Showing papers on "Vertex (graph theory) published in 1990"

Partitioning sparse matrices with eigenvectors of graphs

Abstract: The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is, shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorith...

Random Walks and Percolation on Trees

Abstract: There is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree It equals the exponential of the Hausdorff dimension of the boundary in an appropriate metric Its importance for probabilistic processes on a tree is shown in several ways, including random walk and percolation, where it provides points of phase transition

Rectifiable sets and the Traveling Salesman Problem

Abstract: Let K c C be a bounded set. In this paper we shall give a simple necessary and sufficient condit ion for K to lie in a rectifiable curve. We say that a set is a rectifiable curve if it is the image of a finite interval under a Lipschitz mapping. Recall that for a connected set F c C, F is a rectifiable curve (not necessarily simple) if and only if l(F) < ~ , where l(-) denotes one dimensional Hausdorff measure. This classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge (but not necessarily each vertex!) at most twice. If K is a finite set then we are essentially reduced to the classical Traveling Salesman Problem (TSP): Compute the length of the shortest Hami l ton ian cycle which hits all points of K. This is the same, up to a constant multiple, as asking for the inf imum of l(F) where F is a curve, K c F. (Such a F is called a spanning tree in TSP theory.) For infinite sets K, we cannot hope in general to have K be a subset of a Jordan curve. What we should therefore look at is connected sets which conta in K. Let Fmi n be the shortest (minimal) spanning tree. Then we cannot possibly solve our problem for sets K of infinite cardinality if we cannot find F, I(F) < C O/(Fmin) , for any finite set K. (Here and throughout the paper C, Co, C1, c o , etc. denote various universal constants.) While there are several algorithms for computing l(Fml.), these algorithms work for finite graphs satisfying the triangle inequality, and do not use the Euclidean properties of K. (See [13] for an excellent discussion of some of these algorithms.) Therefore these methods cannot solve our problem for general infinite K. We present a method which is a minor modification of a well-known algorithm ("Farthest Insert ion" see [13]) which yields a F with I(F) < C O l(Fmi,). The Farthest Insert ion algorithm has been extensively studied with large numerical calculations on computers, and is experimentally good in the sense that the F produced satisfy I(F) < C O l(F,,,i,) for all examples which have

Computing the aspect graph for line drawings of polyhedral objects

Abstract: An algorithm for computing the aspect graph for polyhedral objects is described. The aspects graph is a representation of three-dimensional objects by a set of two-dimensional views. The set of viewpoints on the Gaussian sphere is partitioned into regions such that in each region the qualitative structure of the line drawing remains the same. At the boundaries between adjacent regions are the accidental viewpoints where the structure for the line drawing changes. It is shown that for polyhedral objects there are two fundamental visual events: (1) the projections of an edge and a vertex coincide; and (2) the projections of three nonadjacent edges intersect at a point. The geometry of the object is reflected in the locus of the accidental viewpoints. The algorithm computes the partition together with a representative view for each region of the partition. >

Truncating the Schwinger-Dyson equations: How multiplicative renormalizability and the Ward identity restrict the three point vertex in QED

Abstract: Nonperturbative studies of field theory require the Schwinger-Dyson equations to be truncated to make them tractable. Thus, when investigating the behavior of the fermion propagator, for example, an Ansatz for the three-point vertex has to be made. While the well-known Ward identity determines the longitudinal part of this vertex in terms of the fermion propagator as shown by Ball and Chiu, it leaves the transverse part unconstrained. However, Brown and Dorey have recently emphasized that the requirement of multiplicative renormalizability is not satisfied by arbitrary Ansa$iuml---tze for the vertex. We show how this requirement restricts the form of the transverse part. By considering the example of QED in the quenched approximation, we present a form for the vertex that not only satisfies the Ward identity but is multiplicatively renormalizable to all orders in leading and next-to-leading logarithms in perturbation theory and so provides a suitable Ansatz for the full three-point vertex.

