Arrangements Of Pseudolines Research Articles

A classification of peak-pit maximal Condorcet domains

In this paper, we introduce a weaker notion of separability for set-systems and demonstrate that the class of maximal weakly separated systems precisely corresponds to the class of peak-pit maximal Condorcet domains. Additionally, we present a generalisation of arrangements of pseudolines and establish that the sets of chamber sets from them coincide with maximal weakly separated systems, enabling the construction of all peak-pit maximal Condorcet domains. Furthermore, we reveal that peak-pit maximal Condorcet domains coincide with connected maximal Condorcet domains.

Topological Drawings Meet Classical Theorems from Convex Geometry

In this article we discuss classic theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph \(K_n\), any two edges share at most one point: either a common vertex or a point where they cross. Triangles of simple topological drawings can be viewed as convex sets. This gives a link to convex geometry. As our main result, we present a generalization of Kirchberger’s theorem that is of purely combinatorial nature. It turned out that this classic theorem also applies to “generalized signotopes”—a combinatorial generalization of simple topological drawings, which we introduce and investigate in the course of this article. As indicated by their name they are a generalization of signotopes, a structure studied in the context of encodings for arrangements of pseudolines. We also present a family of simple topological drawings with arbitrarily large Helly number, and a new proof of a topological generalization of Carathéodory’s theorem in the plane and discuss further classic theorems from Convex Geometry in the context of simple topological drawings.

Complete Graph Drawings up to Triangle Mutations

The main result of the paper can be stated in the following way: a complete graph drawing in the sphere, where two edges have at most one common point, which is either a crossing or a common endpoint, and no three edges share a crossing, is determined by the circular ordering of edges at each vertex, or equivalently by the set of pairs of edges that cross, up to homeomorphism and a sequence of triangle mutations. A triangle mutation (or switch, or flip) consists in passing an edge across the intersection of two other edges, assuming the three edges cross each other and the region delimited by the three edges has an empty intersection with the drawing. Equivalently, the result holds for a drawing in the plane assuming the drawing is given with a pair of edges indicating where the unbounded region is. The proof is constructive, based on an inductive algorithm that adds vertices and their incident edges one by one (actually yielding an extra property for the sequence of triangle mutations). This result generalizes Ringel’s theorem on uniform pseudoline arrangements (or rank 3 uniform oriented matroids) to complete graph drawings. We also apply this result to plane projections (or visualizations) of a geometric spatial complete graph, in terms of the rank 4 uniform oriented matroid defined by its vertices. Independently, we prove that, for a complete graph drawing, the set of pairs of edges that cross determine, by first order logic formulas, the circular ordering of edges at each vertex, as well as further information.

Arrangements of Approaching Pseudo-Lines

We consider arrangements of n pseudo-lines in the Euclidean plane where each pseudo-line ell _i is represented by a bi-infinite connected x-monotone curve f_i(x), x in mathbb {R}, such that for any two pseudo-lines ell _i and ell _j with i !<! j, the function x !mapsto ! f_j(x) !-! f_i(x) is monotonically decreasing and surjective (i.e., the pseudo-lines approach each other until they cross, and then move away from each other). We show that such arrangements of approaching pseudo-lines, under some aspects, behave similar to arrangements of lines, while for other aspects, they share the freedom of general pseudo-line arrangements. For the former, we prove:There are arrangements of pseudo-lines that are not realizable with approaching pseudo-lines.Every arrangement of approaching pseudo-lines has a dual generalized configuration of points with an underlying arrangement of approaching pseudo-lines. For the latter, we show:There are 2^{Theta (n^2)} isomorphism classes of arrangements of approaching pseudo-lines (while there are only 2^{Theta (n log n)} isomorphism classes of line arrangements).It can be decided in polynomial time whether an allowable sequence is realizable by an arrangement of approaching pseudo-lines. Furthermore, arrangements of approaching pseudo-lines can be transformed into each other by flipping triangular cells, i.e., they have a connected flip graph, and every bichromatic arrangement of this type contains a bichromatic triangular cell.

