Showing papers on "Vertex (graph theory) published in 2002"

Random geometric graphs.

Abstract: We analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size of the largest cluster. We derive an analytical expression for the cluster coefficient, which shows that the graphs are distinctly different from standard random graphs, even for infinite dimensionality. Insights relevant for graph bipartitioning are included.

On the Computational Complexity of Upward and Rectilinear Planarity Testing

Abstract: A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction and no two edges cross An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical segment and no two edges cross Testing upward planarity and rectilinear planarity are fundamental problems in the effective visualization of various graph and network structures For example, upward planarity is useful for the display of order diagrams and subroutine-call graphs, while rectilinear planarity is useful for the display of circuit schematics and entity-relationship diagrams We show that upward planarity testing and rectilinear planarity testing are NP-complete problems We also show that it is NP-hard to approximate the minimum number of bends in a planar orthogonal drawing of an n-vertex graph with an $O(n^{1-\epsilon})$ error for any $\epsilon > 0$

Massive Quasi-Clique Detection

Abstract: We describe techniques that are useful for the detection of dense subgraphs (quasi-cliques) in massive sparse graphs whose vertex set, but not the edge set, fits in RAM. The algorithms rely on efficient semi-external memory algorithms used to preprocess the input and on greedy randomized adaptive search procedures (GRASP) to extract the dense subgraphs. A software platform was put together allowing graphs with hundreds of millions of nodes to be processed. Computational results illustrate the effectiveness of the proposed methods.

RASCAL: Calculation of Graph Similarity using Maximum Common Edge Subgraphs

Abstract: A new graph similarity calculation procedure is introduced for comparing labeled graphs. Given a minimum similarity threshold, the procedure consists of an initial screening process to determine whether it is possible for the measure of similarity between the two graphs to exceed the minimum threshold, followed by a rigorous maximum common edge subgraph (MCES) detection algorithm to compute the exact degree and composition of similarity. The proposed MCES algorithm is based on a maximum clique formulation of the problem and is a significant improvement over other published algorithms. It presents new approaches to both lower and upper bounding as well as vertex selection.

Domination in Graphs Applied to Electric Power Networks

Abstract: The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graphs. We consider the graph theoretical representation of this problem as a variation of the dominating set problem and define a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph G is the power domination number $\gamma_P(G)$. We show that the power dominating set (PDS) problem is NP-complete even when restricted to bipartite graphs or chordal graphs. On the other hand, we give a linear algorithm to solve the PDS for trees. In addition, we investigate theoretical properties of $\gamma_P(T)$ in trees T.

The Complexity of Counting in Sparse, Regular, and Planar Graphs

Abstract: We show that a number of graph-theoretic counting problems remain ${\cal NP}$-hard, indeed $\#{\cal P}$-complete, in very restricted classes of graphs. In particular, we prove that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. We obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2-CNF formulae. To achieve these results, a new interpolation-based reduction technique which preserves properties such as constant degree is introduced.

Parallel static and dynamic multi‐constraint graph partitioning

Abstract: Sequential multi-constraint graph partitioners have been developed to address the static load balancing requirements of multi-phase simulations. These work well when (i) the graph that models the computation fits into the memory of a single processor, and (ii) the simulation does not require dynamic load balancing. The efficient execution of very large or dynamically adapting multi-phase simulations on high-performance parallel computers requires that the multi-constraint partitionings are computed in parallel. This paper presents a parallel formulation of a multi-constraint graph-partitioning algorithm, as well as a new partitioning algorithm for dynamic multi-phase simulations. We describe these algorithms and give experimental results conducted on a 128-processor Cray T3E. These results show that our parallel algorithms are able to efficiently compute partitionings of similar edge-cuts as serial multi-constraint algorithms, and can scale to very large graphs. Our dynamic multi-constraint algorithm is also able to minimize the data redistribution required to balance the load better than a naive scratch-remap approach. We have shown that both of our parallel multi-constraint graph partitioners are as scalable as the widely-used parallel graph partitioner implemented in PARMETIS. Both of our parallel multi-constraint graph partitioners are very fast, as they are able to compute three-constraint 128-way partitionings of a 7.5 million vertex graph in under 7 s on 128 processors of a Cray T3E. Copyright © 2002 John Wiley & Sons, Ltd.

A fast multi-scale method for drawing large graphs.

