We investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod...
Let $$n \in \mathbb {N}$$ and $$k \in \mathbb {N}_0$$ . Given a set P of n points in the plane, a pair $$\{p,q\}$$ of points in P is called k-deep, if there are at least k points from P strictly on each side of the line spanned by p and q. A k-deep clique is a subset of P with all its pairs k-deep. We show that if P is in general position (i.e., no three points on a line), there...
We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and translations. The number of rectangles is the rank of the Dehn invariant of the polygon. The same method can also be used to dissect an axis...
Let $$\textbf{p}$$ be a configuration of n points in $$\mathbb R^d$$ for some n and some $$d \ge 2$$ . Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply...
A grounded L-graph is the intersection graph of a collection of “L” shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number $$\omega $$ has chromatic number at most $$17\omega ^4$$ . This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is...
We introduce a new algorithm computing the characteristic polynomials of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its characteristic polynomial. We showcase our julia implementation, based on OSCAR, on examples coming from hyperplane arrangements with applications to...
Orthogonal drawings, i.e., embeddings of graphs into grids, are a classic topic in Graph Drawing. Often the goal is to find a drawing that minimizes the number of bends on the edges. A key ingredient for bend minimization algorithms is the existence of an orthogonal representation that allows to describe such drawings purely combinatorially by only listing the angles between the...
We consider the Voronoi diagram of points in the real plane when the distance between two points a and b is given by $$L_p(a-b)$$ where $$L_p((x,y)) = (|x|^p+|y|^p)^{1/p}.$$ We prove that the Voronoi diagram has a limit as p converges to zero from above or from below: it is the diagram that corresponds to the distance function $$L_*((x,y)) = |xy|$$ . In this diagram, the bisector...
Fix $$p\in [1,\infty )$$ , $$K\in (0,\infty )$$ , and a probability measure $$\mu $$ . We prove that for every $$n\in \mathbb {N}$$ , $$\varepsilon \in (0,1)$$ , and $$x_1,\ldots ,x_n\in L_p(\mu )$$ with $$\big \Vert \max _{i\in \{1,\ldots ,n\}} |x_i| \big \Vert _{L_p(\mu )} \le K$$ , there exist $$d\le \frac{32e^2 (2K)^{2p}\log n}{\varepsilon ^2}$$ and vectors $$y_1,\ldots , y_n...
We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $${\mathbb {R}}^3$$ . We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains $$K_5$$ , $$K_{5,81}$$ , or any...
Consider a finite collection of affine hyperplanes in $$\mathbb R^d$$ . The hyperplanes dissect $$\mathbb R^d$$ into finitely many polyhedral chambers. For a point $$x\in \mathbb R^d$$ and a chamber P the metric projection of x onto P is the unique point $$y\in P$$ minimizing the Euclidean distance to x. The metric projection is contained in the relative interior of a uniquely...
Given positive integers $$k\le d$$ and a finite field $$\mathbb {F}$$ , a set $$S\subset \mathbb {F}^{d}$$ is (k, c)-subspace evasive if every k-dimensional affine subspace contains at most c elements of S. By a simple averaging argument, the maximum size of a (k, c)-subspace evasive set is at most $$c |\mathbb {F}|^{d-k}$$ . When k and d are fixed, and c is sufficiently large...
We describe a canonical compactification of a polyhedral complex in Euclidean space. When the recession cones of the polyhedral complex form a fan, the compactified polyhedral complex is a subspace of a tropical toric variety. In this case, the procedure is analogous to the tropical compactifications of subvarieties of tori. We give an analysis of the combinatorial structure of...
We study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in recent years under the name of generalized permutahedra. We provide an irredundant description of the deformation cone of the graphical zonotope associated to a graph G, consisting of independent...
We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices $$K_4$$ . For comparison we consider a competing topological class, also with a quotient...
We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more...
The notion of generalized rank in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. However, its efficient computation has not yet been studied in the literature. We show that the generalized rank over a finite interval I of a $$\textbf{Z}^2$$ -indexed persistence...
We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any $$\lambda \ge 1$$ , the critical covering area $$A^*(\lambda )$$ is the minimum value for which any set of disks with total area at least $$A^*(\lambda )$$ can cover a rectangle of dimensions $$\lambda \times...
Planar point sets with many triple lines (which contain at least three distinct points of the set) have been studied for 180 years, started with Jackson and followed by Sylvester. Green and Tao (Discret Comput Geom 50(2):409–468, 2013) have shown that the maximum possible number of triple lines for an n-element set is $$\lfloor n\hspace{0.33325pt}(n - 3)/6\rfloor + 1$$ . Here we...
For a finite set $$S\subset {\mathbb {R}}^2$$ , a map $$\varphi :S\rightarrow {\mathbb {R}}^2$$ is orientation preserving if for every non-collinear triple $$u,v,w\in S$$ the orientation of the triangle u, v, w is the same as that of the triangle $$\varphi (u),\varphi (v),\varphi (w)$$ . Assuming that $$\varphi :G_n\rightarrow {\mathbb {R}}^2$$ is an orientation preserving map...
A d-dimensional lattice polytope P is Gorenstein if it has a multiple rP that is a reflexive polytope up to translation by a lattice vector. The difference $$d+1-r$$ is called the degree of P. We show that a Gorenstein polytope is a lattice pyramid if its dimension is at least three times its degree. This was previously conjectured by Batyrev and Juny. We also present a refined...
A (convex) polytope $$P\subset \mathbb {R}^d$$ and its edge-graph $$G_P$$ can have very distinct symmetry properties, in that the edge-graph can be much more symmetric than the polytope. In this article we ask whether this can be “rectified” by coloring the vertices and edges of $$G_P$$ , that is, whether we can find such a coloring so that the combinatorial symmetry group of...
We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class $${ \forall \exists _{<}\mathbb {R}} $$ . This implies that the problem is NP-, co-NP-, $$\exists \mathbb {R} $$ -, and...
A tournament is an orientation of a graph. Each edge is a match, directed towards the winner. The score sequence lists the number of wins by each team. In this article, by interpreting score sequences geometrically, we generalize and extend classical theorems of Landau (Bull. Math. Biophys. 15, 143–148 (1953)) and Moon (Pac. J. Math. 13, 1343–1345 (1963)), via the theory of...
We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, together with an operation $$\boxplus $$ that leads to a method for obtaining images, kernels and cokernels of tame persistence morphisms. Our focus is on developing efficient methods for...