Showing papers on "Vertex (graph theory) published in 1994"
TL;DR: The notion of a d-cluster is introduced, using it to bound the multiplicity of d in the spectrum of L(G), and the degree sequence and the Laplacian spectrum through majorization are related.
Abstract: Let G be a graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then L(G) = D(G) - A(G) is the Laplacian matrix of G. The first section of this paper is devoted to properties of Laplacian integral graphs, those for which the Laplacian spectrum consists entirely of integers. The second section relates the degree sequence and the Laplacian spectrum through majorization. The third section introduces the notion of a d-cluster, using it to bound the multiplicity of d in the spectrum of L(G).
445 citations
TL;DR: It follows that branch-width is polynomially computable for planar graphs—that too is NP-hard for general graphs.
Abstract: Suppose we expect there to bep(ab) phone calls between locationsa andb, all choices ofa, b from some setL of locations. We wish to design a network to optimally handle these calls. More precisely, a “routing tree” is a treeT with set of leavesL, in which every other vertex has valency 3. It has “congestion” 0 form the edges of a planar graphG, there is an efficient, strongly polynomial algorithm. This is because the problem is equivalent to deciding if a ratcatcher can corner a rat loose in the walls of a house with floor planG, wherep(ab) is a thickness of the wallab. The ratcatcher carries a noisemaker of powerk, and the rat will not move through any wall in which the noise level is too high (determined by the total thickness of the intervening walls between this one and the noisemaker). It follows that branch-width is polynomially computable for planar graphs—that too is NP-hard for general graphs.
394 citations
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TL;DR: The notion of vertex operator superalgebras was introduced in this paper, where it was shown that any local system of vertex operators on a super vector space has a natural vertex (super)algebra structure with $M$ as a module.
Abstract: We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of ``local system of vertex operators'' for a (super) vector space. We first prove that any local system of vertex operators on a (super) vector space $M$ has a natural vertex (super)algebra structure with $M$ as a module. Then we prove that for a vertex (operator) superalgebra $V$, giving a $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of vertex operators on $M$. As applications, we prove that certain lowest weight modules for some well-known infinite-dimensional Lie algebras or Lie superalgebras have natural vertex operator superalgebra structures. We prove the rationality of vertex operator superalgebras associated to standard modules for an affine algebra. We also give an analogue of the notion of the space of linear homomorphisms from one module to another for a Lie algebra by introducing a notion of ``generalized intertwining operators.'' We prove that $G(M^{1},M^{2})$, the space of generalized intertwining operators from one module $M^{1}$ to another module $M^{2}$ for a vertex operator superalgebra $V$, is a generalized $V$-module. Furthermore, we prove that for a fixed vertex operator superalgebra $V$ and
311 citations
TL;DR: The vertex connectivity for the n-dimensional cube is obtained, and the minimal sets of faulty nodes that disconnect the cube are characterized.
Abstract: Introduces a new measure of conditional connectivity for large regular graphs by requiring each vertex to have at least g good neighbors in the graph. Based on this requirement, the vertex connectivity for the n-dimensional cube is obtained, and the minimal sets of faulty nodes that disconnect the cube are characterized. >
270 citations
TL;DR: Algorithms that, for any tree T, compute vs ( T ) in linear time and compute an optimal layout with respect to vertex separation in time O ( n log n) are given.
Abstract: We relate two concepts in graph theory and algorithmic complexity, namely the search number and the vertex separation of a graph. Let s ( G ) denote the search number and vs ( G ) denote the vertex separation of a connected, undirected graph G . We show that vs ( G ) ≤ s ( G ) ≤ vs ( G ) + 2 and we give a simple transformation from G to G′ such that vs ( G′ ) = s ( G ). We characterize those trees having a given vertex separation and describe the smallest such trees. We also note that there exist trees for which the difference between search number and vertex separation is indeed 2. We give algorithms that, for any tree T , compute vs ( T ) in linear time and compute an optimal layout with respect to vertex separation in time O ( n log n ). Vertex separation has previously been related to progressive black / white pebble demand and has been shown to be identical to a variant of search number, node search number , and to path width , which has been related to gate matrix layout cost . All these properties are known to be computationally intractable. For fixed k , an O ( n log 2 n ) algorithm is known which decides whether a graph has path width at most k .
249 citations
TL;DR: It is proved that the recognition of SEG-graphs is of the same complexity as the decision of solvability of a system of strict polynomial inequalities in the reals, i.e., as the decisions of a special existentially quantified sentence in the theory of real closed fields, and thus it belongs to PSPACE.
233 citations
189 citations
TL;DR: In this article, a tensor product theory of classes of modules for vertex operator algebra is presented, which is based on both the formal-calculus approach to vertex algebra theory developed in [FLM2] and [FHL] and the precise geometric interpretation of the notion of vertex algebra established in [H1].
Abstract: In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announcement has also appeared [HL1]. The theory is based on both the formal-calculus approach to vertex operator algebra theory developed in [FLM2] and [FHL] and the precise geometric interpretation of the notion of vertex operator algebra established in [H1].
