Showing papers on "Vertex (graph theory) published in 1991"

Primitive operator digital filters

Abstract: The authors outline a design methodology for the realisation of digital filtering structures with significantly reduced numbers of elementary arithmetic operations. The directed acyclic graphs which result from the design algorithms completely describe the filter arithmetically and may be mapped directly onto hardware or software realisations. Vertex rearrangement, retiming and edge elimination techniques are presented which facilitate the generation of a logical graph with an efficient allocation of pipeline registers. An example of the technique is given for a bit-serial realisation employing a bit-level pipeline. >

Robotic exploration as graph construction

Abstract: Addressed is the problem of robotic exploration of a graphlike world, where no distance or orientation metric is assumed of the world. The robot is assumed to be able to autonomously traverse graph edges, recognize when it has reached a vertex, and enumerate edges incident upon the current vertex relative to the edge via which it entered the current vertex. The robot cannot measure distances, and it does not have a compass. It is demonstrated that this exploration problem is unsolvable in general without markers, and, to solve it, the robot is equipped with one or more distinct markers that can be put down or picked up at will and that can be recognized by the robot if they are at the same vertex as the robot. An exploration algorithm is developed and proven correct. Its performance is shown on several example worlds, and heuristics for improving its performance are discussed. >

Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface

Abstract: We describe all regular tiings of the torus and the Klein bottle. We apply this to describe, for each orientable (respectively nonorientable) surface S, all (but finitely many) vertex-transitive graphs which can be drawn on S but not on any surface of smaller genus (respectively crosscap number). In particular, we prove the conjecture of Babai that, for each g > 3, there are only finitely many vertex-transitive graphs of genus g. In fact, they all have order 2, there are only finitely many groups that act on the surface of genus g . We also derive a nonorientable version of Hurwitz' theorem.

On Disjoint Cycles

Abstract: It is shown, that for each constant k ≥ 1, the following problems can be solved in O(n) 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 \(\mathcal{G}\), that is closed under minor taking, or that is closed under immersion taking, and that does not contain the graph formed by taking the disjoint union of k copies of K3, has an \(\mathcal{O}\)(n) membership test algorithm.

A hierarchical multiple-view approach to three-dimensional object recognition

Abstract: A hierarchical approach is proposed for solving the surface and vertex correspondence problems in multiple-view-based 3D object-recognition systems. The proposed scheme is a coarse-to-fine search process, and a Hopfield network is used at each stage. Compared with 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 scores between the input image and each object model in the database are computed through a Hopfield network and are 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. Experimental results are reported using both synthetic and real range images to corroborate the proposed theory. >

Approximating Treewidth, Pathwidth, and Minimum Elimination Tree Height

Abstract: We show how the value of various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al. that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum elimination tree height, are no more than O(log n) (minimum front size and treewidth) and O(log2n) (pathwidth and minimum elimination tree height) times the optimal values. In addition we examine the existence of bounded approximation algorithms for the parameters, and show that unless P = NP, there are no absolute approximation algorithms for them.

Efficient method to generate collision free paths for an autonomous mobile robot based on new free space structuring approach

Abstract: A technique is developed based on free link concept to construct the available free space between obstacles within robot's environment in terms of free convex area. Then, a new kind of vertex graph called MAKLINK is constructed to support the generation of a collision free path. This graph is constructed using the midpoints of common free links between free convex area as passing points. These points correspond to nodes in a graph and the connection between them within each convex area as arcs in this graph. A collision free path can be efficiently generated using the MAKLINK graph. The complexity of the search for a collision free path is greatly reduced by minimizing the size of the graph to be searched concerning the number of nodes and the number of arcs connecting them. The analysis of the proposed algorithm shows its efficiency in terms of computation ability, safety, optimality, and in supporting robot navigation along the generated path. >

Partitions of graphs into one or two independent sets and cliques: Revised version

Abstract: lt is shown that it can be recognized in polynomial time whether the vertex set of a finite undirected graph can be partitioned into one or two independent sets and cliques. Such graphs generalize bipartite and split graphs and the result also shows that it can be recognized in polynomial time whether a graph can be partitioned into two split graphs. An efficient time bound 0(n2m) is given for the recognition of graphs which can be partitioned into two independent sets and one clique.

Improved lower bounds on k-independence

Abstract: A vertex set Y in a (hyper)graph is called k-independent if in the sub(hyper)-graph induced by Y every vertex is incident to less than k edges. We prove a lower bound for the maximum cardinality of a k-independent set—in terms of degree sequences—which strengthens and generalizes several previously known results, including Turan's theorem.

