Arrangements Of Pseudolines Research Articles

A $$t_k$$ t k Inequality for Arrangements of Pseudolines

Consider arrangements of n pseudolines in the real projective plane. Let $$t_k$$tk denote the number of intersection points where exactly k pseudolines are incident. We present a new combinatorial inequality: $$\begin{aligned} t_2+1.5t_3\ge 8+\sum _{k\ge 4}(2k-7.5)t_k, \end{aligned}$$t2+1.5t3?8+?k?4(2k-7.5)tk,which holds if no more than $$n-3$$n-3 pseudolines intersect at one point. It looks similar but is unrelated to the Hirzebruch inequality for arrangements of complex lines in the complex projective plane. Based on this linear inequality, we construct lower bounds for the number of regions via n and the maximal number of (pseudo)lines passing through one point.

A Pseudoline Counterexample to the Strong Dirac Conjecture

We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of $n$ pseudolines has no member incident to more than $4n/9$ points of intersection. This shows the "Strong Dirac" conjecture to be false for pseudolines.We also raise a number of open problems relating to possible differences between the structure of incidences between points and lines versus the structure of incidences between points and pseudolines.

Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.

Simplicial arrangements revisited

In connection with the publication of the catalogue [1] of known simplicial arrangements of lines in the real projective plane, and the note [2] about small simplicial arrangements of pseudolines, several developments of these topics deserve to be mentioned. The present paper puts these results in perspective, and provides appropriate illustrations. [1] B. Grünbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Mathematica Contemporanea 2 (2009), 1–25. [2] B. Grünbaum, Small unstretchable simplicial arrangements of pseudolines, Geombinatorics 18 (2009), 153–160.

LR Characterization of Chirotopes of Finite Planar Families of Pairwise Disjoint Convex bodies

We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the set of 3-subsets of J is a chirotope of finite planar families of pairwise disjoint convex bodies. Our main tool is the polarity map, i.e., the map that assigns to a convex body the set of lines missing its interior, from which we derive the key notion of arrangements of double pseudolines, introduced for the first time in this paper.

Crossing by lines all edges of a line arrangement

Let L be a family of n blue lines in the real projective plane. Suppose that R is a collection of m red lines, different from the blue lines, and that every edge in the arrangement A(L) is crossed by a line in R. We show that m≥n−13.5. Our result is more general, and applies to pseudo-line arrangements A(L), and even weaker assumptions are required for R. Our result is motivated by the famous conjecture of Dirac about the existence of a line with many intersection points on it in any arrangement of n nonconcurrent lines in the plane. We draw a possible relation between the two problems.

Simplicial Arrangements with up to 27 Lines

We compute all isomorphism classes of simplicial arrangements in the real projective plane with up to 27 lines. It turns out that Grunbaum’s catalogue is complete up to 27 lines except for four new arrangements with 22, 23, 24, 25 lines, respectively. As a byproduct we classify simplicial arrangements of pseudolines with up to 27 lines. In particular, we disprove Grunbaum’s conjecture about unstretchable arrangements with at most 16 lines, and prove the conjecture that any simplicial arrangement with at most 14 pseudolines is stretchable.

Multitriangulations, Pseudotriangulations and Primitive Sorting Networks

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.

The brick polytope of a sorting network

The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud & Pocchiola in their study of flip graphs on pseudoline arrangements with contacts supported by a given sorting network.In this paper, we construct the brick polytope of a sorting network, obtained as the convex hull of the brick vectors associated to each pseudoline arrangement supported by the network. We combinatorially characterize the vertices of this polytope, describe its faces, and decompose it as a Minkowski sum of matroid polytopes.Our brick polytopes include Hohlweg & Lange’s many realizations of the associahedron, which arise as brick polytopes for certain well-chosen sorting networks. We furthermore discuss the brick polytopes of sorting networks supporting pseudoline arrangements which correspond to multitriangulations of convex polygons: our polytopes only realize subgraphs of the flip graphs on multitriangulations and they cannot appear as projections of a hypothetical multiassociahedron.

Coding and Counting Arrangements of Pseudolines

Arrangements of lines and pseudolines are important and appealing objects for research in discrete and computational geometry. We show that there are at most $2^{0.657\> n^{2}}$ simple arrangements of n pseudolines in the plane. This improves on previous work by Knuth who proved an upper bound of $3^{\binom{n}{2}} \cong 2^{0.792\> n^{2}}$ in 1992 and the first author, who obtained $2^{0.697\> n^{2}}$ in 1997. The argument uses surprisingly little geometry. The main ingredient is a lemma that was already central to the argument given by Knuth.

The brick polytope of a sorting network

The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of pseudoline arrangements with contacts supported by a given network. In this paper, we construct the "brick polytope'' of a network, obtained as the convex hull of the "brick vectors'' associated to each pseudoline arrangement supported by the network. We characterize its vertices, describe its faces, and decompose it as a Minkowski sum of simpler polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes of certain well-chosen networks. L'associaèdre est un polytope dont le graphe est le graphe des flips sur les triangulations d'un polygone convexe. Les pseudotriangulations et les multitriangulations généralisent les triangulations dans deux directions différentes, qui ont été unifiées par Pilaud et Pocchiola au travers de leur étude des arrangements de pseudodroites avec contacts couvrant un support donné. Nous construisons ici le "polytope de briques'' d'un support, obtenu comme l'enveloppe convexe des "vecteurs de briques'' associés à chaque arrangement de pseudodroites couvrant ce support. Nous caractérisons les sommets de ce polytope, décrivons ses faces et le décomposons en somme de Minkowski de polytopes élémentaires. Notre construction contient toutes les réalisations de l'associaèdre d'Hohlweg et Lange, qui apparaissent comme polytopes de briques de certains supports bien choisis.