Abstract: We present a multi-scale layout algorithm for the aesthetic drawing of undirected graphs with straight-line edges. The algorithm is extremely fast, and is capable of drawing graphs of substantially larger size than any other algorithm we are aware of. For example, the algorithm achieves optimal drawings of 1000 vertex graphs in about 2 seconds. The paper contains graphs with over 6000 nodes. The proposed algorithm embodies a new multi-scale scheme for drawing graphs, which was motivated by the recently published multi-scale algorithm of Hadany and Harel [7]. It can significantly improve the speed of essentially any force-directed method (regardless of that method's ability of drawing weighted graphs or the continuity of its cost-function).

Generalized Cores

Abstract: Cores are, besides connectivity components, one among few concepts that provides us with efficient decompositions of large graphs and networks In the paper a generalization of the notion of core of a graph based on vertex property function is presented It is shown that for the local monotone vertex property functions the corresponding cores can be determined in $O(m \max (\Delta, \log n))$ time

On the b-Chromatic Number of Graphs

Abstract: The b-chromatic number b(G) of a graph G = (V, E) is the largest integer k such that G admits a vertex partition into k independent sets Xi (i = 1, . . . , k) such that each Xi contains a vertex xi adjacent to at least one vertex of each Xj, j ? i. We discuss on the tightness of some bounds on b(G) and on the complexity of determining b(G). We also determine the asymptotic behavior of b(Gn, p) for the random graph, within the accuracy of a multiplicative factor 2 + o(1) as n ? ?.

The Terwilliger Algebra of the Hypercube

Abstract: We give an introduction to the Terwilliger algebra of a distance-regular graph, focusing on the hypercube QDof dimension D. Let X denote the vertex set ofQD . Fix a vertex x?X, and letT=T(x) denote the associated Terwilliger algebra. We show thatT is the subalgebra of MatX(C) generated by the adjacency matrixA and a diagonal matrix A*=A* (x), where A*has yy entryD? 2 ?(x, y) for all y?X , and where ? denotes the path-length distance function. We show that A andA* satisfy A2A*? 2AA*A+A*A2&=& 4A* , A*2A? 2A*AA*+AA*2&=& 4 A. Using the above equations, we find the irreducible T -modules. For each irreducible T -module W, we display two orthogonal bases, which we call the standard basis and the dual standard basis. We describe the action of A andA* on each of these bases. We give the transition matrix from the standard basis to the dual standard basis for W. We compute the multiplicity with which each irreducible T -module W appears inCX . We give an elementary proof that QDhas the Q -polynomial property. We show that T is a homomorphic image of the universal enveloping algebra of the Lie algebrasl2 (C). We obtain an element ? of T that generates the center ofT . We obtain the central primitive idempotents of T as polynomials in ?.

D-branes, open string vertex operators, and Ext groups

Abstract: In this paper we explicitly work out the precise relationship between Ext groups and massless modes of D-branes wrapped on complex submanifolds of Calabi-Yau manifolds. Specifically, we explicitly compute the boundary vertex operators for massless Ramond sector states, in open string B models describing Calabi-Yau manifolds at large radius, directly in BCFT using standard methods. Naively these vertex operators are in one-to-one correspondence with certain sheaf cohomology groups (as is typical for such vertex operator calculations), which are related to the desired Ext groups via spectral sequences. However, a subtlety in the physics of the open string B model has the effect of physically realizing those spectral sequences in BRST cohomology, so that the vertex operators are actually in one-to-one correspondence with Ext group elements. This gives an extremely concrete physical test of recent proposals regarding the relationship between derived categories and D-branes. We check these results extensively in numerous examples, and comment on several related issues.

On embedding an outer-planar graph in a point set

Abstract: Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(n log3 n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [8,12] that requires O(n2) time. Our algorithm is near-optimal as there is an ω (n log n) lower bound for the problem [4]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where log n ≥ d ≥ 2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(n log n) and O(n2) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ (n log n) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.

