Edge (geometry)

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a polygon side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces (or polyhedron sides) meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. Relation to edges in graphs In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz's the

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Intersection Patterns of Edges in Topological Graphs

Radoslav Fulek

This thesis is devoted to crossing patterns of edges in topological graphs. We consider the following four problems: A thrackle is a graph drawn in the plane such that every pair of edges meet exactly once: either at a common endpoint or in a proper crossing. Conway's Thrackle Conjecture says that a thrackle cannot have more edges than vertices. By a computational approach we improve the previously known upper bound of 1.5n on the maximal number of edges in a thrackle with n vertices to 1.428n. Moreover, our method yields an algorithm with a finite running time that for any ε > 0 either verifies the upper bound of (1 + ε)n on the maximum number of edges in a thrackle or disproves the conjecture. It is not hard to see that any simple graph admits a poly-line drawing in the plane such that each edge is represented by a polygonal curve with at most three bends, and each edge crossings realizes a prescribed angle α. We show that if we restrict the number of bends per edge to two and allow edges to cross in k different angles, a graph on n vertices admitting such a drawing can have at most O(nk2) edges. This generalizes a previous result of Arikushi et al., in which the authors treated a special case of our problem, where k = 1 and the prescribed angle has 90 degrees. The classical result known as Hanani-Tutte Theorem states that a graph is planar if and only if it admits a drawing in the plane in which each pair of non-adjacent edges crosses an even number of times. We prove the following monotone variant of this result, conjectured by J.Pach and G.Tóth. If G has an x-monotone drawing in which every pair of independent edges crosses evenly, then G has an x-monotone embedding (i.e. a drawing without crossings) with the same vertex locations. We show several interesting algorithmic consequences of this result. In a drawing of a graph, two edges form an odd pair if they cross each other an odd number of times. A pair of edges is independent (or non-adjacent) if they share no endpoint. For a graph G we let ocr(G) be the smallest number of odd pairs in a drawing of G and let iocr(G) be the smallest number of independent odd pairs in a drawing of G. We construct a graph G with iocr(G) < ocr(G), answering a question by Székely, and –for the first time– giving evidence that crossings of adjacent edges may not always be trivial to eliminate.

EPFL2012

Thrackles: An improved upper bound

Radoslav Fulek, János Pach

A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is 3/2(n-1), and that this bound is best possible for infinitely many values of n. (C) 2019 Published by Elsevier B.V.

ELSEVIER SCIENCE BV2019

A Computational Approach to Conway's Thrackle Conjecture

Radoslav Fulek, János Pach

A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have. According to a 40 years old conjecture of Conway, t(n) = n for every n >= 3. For any epsilon > 0, we give an algorithm terminating in e(O)((1/epsilon(2))ln(1/epsilon)) steps to decide whether t(n) = 3. Using this approach, we improve the best known upper bound, t(n) 3.(n 1), due to Cairns and Nikolayevsky, to 167/117n< 10428n. (C) 2011 Elsevier B.V. All rights reserved.

Springer2010

Polyhedron

In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is a polyhedron th

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest b

Polygon

In geometry, a polygon (ˈpɒlɪɡɒn) is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its edges or sides. The p

MSE-704: 3D Electron Microscopy and FIB-Nanotomography

The principles of 3D surface (SEM) reconstruction and its limitations will be explained. 3D volume reconstruction and tomography methods by electron microscopy (SEM/FIB and TEM) will be explained and compared with x-ray tomography.