On the Number of Simple Arrangements of Five Double Pseudolines

We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces of one-extensions of double pseudoline arrangements, proved in this paper. Counting results derived from an implementation of our algorithm are also reported.

Segment Orders

We study two kinds of segment orders, using definitions first proposed by Farhad Shahrokhi. Although the two kinds of segment orders appear to be quite different, we prove several results suggesting that the are very much the same. For example, we show that the following classes belong to both kinds of segment orders: (1) all posets having dimension at most 3; (2) interval orders; and for n≥3, the standard example S n of an n-dimensional poset, all 1-element and (n−1)-element subsets of {1,2,…,n}, partially ordered by inclusion. Moreover, we also show that, for each d≥4, almost all posets having dimension d belong to neither kind of segment orders. Motivated by these observations, it is natural to ask whether the two kinds of segment orders are distinct. This problem is apparently very difficult, and we have not been able to resolve it completely. The principal thrust of this paper is the development of techniques and results concerning the properties that must hold, should the two kinds of segment orders prove to be the same. We also derive equivalent statements, one version of which is a stretchability question involving certain sets of pseudoline arrangements. We conclude by proving several facts about continuous universal functions that would transfer segment orders of the first kind into segments orders of the second kind.

Allowable double-permutation sequences and double pseudoline arrangements

Allowable double-permutation sequences and double pseudoline arrangements

Symmetric Simplicial Pseudoline Arrangements

A simplicial arrangement of pseudolines is a collection of topological lines in the projective plane where each region that is formed is triangular. This paper refines and develops David Eppstein's notion of a kaleidoscope construction for symmetric pseudoline arrangements to construct and analyze several infinite families of simplicial pseudoline arrangements with high degrees of geometric symmetry. In particular, all simplicial pseudoline arrangements with the symmetries of a regular $k$-gon and three symmetry classes of pseudolines, consisting of the mirrors of the $k$-gon and two other symmetry classes, plus sometimes the line at infinity, are classified, and other interesting families (with more symmetry classes of pseudolines) are discussed.

The combinatorial encoding of disjoint convex sets in the plane

We introduce a new combinatorial object, the double-permutation sequence, and use it to encode a family of mutually disjoint compact convex sets in the plane in a way that captures many of its combinatorial properties. We use this encoding to give a new proof of the Edelsbrunner-Sharir theorem that a collection of n compact convex sets in the plane cannot be met by straight lines in more than 2n-2 combinatorially distinct ways. The encoding generalizes the authors' encoding of point configurations by "allowable sequences" of permutations. Since it applies as well to a collection of compact connected sets with a specified pseudoline arrangement $$ \mathcal{A} $$ of separators and double tangents, the result extends the Edelsbrunner-Sharir theorem to the case of geometric permutations induced by pseudoline transversals compatible with $$ \mathcal{A} $$ .

Acyclic sets of linear orders via the Bruhat orders

We describe Abello’s acyclic sets of linear orders [SIAM J Discr Math 4(1):1–16, 1991] as the permutations visited by commuting equivalence classes of maximal reduced decompositions. This allows us to strengthen Abello’s structural result: we show that acyclic sets arising from this construction are distributive sublattices of the weak Bruhat order. This, in turn, shows that Abello’s acyclic sets are, in fact, the same as Chameni-Nembua’s distributive covering sublattices (S.T.D.C s). Fishburn’s alternating scheme is shown to be a special case of the Abello/Chameni-Nembua acyclic sets. Any acyclic set that arises in this way can be represented by an arrangement of pseudolines, and we use this representation to derive a simple closed form for the cardinality of the alternating scheme. The higher Bruhat orders prove to be a natural mathematical framework for this approach to the acyclic sets problem.

Oriented matroids and complete-graph embeddings on surfaces

We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable and when one color class of cells consists of triangles only. Precisely for triangular pseudoline arrangements, one element of the image pair of f is a triangular complete-graph embedding on a surface. We obtain all triangular complete-graph embeddings on surfaces this way, when we extend the definition of triangular complete pseudoline arrangements in a natural way to that of triangular curve arrangements on surfaces in which each pair of curves has a point in common where they cross. Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces. Furthermore, we establish the relationship between 2-colorable curve arrangements and Petrie dual maps. A data structure, called intersection pattern is provided for the study of curve arrangements on surfaces. Finally we show that an orientable surface of genus g admits a complete curve arrangement with at most 2 g + 1 curves in contrast to the non-orientable surface where the number of curves is not bounded.

Cubic Partial Cubes from Simplicial Arrangements

We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.

Coloring copoints of a planar point set

To a set of n points in the plane, one can associate a graph that has less than n 2 vertices and has the property that k-cliques in the graph correspond vertex sets of convex k-gons in the point set. We prove an upper bound of 2 k - 1 on the size of a planar point set for which the graph has chromatic number k, matching the bound conjectured by Szekeres for the clique number. Constructions of Erdős and Szekeres are shown to yield graphs that have very low chromatic number. The constructions are carried out in the context of pseudoline arrangements.