes of G. The size of a spanning star forest F of G is defined to be the number of edges of F if G is unweighted and the total weight of all edges of F if G is weighted. We are interested in the problem of finding a Maximum Weight spanning Star Forest (MWSFP) in G. In [C. T. Nguyen, J. Shen, M. Hou, L. Sheng, W. Miller and L. Zhang, Approximating the spanning star forest problem and its applications to genomic sequence alignment, SIAM J. Comput. 38(3) (2008) 946-962], the authors introduced the MWSFP and proved its NP-hardness. They also gave a polynomial time algorithm for the MWSF problem when G is a tree. In this paper, we present a linear time algorithm that solves the MSWF problem when G is a cactus. [ABSTRACT FROM AUTHOR]

In the past decade, we have witnessed the rapid development of a increasing body of research focused on Physarum polycephalum, which has been employed to solve many complicated problems in real-world applications. In this paper, we give an overview towards the developments of Physarum-inspired models for graph-optimization problems. Specifically, we reported the applications of Physarum in the following graph-optimization problems: influential nodes identification, shortest path tree problem, biobjective shortest path problem, improved Physarum algorithm and approximating the transport network. [ABSTRACT FROM AUTHOR]

A subset L ⊆ V of a graph G = (V, E) is called a connected liar's dominating set of G if (i) for all v ∈ V, |NG[v] ∩ L| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(NG[u]∪NG[v])∩L| ≥ 3, and (iii) the induced subgraph of G on L is connected. In this paper, we initiate the algorithmic study of minimum connected liar's domination problem by showing that the corresponding decision version of the problem is NP-complete for general graph. Next we study this problem in subclasses of chordal graphs where we strengthen the NP-completeness of this problem for undirected path graph and prove that this problem is linearly solvable for block graphs. Finally, we propose an approximation algorithm for minimum connected liar's domination problem and investigate its hardness of approximation in general graphs. [ABSTRACT FROM AUTHOR]

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency structure on the half-edges around each vertex, and an embedding of a *-graph is an embedding under which the formal adjacency relation on half-edges corresponds to the adjacency relation induced by the embedding. *-graphs are a natural generalization of four-valent framed graphs, which are four-valent graphs with an opposite half-edge structure. In [Embeddings of four-valent framed graphs into 2-surfaces, Dokl. Akad. Nauk 424(3) (2009) 308-310], the question of whether a four-valent framed graph admits a ℤ2-homologically trivial embedding into a given surface was shown to be equivalent to a problem on matrices. We show that a similar result holds for *-graphs in which all vertices have degree 4 or 6. This gives an algorithm in quadratic time to determine whether a *-graph admits an embedding into the plane. [ABSTRACT FROM AUTHOR]

Problems of motion coordination of multiple entities are addressed in this paper. These problems are dealt on the abstract level where they can be viewed as tasks of constructing a spatial-temporal plan for a set of identical mobile entities. The entities reside in a certain environment where they can move. Each entity need to reach a given goal position supposed it is starting from some initial position. The most abstract formal representations of coordinated motion problems are known as 'pebble motion on a graph' and 'multi-robot path planning'. The existent algorithms for pebble motion and multi-robot problems were suspected of generating solutions containing redundancies. This hypothesis eventually confirmed in this work. We present several techniques for identifying and eliminating redundancies from solutions generated by existent algorithms. An extensive experimental evaluation was performed and it showed that the quality of generated solutions can be improved up to the order of magnitude. We also identify parameters characterizing instances of problems where a significant improvement is expectable. [ABSTRACT FROM AUTHOR]

One of the fundamental problems in social networking with a lot of potential applications is to detect effectively the communities that are created by the users' interaction. Other applications, such as finding web communities, uncovering the structure of social networks, or even analyzing a graph's structure to uncover Internet attacks are equally as important. All these problems converge to a common goal, the development of flexible and efficient local community detection algorithms. In this paper, we demonstrate the performance of an algorithm that uncovers the entire community structure of a network, based solely on local interactions between neighboring nodes and a distributed clustering algorithm. The proposed algorithm, named VCD, is based on the distributed computation of a synthetic coordinates for each graph node. Experimental results and comparisons with another method from the literature (Lancichinetti et al.) are presented. The algorithm is also tested on two real dataset graphs from the SNAP: Stanford Large Network Dataset Collection. In all cases the experimental results demonstrate the high performance of our algorithm in terms of accuracy to detect communities, and its computational efficiency. [ABSTRACT FROM AUTHOR]

