Vertex Adjacencies Research Articles

On Strict (Outer-)Confluent Graphs

A strict confluent (SC) graph drawing is a drawing of a graph with vertices as points in the plane, where vertex adjacencies are represented not by individual curves but rather by unique smooth paths through a planar system of junctions and arcs. If all vertices of the graph lie in the outer face of the drawing, the drawing is called a strict outerconfluent (SOC) drawing. SC and SOC graphs were first considered by Eppstein et al. in Graph Drawing 2013. Here, we establish several new relationships between the class of SC graphs and other graph classes, in particular string graphs and unit-interval graphs. Further, we extend earlier results about special bipartite graph classes to the notion of strict outerconfluency, show that SOC graphs have cop number two, and establish that tree-like ($\Delta$-)SOC graphs have bounded cliquewidth.

Addendum to “Vertex adjacencies in the set covering polyhedron” [Discrete Appl. Math. 218 (2017) 40–56

We study the relationship between the vertices of an up-monotone polyhedron R and those of the polytope P obtained by truncating R with the unit hypercube. When R has binary vertices, we characterize the vertices of P in terms of the vertices of R, show their integrality, and prove that the 1-skeleton of R is an induced subgraph of the 1-skeleton of P. We conclude by applying our findings to settle a claim in the original paper.

Vertex adjacencies in the set covering polyhedron

We describe the adjacency of vertices of the (unbounded version of the) set covering polyhedron, in a similar way to the description given by Chvátal for the stable set polytope. We find a sufficient condition for adjacency, and characterize it with similar conditions in the case where the underlying matrix is row circular. We apply our findings to show a new infinite family of minimally nonideal matrices.

Efficiently navigating a random Delaunay triangulation

Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. However, whilst a number of algorithms and existence proofs have been proposed, very little analysis is available for the properties of the paths generated and the computational resources required to generate them under a random distribution hypothesis for the input. In this paper we analyse a new deterministic planar navigation algorithm with constant competitiveness which follows vertex adjacencies in the Delaunay triangulation. We call this strategy \emph{cone walk}. We prove that given n uniform points in a smooth convex domain of unit area, and for any start point z and query point q; cone walk applied to z and q will access at most O(|zq| √n + log^7 n) sites with complexity O(|zq| √n log log n + log^7 n) with probability tending to 1 as n goes to infinity. We additionally show that in this model, cone walk is (log^(3+x) n)-memoryless with high probability for any pair of start and query point in the domain, for any positive x. We take special care throughout to ensure our bounds are valid even when the query points are arbitrarily close to the border.

A visibility algorithm for hybrid geometry- and image-based modeling and rendering

Hybrid geometry- and image-based modeling and rendering systems use photographs taken of a real-world environment and mapped onto the surfaces of a 3D model to achieve photorealism and visual complexity in synthetic images rendered from arbitrary viewpoints. A primary challenge in these systems is to develop algorithms that map the pixels of each photograph efficiently onto the appropriate surfaces of a 3D model, a classical visible surface determination problem. This paper describes an object-space algorithm for computing a visibility map for a set of polygons for a given camera viewpoint. The algorithm traces pyramidal beams from each camera viewpoint through a spatial data structure representing a polyhedral convex decomposition of space containing cell, face, edge and vertex adjacencies. Beam intersections are computed only for the polygonal faces on the boundary of each traversed cell, and thus the algorithm is output-sensitive. The algorithm also supports efficient determination of silhouette edges, which allows an image-based modeling and rendering system to avoid mapping pixels along edges whose colors are the result of averaging over several disjoint surfaces. Results reported for several 3D models indicate the method is well suited for large, densely occluded virtual environments, such as building interiors.

Galois Graphs: Walks, Trees and Automorphisms

Given a symmetric polynomial Φ(x, y) over a perfect field k of characteristic zero, the Galois graph G(Φ) is defined by taking the algebraic closure \(\bar k\) as the vertex set and adjacencies corresponding to the zeroes of Φ(x, y). Some graph properties of G(Φ), such as lengths of walks, distances and cycles are described in terms of Φ. Symmetry is also considered, relating the Galois group Gal(\(\bar k\)/k) to the automorphism group of certain classes of Galois graphs. Finally, an application concerning modular curves classifying pairs of isogeny elliptic curves is revisited.

Delaunay triangulations in three dimensions with finite precision arithmetic

Geometric algorithms, when implemented, often fail due to the degeneracies in the input data and numerical errors introduced by the finite precision arithmetic computations. In general, these algorithms deal with two types of data-numerical and combinatorial. Combinatorial inferences such as face adjacencies and vertex adjacencies are derived from the numerical data. Thus, inaccuracies in numerical computations may cause inconsistencies in combinatorial data which in effect either produces invalid output or makes the program fail. The ability of geometric algorithms to deal with the degeneracies and the inaccuracies during various numerical computations is referred to as their robustness. Several frameworks for achieving robustness have been proposed by different researchers. [Edelsbrunner & Miick ‘901 and [Yap ‘881 suggest using symbolic perturbation techniques to handle geometric degeneracies. [Sugihara & Iri ‘89a1 and [Dobkin & Silver ‘881 describe an approach to achieve consistent computations in solid modeling by ensuring that computations are carried out with sufficiently higher precision than that used for representing the numerical data. However, there are drawbacks as high precision routines are needed for all primitive numerical computations makin g algorithms highly machine dependent. Furthermore, it is difficult to a priori estimate the required precision for complex problems. Another approach is to live with the finite precision world and tune the arithmetic computations to satisfy certain topological constraints on the combinatorial structure of the problem. This reduces and in some cases completely avoids the chance of inconsistent results.

Computer program for finding all possible cycles in graphs.

A new approach is presented for identifying all possible cycles in graphs. Input data are the total numbers of vertices and edges, as well as the vertex adjacencies using arbitrary vertex numbering. A homeomorphically reduced graph (HRG) is constructed by ignoring vertices of degree less than three. The algorithm is based on successive generation of possible edge-combinations in the HRG. If a combination yields a cycle, it is either printed or stored and then finally printed in a list of all possible cycles arranged in the order of increasing ring size. A unique numbering of the cycle is used. The computer program is listed and exemplified. Computing times are given.