Discrete & Computational Geometry最新文献

We study the k-th nearest neighbor distance function from a finite point-set in $R^d$ . We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-k Delaunay mosaics, and random k-fold coverage.

The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in $R^d$ and any radius satisfies ${\beta }_p=O\left(N^m\right)$ , with $m=min\{p+1,\lceil d/2\rceil \}$ . We construct sets in even and odd dimensions that prove this upper bound is asymptotically tight. For example, we describe a set of $N=2(n+1)$ points in $R^3$ and two radii such that the first Betti number of the Čech complex at one radius is ${(n+1)}^2-1$ , and the second Betti number of the Čech complex at the other radius is $n^2$ .

Given a locally finite set $A\subseteq R^d$ and a coloring $\chi :A\to \{0,1,\dots ,s\}$ , we introduce the chromatic Delaunay mosaic of $\chi$ , which is a Delaunay mosaic in $R^{d+s}$ that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with $n=\#A$ , and the coloring is random, then the chromatic Delaunay mosaic has $O\left(n^{\lceil d/2\rceil }\right)$ cells in expectation. In contrast, for Delone sets and Poisson point processes in $R^d$ , the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in $R^2$ all colorings of a well spread set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.

A famous theorem in polytope theory states that the combinatorial type of a simplicial polytope is completely determined by its facet-ridge graph. This celebrated result was proven by Blind and Mani (Aequationes Math 34(2-3):287-297, 1987, 10.1007/BF01830678), via a non-constructive proof using topological tools from homology theory. An elegant constructive proof was given by Kalai shortly after. In their original paper, Blind and Mani asked whether their result can be extended to simplicial spheres, and a positive answer to their question was conjectured by Kalai (2009, https://gilkalai.wordpress.com/2009/01/16/telling-a-simple-polytope-from-its-graph/). In this paper, we show that Kalai's conjecture holds in the particular case of Knutson and Miller's spherical subword complexes. This family of simplicial spheres arises in the context of Coxeter groups, and is conjectured to be polytopal. In contrast, not all manifolds are reconstructible. We show two explicit examples, namely the torus and the projective plane.

Given a positive integer d, the class d-DIR is defined as all those intersection graphs formed from a finite collection of line segments in $R^2$ having at most d slopes. Since each slope induces an interval graph, it easily follows for every G in d-DIR with clique number at most $\omega$ that the chromatic number $\chi \left(G\right)$ of G is at most $d\omega$ . We show for every even value of $\omega$ how to construct a graph in d-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvořák and Noorizadeh. Furthermore, we show that the $\chi$ -binding function of d-DIR is $\omega \mapsto d\omega$ for $\omega$ even and $\omega \mapsto d(\omega -1)+1$ for $\omega$ odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case $d=2$ .

Mohar recently adapted the classical game of Cops and Robber from graphs to metric spaces, thereby unifying previously studied pursuit-evasion games. He conjectured that finitely many cops can win on any compact geodesic metric space, and that their number can be upper-bounded in terms of the ranks of the homology groups when the space is a simplicial pseudo-manifold. We disprove these conjectures by constructing a metric on $S^3$ with infinite cop number. More problems are raised than settled.

We prove that for any ${\ell }_p$ -norm in the plane with $1<p<\infty$ and for every infinite $M\subset R^2$ , there exists a two-colouring of the plane such that no isometric copy of $M$ is monochromatic. On the contrary, we show that for every polygonal norm (that is, the unit ball is a polygon) in the plane, there exists an infinite $M\subset R^2$ such that for every two-colouring of the plane there exists a monochromatic isometric copy of $M$ .

A bar-joint framework (G, p) is the combination of a finite simple graph $G=(V,E)$ and a placement $p:V\to R^d$ . The framework is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in $R^3$ restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.

We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.

The order type of a point set in (mathbb {R}^d) maps each ((d{+}1))-tuple of points to its orientation (e.g., clockwise or counterclockwise in (mathbb {R}^2)). Two point sets X and Y have the same order type if there exists a bijection f from X to Y for which every ((d{+}1))-tuple ((a_1,a_2,ldots ,a_{d+1})) of X and the corresponding tuple ((f(a_1),f(a_2),ldots ,f(a_{d+1}))) in Y have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an (O(n^d)) algorithm for this task, thereby improving upon the (O(n^{lfloor {3d/2}rfloor })) algorithm of Goodman and Pollack (SIAM J. Comput. 12(3):484–507, 1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.