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k). [ABSTRACT FROM AUTHOR]

A graph G is k-ordered if for any sequence of k distinct vertices of G, there exists a cycle in G containing these k vertices in the specified order. It is k-ordered Hamiltonian if, in addition, the required cycle is Hamiltonian. The question of the existence of an infinite class of 3-regular 4-ordered Hamiltonian graphs was posed in 1997 by Ng and Schultz.13 At the time, the only known examples were K4 and K3,3. Some progress was made in 2008 by Mészáros,12 when the Peterson graph was found to be 4-ordered and the Heawood graph was proved to be 4-ordered Hamiltonian; moreover, an infinite class of 3-regular 4-ordered graphs was found. In 2010 a subclass of generalized Petersen graphs was shown to be 4-ordered by Hsu et al.,10 with an infinite subset of this subclass being 4-ordered Hamiltonian, thus answering the open question. In this paper we find another infinite class of 3-regular 4-ordered Hamiltonian graphs, part of a subclass of the chordal ring graphs. In addition, we classify precisely which of these graphs are 4-ordered Hamiltonian. [ABSTRACT FROM AUTHOR]

Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area. [ABSTRACT FROM AUTHOR]

The graph bisection problem is an important problem in printed circuit board layout and communication networks. Since it is known to be NP-complete, approximation algorithm have been considered. In this paper, we propose a so-called two-state ant colony algorithm for efficiently solving the problem. In the proposed algorithm two kinds of pheromone and two kinds of heuristic information are introduced to reinforce the search ability. The proposed algorithm is tested on a large number of instances and is compared with a heuristic algorithm and a genetic algorithm. The experimental results show that the proposed approach is superior to its competitors. [ABSTRACT FROM AUTHOR]

Many networks, including social and biological networks, are naturally divided into communities. Community detection is an important task when discovering the underlying structure in networks. GN algorithm is one of the most influential detection algorithms based on betweenness scores of edges, but it is computationally costly, as all betweenness scores need to be repeatedly computed once an edge is removed. This paper presents an algorithm which is also based on betweenness scores but more than one edge can be removed when all betweenness scores have been computed. This method is motivated by the following considerations: many components, divided from networks, are independent of each other in their recalculation of betweenness scores and their split into smaller components. It is shown that this method is fast and effective through theoretical analysis and experiments with several real data sets, which have acted as test beds in many related works. Moreover, the version of this method with the minor adjustments allows for the discovery of the communities surrounding a given node without having to compute the full community structure of a graph. [ABSTRACT FROM AUTHOR]

This paper introduces the MaxiMin selection algorithm, a deterministic procedure to achieve maximal spread and almost perfectly distributed Pareto fronts. MaxiMin is successful because the measure of uniformity is performed on the set being constructed, not on the source set as most multi-objective algorithms do. For comparison purposes we present results delivered by several MOEAs with and without the MaxiMin selection. Performance metrics and graphical results show that MaxiMin improves the distribution and spread of Pareto fronts, with no negative effects on the convergence performance of the multi-objective algorithm. [ABSTRACT FROM AUTHOR]

This paper focuses on the self-stabilizing leader election algorithm of Xu and Srimani [10] that finds a leader in a tree graph. The worst case execution time for this algorithm is O(N4), where N is the number of nodes in the tree. We show that the average execution time for this algorithm, assuming two different scenarios, is much lower than O(N4). In the first scenario, the algorithm assumes an equiprobable daemon and it only privileges a single node at a time. The average execution time for this case is O(N2). For the second case, the algorithm can privilege multiple nodes at a time. We eliminate the daemon from this algorithm by making random choices to avoid interference between neighboring nodes. The execution time for this case is O(N). We also show that for specific tree graphs, these results reduce even more. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method, named Greedy Equivalence Search-Expectation Maximization (GES-EM), for learning Bayesian networks from incomplete data. Our method extends the recently proposed Greedy Equivalence Search (GES) algorithm10 to deal with incomplete data. For the quality evaluation of learned networks, we make use of the expected Bayesian Information Criterion (BIC) scoring function. In addition, we propose a new structural evaluation criterion. This so-called SEC criterion is more suitable than existing structural evaluation criteria, since it is based on the comparison of learned networks to the generating ones through Completed Partially Directed Acyclic Graphs (CPDAGs). Experimental results show that GES-EM algorithm yields more accurate structures than the standard Alternating Model Selection-Expectation Maximization (AMS-EM) algorithm.15. [ABSTRACT FROM AUTHOR]