159 citations
TL;DR: In this article, it was shown that the Laplacian spectrum of a maximal graph is the conjugate of its degree sequence, and that, apart from 0, it is the same for all maximal graphs.
144 citations
TL;DR: In this paper, the authors consider systems of particles hopping stochastically on d-dimensional lattices with space-dependent probabilities and derive duality relations, expressing the time evolution of a given initial configuration in terms of correlation functions of simpler dual processes.
Abstract: We consider systems of particles hopping stochastically on d-dimensional lattices with space-dependent probabilities. We map the master equation onto an evolution equation in a Fock space where the dynamics are given by a quantum Hamiltonian (continuous time) or a transfer matrix (discrete time). Using non-Abelian symmetries of these operators we derive duality relations, expressing the time evolution of a given initial configuration in terms of correlation functions of simpler dual processes. Particularly simple results are obtained for the time evolution of the density profile. As a special case we show that for any SU(2) symmetric system the two-point and three-point density correlation functions in the N-particle steady state can be computed from the probability distribution of a single particle moving in the same environment. We apply our results to various models, among them partial exclusion, a simple diffusion-reaction system, and the two-dimensional six-vertex model with space-dependent vertex weights. For a random distribution of the vertex weights one obtains a version of the random-barrier model describing diffusion of particles in disordered media. We derive exact expressions for the averaged two-point density correlation functions in the presence of weak, correlated disorder.
140 citations
TL;DR: The level-ancestor problem is considered and the Euler tour of the tree and the level of each vertex are given and the only change in result (1) above is that preprocessing time increases to O (log n ).
TL;DR: It is proved, in particular, that if a simple graph G = (V, E) without triangles has an interval t-coloring, then t ?
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TL;DR: It is shown, that for each constant k≥1, the following problems can be solved in time: given a graph G, determine whether G has k vertex disjoint cycles, determine Whether G has a feedback vertex set of size ≤k.
Abstract: It is shown, that for each constant k≥1, the following problems can be solved in time: given a graph G, determine whether G has k vertex disjoint cycles, determine whether G has k edge disjoint cycles, determine whether G has a feedback vertex set of size ≤k. Also, every class , that is closed under minor taking, taking, and that does not contain the graph consisting of k disjoint copies of K3, has an membership test algorithm.
TL;DR: An infinite family of vertex-and edge-transitive, but not arc-transveto, graphs of degree 4 was constructed in this article, where the vertex and edge transversality was considered.
Abstract: An infinite family of vertex- and edge-transitive, but not arc-transitive, graphs of degree 4 is constructed.
TL;DR: One of the results states that this decision problem remains NP-complete even if all of the following conditions are met: each “color” xϵ⋃vϵVL(v) occurs in at most three sets L(v), and G is a planar graph.
TL;DR: A constant competitive algorithm for a searcher exploring an initially unknown weighted planar graph G, where the searcher's goal is to visit each vertex of G, incurring as little cost as possible.
TL;DR: It is shown that the following statement is true fork=4 if the authors restrict ourselves to planar graphs and similar statements are considered for weaklys — independent spanning trees and for directed graphs.
Abstract: IfG is a finite undirected graph ands is a vertex ofG, then two spanning treesT 1 andT 2 inG are calleds -- independent if for each vertexx inG the paths fromx tos inT 1 andT 2 are openly disjoint. It is known that the following statement is true fork≤3: IfG isk-connected, then there arek pairwises -- independent spanning, trees inG. As a main result we show that this statement is also true fork=4 if we restrict ourselves to planar graphs. Moreover we consider similar statements for weaklys -- independent spanning trees (i.e., the tree paths from a vertex tos are edge disjoint) and for directed graphs.
TL;DR: In this paper, an efficient deterministic parallel approximation algorithm for minimum-weight vertex-and set-cover problems and their duals (edge/element packing) is presented. But it does not assume knowledge of primal-dual approximation techniques.
TL;DR: The notion of vertex decomposability is used in this article to prove shellability of combinatorially defined simplicial complexes, such as chessboard complexes, which are the complexes of all nontaking rook positions on chess boards of various shapes.
Abstract: The matchings in a complete bipartite graph form a simplicial complex, which in many cases has strong structural properties. We use an equivalent description aschessboard complexes: the complexes of all nontaking rook positions on chessboards of various shapes. In this paper we construct ‘certificatek-shapes’ Σ(m, n, k) such that if the shapeA contains some Σ(m, n, k), then the (k−1)-skeleton of the chess-board complexδ(A) isvertex decomposable in the sense of Provan & Billera. This covers, in particular, the case of rectangular chessboardsA=[m]×[n], for which Δ(A) is vertex decomposable ifn≥2m−1, and the
$$([\frac{{m + n + 1}}{3}] - 1)$$
-skeleton is vertex decomposable in general. The notion of vertex decomposability is a very convenient tool to prove shellability of such combinatorially defined simplicial complexes. We establish a relation between vertex decomposability and the CL-shellability technique (for posets) of Bjorner & Wachs.
TL;DR: A different dynamic programming scheme is employed to reduce the time complexity to O( nm ) for this problem in permutation graphs.
