Vertex (geometry)

About: Vertex (geometry) is a research topic. Over the lifetime, 18765 publications have been published within this topic receiving 294216 citations. The topic is also known as: 0-polytope & 0-simplex.

Algorithmic Aspects of Vertex Elimination on Graphs

Abstract: We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse symmetric positive definite systems of linear equations. We give a new linear-time algorithm to calculate the fill-in produced by any elimination ordering, and we give two new related algorithms for finding orderings with special properties. One algorithm, based on breadth-first search, finds a perfect elimination ordering, if any exists, in $O(n + e)$ time, if the problem graph has n vertices and e edges. An extension of this algorithm finds a minimal (but not necessarily minimum) ordering in $O(ne)$ time. We conjecture that the problem of finding a minimum ordering is NP-complete

A Separator Theorem for Planar Graphs

Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.

Methods for Visual Understanding of Hierarchical System Structures

Abstract: Two kinds of new methods are developed to obtain effective representations of hierarchies automatically: theoretical and heuristic methods. The methods determine the positions of vertices in two steps. First the order of the vertices in each level is determined to reduce the number of crossings of edges. Then horizontal positions of the vertices are determined to improve further the readability of drawings. The theoretical methods are useful in recognizing the nature of the problem, and the heuristic methods make it possible to enlarge the size of hierarchies with which we can deal. Performance tests of the heuristic methods and several applications are presented.

A separator theorem for planar graphs

Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A,B,C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2\sqrt{2}\sqrt{2}$ vertices. We exhibit an algorithm which finds such a partition A,B,C in O(n) time.

Estimation and prediction for stochastic blockstructures

Abstract: A statistical approach to a posteriori blockmodeling for digraphs and valued digraphs is proposed. The probability model assumes that the vertices of the digraph are partitioned into several unobserved (latent) classes and that the probability distribution of the relation between two vertices depends only on the classes to which they belong. A Bayesian estimator based on Gibbs sampling is proposed. The basic model is not identified, because class labels are arbitrary. The resulting identifiability problems are solved by restricting inference to the posterior distributions of invariant functions of the parameters and the vertex class membership. In addition, models are considered where class labels are identified by prior distributions for the class membership of some of the vertices. The model is illustrated by an example from the social networks literature (Kapferer's tailor shop).