This paper is devoted to the following incremental problem. Initially, a graph and a distinguished subset of vertices, called initial group, are given. This group is connected by an initial tree. The incremental part of the input is given by an on-line sequence of vertices of the graph, not yet in the current group, revealed on-line one after one. The goal is to connect each new member to the current tree, while satisfying a quality constraint: the average distance between members in each constructed tree must be kept in a given range compared to the best possible one. Under this quality constraint, our objectives are to minimize the number of critical stages and the number of elementary changes of the sequence of constructed trees. We call "critical" a stage where the inclusion of a new member implies heavy changes in the current tree. Otherwise, the new member is just added by connecting it with a (well chosen) path to the current tree. In both cases, updating a tree implies a certain number of elementary changes (that we define). We propose a strategy leading to at most O(log i) critical stages (i is the number of new members) and to at most a constant average number of elementary changes per stage. We also prove that there exists situations where Ω(log i) critical stages are necessary to any algorithm to maintain the quality constraint. Our strategy is then worst case optimal in order of magnitude for the number of critical stages criterion and induces a constant number of elementary changes in average per stage. [ABSTRACT FROM AUTHOR]

It is an open problem to find a good characterization for independence or, more generally, the rank function in the d-dimensional rigidity matroid of a graph when d ≥ 3. In this paper we give a brief survey of existing lower and upper bounds on the rank of the 3-dimensional rigidity matroid of a graph and introduce a new upper bound, which may lead to the desired good characterization. [ABSTRACT FROM AUTHOR]

A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of testing an arbitrary digraph G = (V,E) for upward planarity. In particular we describe two fixed-parameter tractable algorithms for testing the upward planarity of G. The first algorithm that we present can test the upward planarity of G in O(2t · t! · |V|2)-time where t is the number of triconnected components of G. The second algorithm that we present uses a standard technique known as kernelisation and can test the upward planarity of G in O(|V|2 + k4 · (2k)!)-time, where k = |E| - |V|. [ABSTRACT FROM AUTHOR]

This paper investigates geometric and algorithmic properties of the Voronoi diagram for a transportation network on the Euclidean plane. In the presence of a transportation network, the distance is measured as the length of the shortest (time) path. In doing so, we introduce a needle, a generalized Voronoi site. We present an O(nm2 + m3 + nm log n) algorithm to compute the Voronoi diagram for a transportation network on the Euclidean plane, where n is the number of given sites and m is the complexity of the given transportation network. Moreover, in the case that the roads in a transportation network have only a constant number of directions and speeds, we propose two algorithms; one needs O(nm + m2 + n log n) time with O(m(n + m)) space and the other O(nm log n + m2log m) time with O(n + m) space. [ABSTRACT FROM AUTHOR]

A drawing of a plane graph is called an inner rectangular drawing if every edge is drawn as a horizontal or vertical line segment so that every inner face is a rectangle. In this paper we show that a plane graph G has an inner rectangular drawing D if and only if a new bipartite graph constructed from G has a perfect matching. We also show that D can be found in time O(n1.5/ log n) if G has n vertices and a sketch of the outer face is prescribed, that is, all the convex outer vertices and concave ones are prescribed. [ABSTRACT FROM AUTHOR]

The 3-D static leaf sequencing (SLS) problem arises in radiation therapy for cancer treatments, aiming to deliver a prescribed radiation dose to a target tumor accurately and efficiently. The treatment time and machine delivery error are two crucial factors to the solution (i.e., a treatment plan) for the SLS problem. In this paper, we prove that the 3-D SLS problem is NP-hard, and present the first ever algorithm for the 3-D SLS problem that can determine a tradeoff between the treatment time and machine delivery error (also called the "tongue-and-groove" error in medical literature). Our new 3-D SLS algorithm with error control gives the users (e.g., physicians) the option of specifying a machine delivery error bound, and subject to the given error bound, the algorithm computes a treatment plan with the minimum treatment time. We formulate the SLS problem with error control as computing a k-weight shortest path in a directed graph and build the graph by computing g-matchings and minimum cost flows. Further, we extend our 3-D SLS algorithm to all the popular radiotherapy machine models with different constraints. In our extensions, we model the SLS problems for some of the radiotherapy systems as computing a minimum g-path cover of a directed acyclic graph. We implemented our new 3-D SLS algorithm suite and conducted an extensive comparison study with commercial planning systems and well-known algorithms in medical literature. Some of our experimental results based on real medical data are presented. [ABSTRACT FROM AUTHOR]