Geometric interpretation of vertex operator algebras.

Abstract: In this paper, Vafa's approach to the formulation of conformal field theories is combined with the formal calculus developed in Frenkel, Lepowsky, and Meurman's work on the vertex operator construction of the Monster to give a geometric definition of vertex operator algebras. The main result announced is the equivalence between this definition and the algebraic one in the sense that the categories determined by these definitions are isomorphic.

Reducing edge connectivity to vertex connectivity

Abstract: We show how to reduce edge connectivity to vertex connectivity. Using this reduction, we obtain a linear-time algorithm for deciding whether an undirected graph is 3-edge-connected, and for computing the 3-edge-connected components of an undirected graph.

Two-graphs, a second survey

Abstract: Publisher Summary A two-graph ( Ω , Δ ) is a vertex set Ω and a collection Δ of 3-subsets of Ω such that every 4-subset of Ω contains an even number of 3-subsets from Δ . The two-graph is regular if every 2-subset of Ω is contained in the same number of 3-subsets from Δ . The chapter describes two-graphs in terms of exterior algebra and presents Cameron's simple proof as per which two-graphs and Euler graphs are equal in number. It also presents the relations with equiangular lines, and highlights certain generalizations and Cameron's cohomology classes associated with a group of automorphisms of a two-graph by a number of simple examples. The example with nontrivial first and second invariant is completely worked out. The chapter also explains the state of affairs for regular two-graphs for n ≤ 46 and the construction of conference two-graphs of order pq 2 + 1 .

Vertex-transitive graphs and vertex-transitive maps

Abstract: We consider vertex-transitive graphs embeddable on a fixed surface. We prove that all but a finite number of them admit embeddings as vertex-transitive maps on surfaces of nonnegative Euler characteristic (sphere, projective plane, torus, or Klein bottle). It follows that with the exception of the cycles and a finite number of additional graphs, they are factor graphs of semiregular plane tilings. The results generalize previous work on the genus of minimal Cayley graphs by V. Proulx and T. W. Tucker and were obtained independently by C. Thomassen, with significant differences in the methods used. Our method is based on an excursion into the infinite. The local structure of our finite graphs is studied via a pointwise limit construction, and the infinite vertex-transitive graphs obtained as such limits are classified by their connectivity and the number of ends. In two appendices, we derive a combinatorial version of Hurwitz's Theorem, and classify the vertex-transitive maps on the Klein bottle.

Some extremal results in cochromatic and dichromatic theory

Abstract: For a graph G, the cochromatic number of G, denoted z(G), is the least m for which there is a partition of the vertex set of G having order m. where each part induces a complete or empty graph. We show that if {Gn} is a family of graphs where Gn has o(n2 log2(n)) edges, then z(Gn) = o(n). We turn our attention to dichromatic numbers. Given a digraph D, the dichromatic number of D is the minimum number of parts the vertex set of D must be partitioned into so that each part induces an acyclic digraph. Given an (undirected) graph G, the dichromatic number of G, denoted d(G), is the maximum dichromatic number of all orientations of G. Let m be an integer; by d(m) we mean the minimum size of all graphs G where d(G) = m. We show that d(m) = θ(m2 ln2(m)).

Vertex operators and quantum hall effect

Abstract: F.V. vertex operator which allows a consistent bosonization of fermions, bosons and anyons is shown. It thus plays an essential role in the general theory of Fractional Quantum Hall Effect (F.Q.H.E.).

Algorithms for parallel k-vertex connectivity and sparse certificates

Abstract: A certificate for the k-vertex connectivity of a graph G = (V, E) is a subset E’ of E such that (V, E’) is k-vertex connected iff G is k-vertex connected. Let n = IV[ and m = IEI. A certificate is called sparse if it has size lE’I = O(kn). We show that sparse certificates for undirected graphs can be computed by executing k breadth first searches in sequence. This directly gives an efficient algorithm on the distributed model of computation. Using certificates of size O(k2 n), we show that the k-vertex connectivity of an undirected graph can be tested on the CRCW PRAM model in a parallel running time of O(k2 log n) using k3n2cr(k2n, n)/(log n) processors. For k >3 and m > k2n, this improves on previous deterministic parallel algorithms (for dense graphs and for constant k >4 the number of processors improves by a factor of roughly n). We also give sequential algorithms for finding (undirected) sparse certificates ‘(on-line”, and for finding -. certificate es of size <2 k2 n for directed graphs,