Showing papers on "Vertex (graph theory) published in 1998"
18 Jun 1998
TL;DR: This paper develops applications to several classes of graphs that include cographs and are defined by forbidding subgraphs with ``too many'' induced paths with four vertices, and proves that this is also the case for graphs of clique-width at most k.
Abstract: Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we show that the (q,q-4) graphs are of clique width at most q and P4-tidy graphs are of clique-width at most 4. Furthermore, the k-expression (for k=4 or k=q) associated with such a graph can be found in linear time.
300 citations
249 citations
01 Jan 1998
TL;DR: In this paper, the authors considered the space of graph connections which can be endowed with a Poisson structure in terms of a ciliated fat graph, which is a graph with a fixed linear order of ends of edges at each vertex.
Abstract: We consider the space of graph connections (lattice gauge fields) which can be endowed with a Poisson structure in terms of a ciliated fat graph. (A ciliated fat graph is a graph with a fixed linear order of ends of edges at each vertex.) Our aim is however to study the Poisson structure on the moduli space of locally flat vector bundles on a Riemann surface with holes (i.e. with boundary). It is shown that this moduli space can be obtained as a quotient of the space of graph connections by the Poisson action of a lattice gauge group endowed with a PoissonLie structure. The present paper contains as a part an updated version of a 1992 preprint [11] which we decided still deserves publishing. We have removed some obsolete inessential remarks and added some newer ones.
221 citations
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TL;DR: In this article, a quantum vertex operator algebra from a rational, trigonometric, or elliptic R-matrix is constructed, which is a quantum deformation of the affine vertex algebra, and the simplest vertex operator in this algebra is the quantum current of Reshetikhin and Semenov-Tian-Shansky.
Abstract: This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this paper is to define and study the notion of a quantum vertex operator algebra in the setting of the formal deformation theory and give interesting examples of such algebras. In particular, we construct a quantum vertex operator algebra from a rational, trigonometric, or elliptic R-matrix, which is a quantum deformation of the affine vertex operator algebra. The simplest vertex operator in this algebra is the quantum current of Reshetikhin and Semenov-Tian-Shansky.
212 citations
Patent•
16 Jul 1998TL;DR: In this paper, a routing description that includes connection segments and a vertex where at least two of the connection segments connect to each other is obtained, and a penalty is determined for the vertex based on a potential layer assignment combination for the connect segments that connect at the vertex, and routing layers are assigned to the connected segments based on the determined penalty.
Abstract: Routing layers are assigned to connection segments in integrated circuit design. A routing description that includes connection segments and a vertex where at least two of the connection segments connect to each other is obtained. A penalty is determined for the vertex based on a potential layer assignment combination for the connection segments that connect at the vertex, and routing layers are assigned to the connection segments based on the determined penalty.
207 citations
TL;DR: In this article, the authors describe the normalization of K [G ] explicitly and give a combinatorial criterion for K [ G ] to be normal, where G is a finite connected graph on the vertex set {1,…, d } allowing loops and having no multiple edge.
173 citations
TL;DR: This work gives an algorithm for the Vertex Cover problem that runs in time O(kn + (1.324718)nn2) to find the minimum vertex cover in the graph.
162 citations
Book•
30 Sep 1998
TL;DR: In this article, the authors introduce the concept of Self-Dual Lattices and codes and define a hierarchy of superalgebras and their properties, including their properties and properties.
Abstract: Preface. Introduction: Notational Conventions.I. Self-Dual Lattices and Codes. 1. Self-Dual Codes. 2. Self-Dual Lattices. II. Vertex Operator Superalgebras and Their Modules. 3. Definitions and General Properties. 4. Conformal Superalgebras, Affine Kac-Moody Algebras and KZ Equations. 5. Analogue of the Highest-Weight Theory. 6. Lattice Vertex Operator Superalgebras. 7. VOSAs Generated by Their Subspaces of Small Weights. Bibliography. Index.
121 citations
TL;DR: In this paper, the integral closure of the monomial subring of a polynomial ring over a field k is described, where the squarefree monomials of degree two defining the edges of the graph are defined.
115 citations
TL;DR: In this paper, an upper bound for the spectral radius of Laplacian matrix of a graph in terms of a 2-degree of a vertex is presented. But this upper bound is not applicable to the case of graphs.
106 citations
TL;DR: An efficient algorithm listing all minimal vertex separators of an undirected graph is given and needs polynomial time per separator that is found.
Abstract: An efficient algorithm listing all minimal vertex separators of an undirected graph is given. The algorithm needs polynomial time per separator that is found.
