Discrete & Computational Geometry

https://link.springer.com/journal/454

List of Papers (Total 555)

Poisson–Delaunay Mosaics of Order k

The order-k Voronoi tessellation of a locally finite set \(X \subseteq {\mathbb {R}}^n\) decomposes \({\mathbb {R}}^n\) into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a...

Barcodes of Towers and a Streaming Algorithm for Persistent Homology

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG, 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger...

Coloring Curves that Cross a Fixed Curve

We prove that for every integer \(t\geqslant 1\), the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is \(\chi \)-bounded. This is essentially the strongest \(\chi \)-boundedness result one can get for those kind of graph classes. As a corollary, we prove that for any fixed integers \(k\geqslant 2\) and...

A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations

Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a...

The Rigidity of Infinite Graphs

A rigidity theory is developed for countably infinite simple graphs in \({\mathbb {R}}^d\). Generalisations are obtained for the Laman combinatorial characterisation of generic infinitesimal rigidity for finite graphs in \({\mathbb {R}}^2\) and Tay’s multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in \({\mathbb {R}}^d\). Analogous...

Lines in Euclidean Ramsey Theory

Let \(\ell _m\) be a sequence of m points on a line with consecutive points of distance one. For every natural number n, we prove the existence of a red/blue-coloring of \({\mathbb {E}}^n\) containing no red copy of \(\ell _2\) and no blue copy of \(\ell _m\) for any \(m \ge 2^{cn}\). This is best possible up to the constant c in the exponent. It also answers a question of Erdős...

A Zonotope and a Product of Two Simplices with Disconnected Flip Graphs

We give an example of a three-dimensional zonotope whose set of tight zonotopal tilings is not connected by flips. Using this, we show that the set of triangulations of \(\Delta ^4 \times \Delta ^n\) is not connected by flips for large n. Our proof makes use of a non-explicit probabilistic construction.

Zigzag Structure of Thin Chamber Complexes

Zigzags in thin chamber complexes are investigated, in particular, all zigzags in the Coxeter complexes are described. Using this description, we show that the lengths of all zigzags in the simplex \(\alpha _{n}\), the cross-polytope \(\beta _{n}\), the 24-cell, the icosahedron and the 600-cell are equal to the Coxeter numbers of \(\mathsf{A}_{n}\), \(\mathsf{B}_{n}=\mathsf{C}_{n...

Polynomial-Sized Topological Approximations Using the Permutahedron

Classical methods to model topological properties of point clouds, such as the Vietoris–Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in \(\mathbb {R}^d\), we obtain a O(d)-approximation whose k-skeleton has size...

Shellability and Sphericity of Finite Quasi-arc Complexes

Quasi-triangulations of a non-orientable surface were introduced by Dupont and Palesi (J Algebr Comb 42(2):429–472, 2015). The quasi-arc complex provides an intricate description of the combinatorics of these quasi-triangulations. This is the simplicial complex where vertices correspond to quasi-arcs and maximal simplices to quasi-triangulations. We prove that when the quasi-arc...

String-Node Nets and Meshes

New classes of infinite bond-node structures are introduced, namely string-node nets and meshes, a mesh being a string-node net for which the nodes are dense in the strings. Various construction schemes are given including the minimal extension of a (countable) line segment net by a countable scaling group. A linear mesh has strings that are straight lines and nodes given by the...

Intrinsic Volumes of Polyhedral Cones: A Combinatorial Perspective

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the general Steiner formula, the conic analogues of the Brianchon–Gram–Euler and the Gauss...

Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings

An embedding \(i \mapsto p_i\in \mathbb {R}^d\) of the vertices of a graph G is called universally completable if the following holds: For any other embedding \(i\mapsto q_i~\in \mathbb {R}^{k}\) satisfying \(q_i^{T}q_j = p_i^{T}p_j\) for \(i = j\) and i adjacent to j, there exists an isometry mapping \(q_i\) to \(p_i\) for all \( i\in V(G)\). The notion of universal...

On Sets Defining Few Ordinary Circles

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least \(n^2/4 - O(n)\) ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this...

Regular Maps of High Density

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a simple graph with ratio between vertex-degree and number of vertices strictly exceeding \(\frac{1}{2}\). We conclude that all regular maps of this type belong to a family of maps...

On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman

In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii \(r_1\), \(\ldots \), \(r_n\) in the plane, it is always possible to cover them by a disk of radius \(R = \sum r_i\), provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially...

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fréchet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original \(O(n^2 \log n)\) algorithm by Alt and Godau for computing the Fréchet...

Hyperbolicity vs. Amenability for Planar Graphs

The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.

On the Beer Index of Convexity and Its Variants

Let S be a subset of \(\mathbb {R}^d\) with finite positive Lebesgue measure. The Beer index of convexity \({\text {b}}(S)\) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio \({\text {c}}(S)\) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We...

Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

We give a concise definition of mitered offset surfaces for nonconvex polytopes in \({\mathbbm {R}}^3\), along with a proof of existence and a discussion of basic properties. These results imply the existence of 3D straight skeletons for general nonconvex polytopes. The geometric, topological, and algorithmic features of such skeletons are investigated, including a classification...

Computing the Fréchet Distance with a Retractable Leash

All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet...

On Computability and Triviality of Well Groups

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map \(f:K\rightarrow {\mathbb {R}}^n\) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within \(L_\infty \) distance r from f for a given \(r>0\). The main drawback of the approach is...