Perfect dominating sets

Abstract: A dominating set of a graph is perfect if each vertex of is dominated by exactly one vertex in . We study the existence and construction of PDSs in families of graphs arising from the interconnection networks of parallel computers. These inc lude trees, dags, series-parallel graphs, meshes, tori, hypercubes, cube-connected cycles, cube-connected paths, and de Bruijn graphs. For trees, dags, and series-parallel graphs we give linear t ime algorithms that determine if a PDS exists, and generate a PDS when one does. For 2- and 3-dimensional meshes, 2-dimensional tori, hypercubes, and cube-connected paths we completely characterize which graphs have a PDS, and the structure of all PDSs. For higher dimensional meshes and tori, cube-connected cycles, and de Bruijn graphs, we show the existence of a PDS in infinitely man y cases, but our characterization is not complete. Our results include distance -domination for arbitrary .

Origins of the simplex method

Abstract: : In the summer of 1947, when the author first began to work on the simplex method for solving linear programs, the first idea that occurred to him is one that would occur to any trained mathematician, namely the idea of step by step descent (with respect to the objective function) along edges of the convex polyhedral set from one vertex to an adjacent one. He rejected this algorithm outright on intuitive grounds - it had to be inefficient because it proposed to solve the problem by wandering along some path of outside edges until the optimal vertex was reached. He therefore began to look for other methods which gave more promise of being efficient, such as those that went directly through interior. Today we know that before 1947 that five isolated papers had been published on special cases of the linear programming problem by Monge (1781), Fourier (1824), de la Vallee Poussin (1911), Kantorovich (1939), and Hitchcock (1941). Fourier, Poussin, and Hitchcock proposed as a solution method descent along the outside edges of the polyhedral set which is the way we describe the simplex method today. There is no evidence that these papers had any influence on each other. Evidently they sparked zero interest on the part of other mathematicians, an exception being a paper Appell (1928) on Monge's translocation of masses problem. These references were unknown to the author when he first proposed the simplex method. As we shall see the simplex algorithm evolved from a very different geometry, one in which it appeared to be very efficient.

Multiple-view approach to three-dimensional object recognition

Abstract: In this thesis, a model-based three-dimensional (polyhedral) object recognition system using multiple-view approach is presented. A systematic method to automatically generate all topologically different characteristic views (CVs) of a polyhedron and organize these CVs as aspect graphs is first devised. Then, a coarse-to-fine search strategy based on Hopfield networks is proposed to locate the most similar view in the aspect graph. Compared with the conventional object matching schemes, the proposed technique provides a more general and compact formulation of the problem and a solution more suitable for parallel implementation. At the coarse search stage, the surface matching rates between the input image and each object model in the database are computed through a Hopfield network and used to select the candidates for further consideration. At the fine search stage, the object models selected from the previous stage are fed into another Hopfield network for vertex matching. The object model that has the best surface and vertex correspondences with the input image is finally singled out as the best matched model. Once an object model is identified, the viewing geometry in the modeling phase is utilized to derive the pose of the unknown object.

On the linear vertex-arboricity of a planar graph

Abstract: We prove in this note that the linear vertex-arboricity of any planar graph is at most three, which confirms a conjecture due to Broere and Mynhardt, and others.

Electromagnetic form factors of spin-3/2 baryons.

Abstract: Relationships are established among current matrix elements, covariant vertex functions, and multipole form-factor decompositions for the case of electromagnetic interactions of spin-3/2 systems The electromagnetic current matrix element for spin-3/2 baryons is defined in terms of the minimum required four independent covariant vertex functions Explicit Lorentz-invariant relations for the multipole form factors are derived in terms of the previously defined covariant vertex function coefficients The derivation does not involve any nonrelativistic approximations Finally, the multipole form factors are isolated and expressed in terms of the electromagnetic current matrix elements These results are particularly useful in lattice QCD calculations

Finding critical independent sets and critical vertex subsets are polynomial problems

Abstract: An independent set $J_c $ of a graph G is called critical if \[ | J_c | - | N ( J_c ) | = \max \{ | J | - | N ( J ) |:J\,\text{is an independent set of }G \}, \] and a vertex subset $U_c $ is called critical if \[ | U_c | - | N ( U_c ) | = \max \{ | U | - | N ( U ) |:U\,\text{is a vertex subset of }G \} . \] In this paper, it will be shown that finding a critical independent set and a critical vertex subset of a graph are solvable in polynomial time.