TL;DR: In this paper, a positive definite even lattice of rank one and a set of generators and the full automorphism group of the lattice were determined, and the fixed points of a lattice VOAVL associated to an automomorphism ofVLlifting the −1 isometry ofL were determined.
TL;DR: In this article, the representation theory of code vertex operator algebras (VOAs) constructed from an even binary linear code was studied, and the structure of VOAs containing a set of mutually orthogonal rational conformal vectors with central charge was analyzed.
TL;DR: Inspired by the approach of Aggarwal, Orlin and Tai, this work examines variations of each element of the genetic algorithm—selection, population replacement and mutation—and develops a steady-state genetic algorithm that performs better than its competitors on most problems.
Abstract: In Balas and Niehaus (1996), we have developed a heuristic for generating large cliques in an arbitrary graph, by repeatedly taking two cliques and finding a maximum clique in the subgraph induced by the union of their vertex sets, an operation executable in polynomial time through bipartite matching in the complement of the subgraph. Aggarwal, Orlin and Tai (1997) recognized that the latter operation can be embedded into the framework of a genetic algorithm as an optimized crossover operation. Inspired by their approach, we examine variations of each element of the genetic algorithm—selection, population replacement and mutation—and develop a steady-state genetic algorithm that performs better than its competitors on most problems.
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TL;DR: In this paper, the authors define free objects in the categories of conformal and vertex algebras, and explicitly build their Groebner bases, in some cases explicitly.
Abstract: Any variety of classical algebras has a so-called conformal counterpart. For example one can consider Lie conformal or associative conformal algebras. Lie conformal algebras are closely related to vertex algebras. We define free objects in the categories of conformal and vertex algebras. In some cases we can explicitly build their Groebner bases.
TL;DR: In this article, the vertex representations of the quantum toroidal algebras Uq ( §U+i,ro?) were constructed in the classical case and in the quantum case they are irreducible.
Abstract: We construct the vertex representations of the quantum toroidal algebras Uq (§U+i,ro?) In the classical case the vertex representations are not irreducible However in the quantum case they are irreducible
TL;DR: In this article, it was shown that there exists a point O and suitable subsets Qi ⊆ Pi (i = 1, 2,…, d + 1) such that an every d-dimensional simplex with exactly one vertex in each Qi contains Q in its interior.
Abstract: Let P1, P2,…, Pd+1 be pairwise disjoint n-element point sets in general position in d-space. It is shown that there exist a point O and suitable subsets Qi ⊆ Pi (i = 1, 2,…, d + 1) such that Qi ≥ cdPi, an every d-dimensional simplex with exactly one vertex in each Qi contains Q in its interior. Here cd is a positive constant depending only on d.
TL;DR: It is shown that the problem of determining whether G has an efficient edge dominating set is NP-complete when G is restricted to a bipartite graph, and a linear time algorithm is presented to solve the weighted efficient edge domination problem on bipartITE permutation graphs, which form a subclass of bipartites graphs using the technique of dynamic programming.
TL;DR: It is proved that a vertex of the network state hypercube is asymptotically stable if and only if it is an optimal solution to the problem, so this network can be called the "optimal" Hopfield network.
Abstract: An "optimal" Hopfield network is presented for combinatorial optimization problems with linear cost function. It is proved that a vertex of the network state hypercube is asymptotically stable if and only if it is an optimal solution to the problem. That is, one can always obtain an optimal solution whenever the network converges to a vertex. In this sense, this network can be called the "optimal" Hopfield network. It is also shown through simulations of assignment problems that this network obtains optimal or nearly optimal solutions more frequently than other familiar Hopfield networks.
TL;DR: In this paper, a new set of generating series in the quantum affine algebra U_q(^sl_2), which are elliptic analogs of the Drinfeld currents, are introduced.
Abstract: We investigate the structure of the elliptic algebra U_{q,p}(^sl_2) introduced earlier by one of the authors. Our construction is based on a new set of generating series in the quantum affine algebra U_q(^sl_2), which are elliptic analogs of the Drinfeld currents. They enable us to identify U_{q,p}(^sl_2) with the tensor product of U_q(^sl_2) and a Heisenberg algebra generated by P,Q with [Q,P]=1. In terms of these currents, we construct an L operator satisfying the dynamical RLL relation in the presence of the central element c. The vertex operators of Lukyanov and Pugai arise as `intertwiners' of U_{q,p}(^sl_2) for level one representation, in the sense to be elaborated on in the text. We also present vertex operators with higher level/spin in the free field representation.
TL;DR: The study of eight representative physical properties of alkanes was used to compare the ability of both series of indices to produce significant quantitative structure−property relationship (QSPR) models.