PALP: A Package for Analyzing Lattice Polytopes with Applications to Toric Geometry

Abstract: We describe our package PALP of C programs for calculations with lattice polytopes and applications to toric geometry, which is freely available on the internet. It contains routines for vertex and facet enumeration, computation of incidences and symmetries, as well as completion of the set of lattice points in the convex hull of a given set of points. In addition, there are procedures specialised to reflexive polytopes such as the enumeration of reflexive subpolytopes, and applications to toric geometry and string theory, like the computation of Hodge data and fibration structures for toric Calabi-Yau varieties. The package is well tested and optimised in speed as it was used for time consuming tasks such as the classification of reflexive polyhedra in 4 dimensions and the creation and manipulation of very large lists of 5-dimensional polyhedra. While originally intended for low-dimensional applications, the algorithms work in any dimension and our key routine for vertex and facet enumeration compares well with existing packages.

Bethe-Salpeter equation and a nonperturbative quark-gluon vertex

Abstract: A Ward-Takahashi identity preserving Bethe-Salpeter kernel can always be calculated explicitly from a dressed-quark-gluon vertex whose diagrammatic content is enumerable. We illustrate that fact using a vertex obtained via the complete resummation of dressed-gluon ladders. While this vertex is planar, the vertex-consistent kernel is nonplanar and that is true for any dressed vertex. In an exemplifying model the rainbow-ladder truncation of the gap and Bethe-Salpeter equations yields many results; e.g., \ensuremath{\pi}- and \ensuremath{\rho}-meson masses, that are changed little by including higher-order corrections. Repulsion generated by nonplanar diagrams in the vertex-consistent Bethe-Salpeter kernel for quark-quark scattering is sufficient to guarantee that diquark bound states do not exist.

Dynamic Generators of Topologically Embedded Graphs

Abstract: We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(log n + log g(log log g)^3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log^4 g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for low-genus graphs, and to find in linear time a tree-decomposition of low-genus low-diameter graphs.

Differential equations and intertwining operators

Abstract: We show that if every module W for a vertex operator algebra V satisfies the condition that the dimension of W/C_1(W) is less than infinity, where C_1(W) is the subspace of W spanned by elements of the form u_{-1}w for u in V of positive weight and w in W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exist such systems of differential equations such that the prescribed singular points are regular. The finiteness of the fusion rules is an immediate consequence of a result used to establish the existence of such systems. Using these systems of differential equations and some additional reducibility conditions, we prove that products of intertwining operators for V satisfy the convergence and extension property needed in the tensor product theory for V-modules. Consequently, when a vertex operator algebra V satisfies all the conditions mentioned above, we obtain a natural structure of vertex tensor category (consequently braided tensor category) on the category of V-modules and a natural structure of intertwining operator algebra on the direct sum of all (inequivalent) irreducible V-modules.

Edit distance with move operations

Abstract: The traditional edit-distance problem is to find the minimum number of insert-character and delete-character (and sometimes change character) operations required to transform one string into another. Here we consider the more general problem of strings being represented by a singly linked list (one character per node) and being able to apply these operations to the pointer associated with a vertex as well as the character associated with the vertex. That is, in O(1) time, not only can characters be inserted or deleted, but also substrings can be moved or deleted. We limit our attention to the ability to move substrings and leave substring deletions for future research. Note that O(1) time substring move operations imply O(1) substring exchange operations as well, a form of transformation that has been of interest in molecular biology. We show that this problem is NP-complete, show that a recursive sequence of moves can be simulated with at most a constant factor increase by a non-recursive sequence, and present a polynomial time greedy algorithm for non-recursive moves with a worst-case log factor approximation to optimal. The development of this greedy algorithm shows how to reduce moves of substrings to moves of characters, and how to convert moves with characters to only insert and deletes of characters.

D-branes, open string vertex operators, and Ext groups

Abstract: In this paper we explicitly work out the precise relationship between Ext groups and massless modes of D-branes wrapped on complex submanifolds of Calabi-Yau manifolds. Specifically, we explicitly compute the boundary vertex operators for massless Ramond sector states, in open string B models describing Calabi-Yau manifolds at large radius, directly in BCFT using standard methods. Naively these vertex operators are in one-to-one correspondence with certain sheaf cohomology groups (as is typical for such vertex operator calculations), which are related to the desired Ext groups via spectral sequences. However, a subtlety in the physics of the open string B model has the effect of physically realizing those spectral sequences in BRST cohomology, so that the vertex operators are actually in one-to-one correspondence with Ext group elements. This gives an extremely concrete physical test of recent proposals regarding the relationship between derived categories and D-branes. We check these results extensively in numerous examples, and comment on several related issues.