Extending Drawings of Graphs to Arrangements of Pseudolines

In the recent study of crossing numbers, drawings of graphs that can be extended to an arrangement of pseudolines (pseudolinear drawings) have played an important role as they are a natural combinatorial extension of rectilinear (or straight-line) drawings. A characterization of the pseudolinear drawings of $K_n$ was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.

Two extensions of the Erdős–Szekeres problem

According to Suk's breakthrough result on the Erdős–Szekeres problem, any point set in general position in the plane, which has no n elements that form the vertex set of a convex n -gon, has at most 2^{n+O\left({n^{2/3}\mathrm log n}\right)} points. We strengthen this theorem in two ways. First, we show that the result generalizes to convexity structures induced by pseudoline arrangements. Second, we improve the error term. A family of n convex bodies in the plane is said to be in convex position if the convex hull of the union of no n-1 of its members contains the remaining one. If any three members are in convex position, we say that the family is in general position . Combining our results with a theorem of Dobbins, Holmsen, and Hubard, we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes Tóth and by Pach and Tóth, respectively. Let c(n) (and c'(n) ) denote the smallest positive integer N with the property that any family of N pairwise disjoint convex bodies in general position (resp., N convex bodies in general position, any pair of which share at most two boundary points) has an n -membered subfamily in convex position. We show that c(n)\leq c'(n)\leq 2^{n+O\left(\sqrt{n\mathrm log n}\right)} .

Two-Dimensional Partial Cubes

We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from being solved even for general set systems of VC-dimension 2. Moreover, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.

An Ongoing Project to Improve the Rectilinear and the Pseudolinear Crossing Constants

A drawing of a graph in the plane is pseudolinear if the edges of the drawing can be extended to doubly-infinite curves that form an arrangement of pseudolines, that is, any pair of these curves crosses precisely once. A special case is rectilinear drawings where the edges of the graph are drawn as straight line segments. The rectilinear (pseudolinear) crossing number of a graph is the minimum number of pairs of edges of the graph that cross in any of its rectilinear (pseudolinear) drawings. In this paper we describe an ongoing project to continuously obtain better asymptotic upper bounds on the rectilinear and pseudolinear crossing number of the complete graph $K_n$.

Subquadratic encodings for point configurations

For many algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of a realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows an efficient query of the orientation of any triple: the encoding uses \( O(n^2) \) bits and an orientation query takes \(O(\log n)\) time in the word-RAM model with word size \(w \geq \log n\). This encoding is space-optimal for abstract order types. We show how to shorten the encoding to \(O(n^2 {(\log\log n)}^2 / \log n)\) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain \(O(\log n/\log\log n)\) query time at the expense of a negligibly larger space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension.

String cone and superpotential combinatorics for flag and Schubert varieties in type A

We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary.We prove that the two cones are unimodularly equivalent. As a corollary of our combinatorial result we realize Caldero's toric degenerations of Schubert varieties as GHKK-degeneration using cluster theory.

The unavoidable arrangements of pseudocircles

A fact closely related to the classical Erdos-Szekeres theorem is that cyclic arrangements are the only unavoidable simple arrangements of pseudolines: for each fixed m ≥ 1, every sufficiently large simple arrangement of pseudolines has a cyclic subarrangement of size m. In the same spirit, we show that there are three unavoidable arrangements of pseudocircles.

Aligned Drawings of Planar Graphs

Let $G$ be a graph that is topologically embedded in the plane and let $\mathcal A$ be an arrangement of pseudolines intersecting the drawing of $G$. An aligned drawing of $G$ and $\mathcal A$ is a planar polyline drawing $\Gamma$ of $G$ with an arrangement $A$ of lines so that $\Gamma$ and $A$ are homeomorphic to $G$ and $\mathcal A$. We show that if $\mathcal A$ is stretchable and every edge $e$ either entirely lies on a pseudoline or it has at most one intersection with $\mathcal A$, then $G$ and $\mathcal A$ have a straight-line aligned drawing. In order to prove this result, we strengthen a result of Da Lozzo et al. [Da Lozzo et al. GD 2016], and prove that a planar graph $G$ and a single pseudoline $\mathcal L$ have an aligned drawing with a prescribed convex drawing of the outer face. We also study the less restrictive version of the alignment problem with respect to one line, where only a set of vertices is given and we need to determine whether they can be collinear. We show that the problem is $\mathcal{NP}$-complete but fixed-parameter tractable.