01 Oct 1994
TL;DR: An approximation algorithm is proposed for this problem, which guaranteed a solution whose cost is greater than the optimum by a factor of at most log n (and constant in the planar case).
Abstract: This paper considers the problem of designing a minimum cost network meeting a given set of traffic requirements between n sites, using one type of channels of a given capacity, with varying set-up costs for different vertex pairs (comprised of a fixed part plus a part dependent on the pair). An approximation algorithm is proposed for this problem, which guaranteed a solution whose cost is greater than the optimum by a factor of at most log n (and constant in the planar case). The algorithm is based on an application of the recent construction of light-weight distance-preserving spanners.
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TL;DR: In this paper, the authors describe a natural structure of an abelian intertwining algebra on the direct sum of the untwisted vertex operator algebra constructed from the Leech lattice and its unique irreducible twisted module.
Abstract: We describe a natural structure of an abelian intertwining algebra (in the sense of Dong and Lepowsky) on the direct sum of the untwisted vertex operator algebra constructed {}from the Leech lattice and its (unique) irreducible twisted module. When restricting ourselves to the moonshine module, we obtain a new and conceptual proof that the moonshine module has a natural structure of a vertex operator algebra. This abelian intertwining algebra also contains an irreducible twisted module for the moonshine module with respect to the obvious involution. In addition, it contains a vertex operator superalgebra and a twisted module for this vertex operator superalgebra with respect to the involution which is the identity on the even subspace and is $-1$ on the odd subspace. It also gives the superconformal structures observed by Dixon, Ginsparg and Harvey.
01 Jan 1994
TL;DR: The first algorithm computes a perfect edge without vertex elimination ordering and the second one lists all maximal complete bipartite subgraphs.
Abstract: We present efficient algorithms for chordal bipartite graphs. Both algorithms use a doubly lexical ordering of the bipartite adjacency matrix. The first algorithm computes a perfect edge without vertex elimination ordering and the second one lists all maximal complete bipartite subgraphs.
TL;DR: A complete classification of all vertex-transitive graphs whose order is a product of two primes with a primitive automorphism group containing no imprimitive subgroup is obtained.
Abstract: Vertex-transitive graphs whose order is a product of two primes with a primitive automorphism group containing no imprimitive subgroup are classified. Combined with the results of [15] a complete classification of all vertex-transitive graphs whose order is a product of two primes is thus obtained (Theorem 2.1).
TL;DR: It is shown that if k⩾2 and k∤d(v) (∀vϵV(G)) and G is a simple graph, then G has an equitable edge-colouring with k colours, and this result is extended to one about I-regular edge-colourings of simple graphs.
TL;DR: In this paper, an approach towards construction of conformal field theory starting from the basic axioms of vertex operator algebras is presented. But this approach is restricted to the case of vertex operators.
Abstract: We develop an approach towards construction of conformal field theory starting from the basic axioms of vertex operator algebras.
TL;DR: The authors' algorithms can be used to determine the number of nonzero entries in each row and column of the triangular factor of a matrix from the zero/nonzero structure of the matrix.
Abstract: Let an undirected graph $G$ be given, along with a specified depth-first spanning tree $T$. Almost-linear-time algorithms are given to solve the following two problems. First, for every vertex $v$, compute the number of descendants $w$ of $v$ for which some descendant of $w$ is adjacent (in $G$) to $v$. Second, for every vertex $v$, compute the number of ancestors of $v$ that are adjacent (in $G$) to at least one descendant of $v$.
These problems arise in Cholesky and $QR$ factorizations of sparse matrices. The authors' algorithms can be used to determine the number of nonzero entries in each row and column of the triangular factor of a matrix from the zero/nonzero structure of the matrix. Such a prediction makes storage allocation for sparse matrix factorizations more efficient. The authors' algorithms run in time linear in the size of the input times a slowly growing inverse of Ackermann's function. The best previously known algorithms for these problems ran in time linear in the sum of the nonzero counts, which is usually much larger. Experimental results are given demonstrating the practical efficiency of the new algorithms.
06 Jun 1994
TL;DR: The contribution of this paper is an implicit method for computing the minimum cost feedback vertex set for a graph that efficiently derives a Boolean function whose satisfying assignments directly correspond to feedback vertex sets of the graph.
Abstract: The contribution of this paper is an implicit method for computing the minimum cost feedback vertex set for a graph. For an arbitrary graph,we efficiently derive a Boolean function whose satisfying assignments directly correspond to feedback vertex sets of the graph. Importantly, cycles in the graph are never explicitly enumerated, but rather, are captured implicitly in this Boolean function. This function is then used to determine theminimum cost feedbackvertex set. Even though computing theminimumcost satisfying assignment for a Boolean function remains an NP-hard problem, we can exploit the advances made in the area of Boolean function representation in logic synthesis to tackle this problem efficiently in practice for even reasonably large sized graphs. The algorithm has obvious application in flip-flop selection for partial scan. Our algorithm was the first to obtain the MFVS solutions for many benchmark circuits.