A partition property of simplices in Euclidean space

Abstract: In a series of papers, Erdos et al. [E] have investigated this property. They have shown that all Ramsey sets are spherical, that is, every Ramsey set is contained in an appropriate sphere. On the other hand, they have shown that the vertex set (and, therefore, all its subsets) of bricks (d-dimensional parallelepipeds) is Ramsey. The simplest sets that are spherical but cannot be embedded into the vertex set of a brick are the sets of obtuse triangles. In [FR1], it is shown that they are indeed Ramsey, using Ramsey's Theorem (cf. [G2]) and the Product Theorem of [E]. The aim of the present paper is twofold. First, we want to show that the vertex set of every nondegenerate simplex in any dimension is Ramsey. Second, we want to show that for both simplices and bricks, and even for their products, one can in fact choose n(r, B) = c(B)logr, where c(B) is an appropriate positive constant. The paper is organized as follows. In ?2, super-Ramsey property is introduced. This notion is stronger than being Ramsey. It is shown that the direct product of super-Ramsey sets is super-Ramsey. In ?3, it is shown that if every edge of an n-dimensional simplex is between 1 e and 1 + e with e = e(n) being a sufficiently small positive number, then it can be embedded into a brick, i.e., into the direct product of two-element sets. In ?4, it is proved that given a nondegenerate simplex with edge lengths aij, 1 0, there exists some super-Ramsey simplex whose edge lengths vi verify la 2_-v2jI

A correlation inequality and a poisson limit theorem for nonoverlapping balanced subgraphs of a random graph

Abstract: We consider non-overlapping subgraphs of fixed order in the random graph Kn, p(n). Fix a strictly strongly balanced graph G. A subgraph of Kn, p(n) isomorphic to G is called a G-subgraph. Let Xn be the number of G-subgraphs of Kn, p(n) vertex disjoint to all other G-subgraphs. We show that if E[Xn]→∞ as n→, then Xn/E[Xn] converges to 1 in probability. Also, if E[Xn]→c as n→∞, then Xn satisfies a Poisson limit theorem. the Poisson limit theorem is shown using a correlation inequality similar to those appeared in Janson, Łuczak, and Rucinski[8] and Boppana and Spencer [4].

A characterization of smooth Cantor bouquets

Abstract: We prove that all smooth fans having a dense set of endpoints are topologically equivalent. Let X be a smooth fan whose set of endpoints is dense in X. Such fans have been constructed, e.g., by J. H. Roberts [6], who proved that the space of rational sequences of the Hilbert cube can be embedded in the Cantor fan, and by A. Lelek [3], who showed the existence of a fan whose (one-dimensional) set of endpoints can be connectified by adding the vertex. Lately, spaces similar to X \ {v}, where v is the vertex of X, were discovered to be Julia sets of some nice analytic functions (see R. L. Devaney and M. Krych [2]; see also J. C. Mayer [4]). We are going to prove that all such examples are homeomorphic. Theorem. All smooth fans having dense set of endpoints are topologically equivalent. Let us recall that a continuum X is said to be hereditarily unicoherent if K n L is connected for every pair K, L of subcontinua of X. A continuum X is called a dendroid if it is arcwise connected and hereditarily unicoherent. By afan we will mean a dendroid having exactly one ramification point; we will call this point the vertex of X. A fan X is said to be smooth if the sequence of arcs [v , xn ] converges to the arc [v , x] for every sequence xn converging to x, where x, xn E X and v is the vertex of X. If X is a fan, then E(X) will denote the set of endpoints of X. If x, y E 9i x 9i, then by Ix yI we will denote the Euclidean distance between points x and y and by [x , y] we will mean the linear segment with endpoints x and y. MAPPINGS BETWEEN INVERSE SYSTEMS The following lemma is similar to [5, Theorem 2']. The proof is a standard inductive argument and is left to the reader. Received by the editors October 18, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F20, 54F65.