Covering problems with hard capacities

Abstract: We consider the classical vertex cover and set cover problems with the addition of hard capacity constraints. This means that a set (vertex) can only cover a limited number of its elements (adjacent edges) and the number of available copies of each set (vertex) is bounded. This is a natural generalization of the classical problems that also captures resource limitations in practical scenarios. We obtain the following results. For the unweighted vertex cover problem with hard capacities we give a 3-approximation algorithm which is based on randomized rounding with alterations. We prove that the weighted version is at least as hard as the set cover problem. This is an interesting separation between the approximability of weighted and unweighted versions of a "natural" graph problem. A logarithmic approximation factor for both the set cover and the weighted vertex cover problem with hard capacities follows from the work of Wolsey (1982) on submodular set cover. We provide in this paper a simple and intuitive proof for this bound.

Optimization over the efficient set: overview

Abstract: Over the past several decades, the optimization over the efficient set has seen a substantial development. The aim of this paper is to provide a state-of-the-art survey of the development. Given p linear criteria c1x,ccc,cp x and a feasible region X of Rn, the linear multicriteria problem is to find a point x of X such that no point x' of X satisfies (c1 x',ccc,cp x')≥(c1 x,ccc,cp x) and (c1x',ccc,cp x')≠q (c1 x ,ccc,cp x). Such a point is called an efficient point. The optimization over the efficient set is the maximization of a given function φ over the set of efficient points. The difficulty of this problem is mainly due to the nonconvexity of this set. The existing algorithms for solving this problem could be classified into several groups such as adjacent vertex search algorithm, nonadjacent vertex search algorithm, branch-and-bound based algorithm, Lagrangian relaxation based algorithm, dual approach and bisection algorithm. In this paper we review a typical algorithm from each group and compare them from the computational point of view.

Simple and fast: Improving a branch-and-bound algorithm for maximum clique

Abstract: We consider a branch-and-bound algorithm for maximum clique problems. We introduce cost based filtering techniques for the so-called candidate set (i.e. a set of nodes that can possibly extend the clique in the current choice point). Additionally, we present a taxonomy of upper bounds for maximum clique. Analytical results show that our cost based filtering is in a sense as tight as most of these well-known bounds for the maximum clique problem. Experiments demonstrate that the combination of cost based filtering and vertex coloring bounds outperforms the old approach as well as approaches that only apply either of these techniques. Furthermore, the new algorithm is competitive with other recent algorithms for maximum clique.

Rational Vertex Operator Algebras and the Effective Central Charge

Abstract: We establish that the Lie algebra of weight one states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank is bounded above by the effective central charge. We show that lattice vertex operator algebras may be characterized by the equalities of the effective central charge, the Lie rank and the central charge, and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator algebras by the equality of the Lie rank and the central charge.

Twisted sectors for tensor product vertex operator algebras associated to permutation groups

Abstract: Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V ⊗ k . We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the tensor product vertex operator algebra V ⊗ k are isomorphic to the categories of weak, weak admissible and ordinary V-modules, respectively. The main result is an explicit construction of the weak g-twisted V ⊗ k -modules from weak V-modules. For an arbitrary permutation automorphism g of V ⊗ k the category of weak admissible g-twisted modules for V ⊗ k is semisimple and the simple objects are determined if V is rational. In addition, we extend these results to the more general setting of γg-twisted V ⊗ k -modules for γ a general automorphism of V acting diagonally on V ⊗ k and g a permutation automorphism of V ⊗ k .

Rationality, Regularity, and C_2-cofiniteness

Abstract: We demonstrate that, for CFT vertex operator algebras, C_2-cofiniteness and rationality is equivalent to regularity. In addition, we show that, for C_2-cofinite vertex operators algebras, irreducible weak modules are ordinary modules and C_2-cofinite, and V_L^+ are C_2-cofinite.

1-perfect codes in Sierpiński graphs

Abstract: Sierpinski graphs S (n, κ) generalise the Tower of Hanoi graphs—the graph S (n, 3) is isomorphic to the graph Hn of the Tower of Hanoi with n disks. A 1-perfect code (or an efficient dominating set) in a graph G is a vertex subset of G with the property that the closed neighbourhoods of its elements form a partition of V (G). It is proved that the graphs S (n, κ) possess unique 1-perfect codes, thus extending a previously known result for Hn. An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to Hn.