A large class of neural network models have their units organized in a lattice with fixed topology or generate their topology during the learning process. These network models can be used as neighborhood preserving map of the input manifold, but such a structure is difficult to manage since these maps are graphs with a number of nodes that is just one or two orders of magnitude less than the number of input points (i.e., the complexity of the map is comparable with the complexity of the manifold) and some hierarchical algorithms were proposed in order to obtain a high-level abstraction of these structures. In this paper a general structure capable to extract high order information from the graph generated by a large class of self–organizing networks is presented. This algorithm will allow to build a two layers hierarchical structure starting from the results obtained by using the suitable neural network for the distribution of the input data. Moreover the proposed algorithm is also capable to build a topology preserving map if it is trained using a graph that is also a topology preserving map. [ABSTRACT FROM AUTHOR]

We study the Hausdorff Voronoi diagram of a set 5 of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of 5 under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conquer algorithm for the construction of this diagram that improves upon previous results. As a byproduct we introduce the Hausdorff hull, a structure that relates to the Hausdorff Voronoi diagram in the same way as a convex hull relates to the ordinary Voronoi diagram. The Hausdorff Voronoi diagram finds direct application in the problem of computing the critical area of a VLSI Layout, a measure reflecting the sensitivity of a VLSI design to random manufacturing defects, described in a companion paper. [ABSTRACT FROM AUTHOR]

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of $(3+\sqrt5)/2$. [ABSTRACT FROM AUTHOR]

Classical topology-based operations such as extrusion (or more generally Minkowski sums) are defined by a cartesian product of combinatorial structures describing the topology of geometric space subdivisions. In this paper, we define the cartesian product operation for four different structures: semi-simplicial sets, generalized maps, oriented maps and chains of maps. We follow a homogeneous framework allowing to define the cartesian product for any simplicial and cellular structures derived from simplicial sets and combinatorial maps. This results from the relationships already established between these structures and those used in computational geometry and geometric modeling. For esch structure we have studied, we additionally provide a time-optimal computation algorithm. [ABSTRACT FROM AUTHOR]

In this paper we describe a classification method that allows the use of graph-based representations of data. instead of traditional vector-based representations. We compare the vector approach combined with the k-Nearest Neighbor (k-NN) algorithm to the graph-matching approach when classifying three different web document collections, using the leave-one-out approach for measuring classification accuracy. We also compare the performance of different graph distance measures as well as various document representations that utilize graphs. The results show the graph-based approach can outperform traditional vector-based methods in terms of accuracy, dimensionality and execution time. [ABSTRACT FROM AUTHOR]

The P[sub 4] is the induced path of four vertices. The gem consists of a P[sub 4] with an additional universal vertex being completely adjacent to the P[sub 4], and the co-gem is its complement graph. Gem- and co-gem-free graphs generalize the popular class of cographs (i. e. P[sub 4]-free graphs). The tree structure and algebraic generation of cographs has been crucial for several concepts of graph decomposition such as modular and homogeneous decomposition. Moreover, it is fundamental for the recently introduced concept of clique-width of graphs which extends the famous concept of treewidth. It is well-known that the cographs are exactly those graphs of clique-width at most 2. In this paper, we show that the clique-width of gem- and co-gem-free graphs is at most 16. [ABSTRACT FROM AUTHOR]

One of the most common ways of representing classes of equivalent Bayesian networks is the use of essential graphs which are also known in the literature as completed patterns or completed pdags. The name essential graph was proposed by Andersson, Madigan and Perlman who also gave a graphical characterization of essential graphs. In this paper an alternative characterization of essential graphs is presented. The main observation is that every essential graph is the largest chain graph within a special class of chain graphs. More precisely, every equivalence class of Bayesian networks is contained in an equivalence class of chain graphs without flags (= certain induced subgraphs). A special operation of legal merging of (connectivity) components for a chain graph without flags is introduced. This operation leads to an algorithm for finding the essential graph on the basis of any graph in that equivalence class of chain graphs without flags which contains the equivalence class of a Bayesian network. In particular, the algorithm may start with any Bayesian network. [ABSTRACT FROM AUTHOR]

With the recent advances in computed tomography and magnetic resonance devices, cross-sectional images are now commonly used for diagnosis. However, how contours between cross-sections should be connected is often ambiguous. In this paper, we propose an algorithm that enumerates all possible cases of the correspondence of contours. This is useful for achieving fully automatic interpolation of contours, although our current implementation still requires some degree of manual interaction. [ABSTRACT FROM AUTHOR]