Abstract: The concept of edge connectivity index is extended to a series of indices based on adjacency between edges in various fragments of the molecular graph. The analogous concept of vertex adjacency indices of the line graph of the molecular graph is also introduced. Some mathematical relations between both series of indices are found, showing that line-graph-based connectivity indices are linear combinations of edge-based descriptors. The study of eight representative physical properties of alkanes was used to compare the ability of both series of indices to produce significant quantitative structure−property relationship (QSPR) models.
14 Dec 1998
TL;DR: This work proves that for each of these problems, there exists a constant Ɛ > 0, such that no polynomial time algorithm can guarantee an approximation ratio of 1 + Ɠ unless P = NP.
Abstract: The three art gallery problems VERTEX GUARD, EDGE GUARD and POINT GUARD are known to be NP-hard [8]. Approximation algorithms for VERTEX GUARD and EDGE GUARD with a logarithmic ratio were proposed in [7]. We prove that for each of these problems, there exists a constant Ɛ > 0, such that no polynomial time algorithm can guarantee an approximation ratio of 1 + Ɛ unless P = NP. We obtain our results by proposing gap-preserving reductions, based on reductions from [8]. Our results are the first inapproximability results for these problems.
TL;DR: A “Complementation Theorem” for the number of matchings of certain subgraphs of cellular graphs is presented, which generalizes the main result of M. Ciucu and is a weighted generalization of a result of Knuth.
TL;DR: This work presents the first known deterministic sublinear space, polynomial time algorithm for directed s-t connectivity, which can use as little as n/2^{\Theta(\sqrt{\log n})}$ space while still running in polynometric time.
Abstract: Directed s-t connectivity is the problem of detecting whether there is a path from vertex s to vertex t in a directed graph. We present the first known deterministic sublinear space, polynomial time algorithm for directed s-t connectivity. For $n$-vertex graphs, our algorithm can use as little as $n/2^{\Theta(\sqrt{\log n})}$ space while still running in polynomial time.
23 May 1998
TL;DR: This work presents simple semidefinite programming relaxations for the m-hard minimum bandwidth and minimum length linear ordering problems and shows how these relaxations can be rounded in a natural way to obtain new approximation guarantees.
Abstract: We present simple semi-definite programming relaxations for the NP-hard minimum bandwidth and minimum length linear ordering problems. We then show how these relaxations can be rounded in a natural way (via random projection) to obtain approximation guarantees for both of these vertex-ordering problems.
TL;DR: In this paper, the irreducible modules for the fixed point vertex operator subalgebra V_L^+ of the vertex operator algebra associated to a positive definite even lattice of rank 1 under the automorphism lifted from the -1 isometry of L.
Abstract: We classify the irreducible modules for the fixed point vertex operator subalgebra V_L^+ of the vertex operator algebra V_L associated to a positive definite even lattice of rank 1 under the automorphism lifted from the -1 isometry of L.
TL;DR: This paper proves a characterization of these separators in terms of the monotone adjacency sets of the vertices of the graph, numbered by the maximum cardinality search (MCS) scheme, and introduces the notion of multiplicity of a minimal vertex separator which indicates the number of different pairs of vertices separated by it.
TL;DR: It is proved that ifGis a planar graph with total (vertex?edge) chromatic number ??, maximum degree ? and girthg, then ??
Abstract: It is proved that ifGis a planar graph with total (vertex?edge) chromatic number ??, maximum degree ? and girthg, then ??=?+1 if ??5 andg?5, or ??4 andg?6, or ??3 andg?10. These results hold also for graphs in the projective plane, torus and Klein bottle.
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TL;DR: In this paper, the irreducible modules for the fixed point vertex operator subebra of the rank 1 free bosonic VOA under the -1 automorphism were classified.
Abstract: We classify the irreducible modules for the fixed point vertex operator subebra of the rank 1 free bosonic VOA under the -1 automorphism.
04 Oct 1998
TL;DR: A new compression scheme which encode topological data by using the dual graph of the original mesh is proposed, and it is found that the dualgraph can be represented as a degraded binary tree.
Abstract: The triangular mesh provides one of the most popular representations for 3D graphic models. A typical triangular mesh consists of two different types of data: topological data which specify the connectivity of the mesh and geometrical data which describe information associated with each individual vertex or triangle. We propose a new compression scheme which encode topological data by using the dual graph of the original mesh. It is found that the dual graph can be represented as a degraded binary tree. Furthermore, geometrical data can be coded progressively with local prediction and embedded entropy coding. Experimental results show that an acceptable quality level can be reached at a compression ratio of 60 to 1 for general test models.