On the road coloring problem

Abstract: Let G=(V, E) be a strongly connected, aperiodic, directed graph having outdegree 2 at each vertex. A red-blue coloring of G is a coloring of the edges with the colors red and blue such that each vertex has one red edge and one blue edge leaving it. Given such a coloring, we define R:V→V by R(v)=w if there is a red edge from v to w. Similarly we define B:V→V. G is said to be collapsible if some composition of R's and B's maps V to a single vertex. The road coloring problem is to determine whether G has a collapsible coloring. The adjacency matrix, A, of G has a positive left eigenvector w=(w(v 1 ),..., w(v n )) with eigenvalue 2. We can assume that w's components are integers with no common factor. We call w(v) the weight of v. Let W≡Σ v∈V w(v), defined to be the weight of the graph. We prove that if G has a simple cycle of length relatively prime to W, then G is collapsibly colorable

On the structure of minimum-weight k-connected spanning networks

Abstract: The problem of finding a minimum-weight k-connected spanning subgraph of a complete graph, assuming that the edge weights satisfy the triangle inequality, is studied. It is shown that the class of minimum-weight k-edge connected spanning subgraphs can be restricted to those subgraphs which, in addition to the connectivity requirements, satisfy the following two conditions: (I) Every vertex has degree k or $k + 1$; (II) Removing any $1, 2, \cdots ,$ or k edges does not leave the resulting connected components all k-edge connected. For the k-vertex connected case, the parallel result is obtained with “k-edge” replaced by “k-vertex,” with the added technical restriction that $| V |\geqq 2k$ for condition (I) to hold. This generalizes recent work of Monma, Munson, and Pulleyblank for the case $k = 2$.

Transversals of Vertex Partitions in Graphs

Abstract: This paper studies graph properties of the following forms: For every partition of the vertex set that satisfies an upper (or lower) bound on the number of elements in each partition class, there is a transversal of the partition that is an independent (or dominating) set. A possible application to fault-tolerant data storage is discussed, and bounds for the parameters that are functions of minimum and maximum degree are established. The complexity of associated decision problems is also addressed.

Dynamical mass generation in three-dimensional QED: Improved vertex function.

Abstract: We study chiral-symmetry breaking for three-dimensional QED with N fermion flavors, just above the critical threshold. By analysis of a consistently truncated Schwinger-Dyson system for the fermion propagator and the fermion-boson vertex, we argue that the critical coupling must be strictly positive.

Vertex operator superalgebras

Abstract: In this paper, we construct the action of super vertex operators In section 1, we construct a Fock space V for a 2l-dimensional C-vector space A. In physics, V appears as the Fock space of Neveu-Schwarz model of dual poins. See Neveu-Schwarz [5]. In mathematics, V appears as the spin representation space of the affine Lie algebra&ŝ2l. See Frenkel [2] and [3]. In section 2, we define a super vertex operator A V,Z : V→V[[z,z-1]] for each element v of the Fock space V. The most important property of the super vertex operators is the so-called associative law. Using this law, we can see that the coefficients v n : V→V in A v,z span a Lie superalgebra and explicitly write down the Lie superbackets in section 3. It is a huge Lie superalgebra called the vertex operator superalgebra of A.

Local Tournaments and Proper Circular Arc Gaphs

Abstract: A local tournament is a digraph in which the out-set as well as the in-set of every vertex is a tournament. These digraphs have recently been found to share many desirable properties of tournaments. We illustrate this by giving O(m+n logn) algorithms to find a hamiltonian path and cycle in a local tournament. We mention several characterizations and recognition algorithms of graphs orientable as local tournaments. It turns out that they are precisely the graphs previously studied as proper circular arc graphs. Thus we obtain new recognition algorithms for proper circular arc graphs. We also give a more detailed structural characterization of chordal graphs that are orientable as local tournaments, i.e., that are proper circular arc graphs.

Graphs with homeomorphically irreducible spanning trees

Abstract: It is an NP-complete problem to decide whether a graph contains a spanning tree with no vertex of degree 2. We show that these homeomorphically irreducible spanning trees are contained in all graphs with minimum degree at least c√n and in triangulations of the plane. They are nearly present in all graphs of diameter 2. They do not necessarily occur in r-regular or r-connected graphs.