On the Sylvester–Gallai and the orchard problem for pseudoline arrangements

We study a non-trivial extreme case of the orchard problem for $12$ pseudolines and we provide a complete classification of pseudoline arrangements having $19$ triple points and $9$ double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of B\"or\"oczky. Since Melchior's inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Gr\"unbaum's problems. We formulate some open problems which involve our new examples of line arrangements.

Levi's Lemma, pseudolinear drawings of , and empty triangles

AbstractThere are three main thrusts to this article: a new proof of Levi's Enlargement Lemma for pseudoline arrangements in the real projective plane; a new characterization of pseudolinear drawings of the complete graph; and proofs that pseudolinear and convex drawings of have O and O(n2), respectively, empty triangles. All the arguments are elementary, algorithmic, and self‐contained.

On finding convex cuts in general, bipartite and plane graphs

The general notion of convexity also applies to a graph G=(V,E). A subset Vc of V is called a convex set if all shortest paths with end vertices in Vc remain within Vc. A convex cut of G is a partition (V1,V2) of V such that V1 and V2 are convex sets (halfspaces). Finding convex cuts is NP-hard for general graphs.In this paper we first characterize the convex cuts of a connected bipartite graph G′ in terms of the Djoković relation, a reflexive and symmetric relation on the edges of a graph that is based on shortest paths between the edges' end vertices. Specifically, we show that a cut of G′ is convex if and only if the Djoković relation holds for any pair of edges in its cut-set. As a consequence, all convex cuts of G′={V′,E′} can be found in O(|V′||E′|).We then characterize the convex cuts of a general connected graph G using the Djoković–Winkler relation θ and another relation τ on the edges of G. Based on this general characterization, we show how one can find all convex cuts of G in polynomial time. The key parts here are (i) describing the Djoković–Winkler relation on the edges of G in terms of the Djoković relation θ′ on the bipartite graph obtained from G by subdividing each edge of G into two edges, (ii) studying the interplay of θ′ and τ on plane curves representing convex cuts, and (iii) a running time analysis of our algorithm for finding the convex cuts.Our method for characterizing and finding convex cuts of a connected plane graph G is motivated by the concept of alternating cuts and conditions on the latter to be convex. In the last part of this paper we represent alternating cuts as plane curves and focus on their intersection pattern. If the plane curves form an arrangement of pseudolines, G is scale embedded in its dual, and any edge of G is contained in the cut-set of a convex cut of G.

Combinatorial configurations, quasiline arrangements, and systems of curves on surfaces

It is well known that not every combinatorial configuration admits a geometric realization with points and lines. Moreover, some of them do not even admit realizations with pseudoline arrangements, i.e., they are not topological. In this paper we generalize the concept of topological configurations to a more general one (in a least possible way) such that every combinatorial configuration is realizable in this way. In particular, we generalize the notion of a pseudoline arrangement to the notion of a quasiline arrangement by relaxing the condition that two pseudolines meet exactly once. We also generalize well-known tools from pseudoline arrangements such as sweeps and wiring diagrams. We introduce monotone quasiline arrangements as a subfamily of quasiline arrangements that can be represented with generalized wiring diagrams. We show that every incidence structure (and therefore also every combinatorial configuration) can be realized as a monotone quasiline arrangement in the real projective plane. A quasiline arrangement with selected vertices belonging to an incidence structure can be viewed as a map on a closed surface. Such a map can be used to distinguish between two “distinct” realizations of an incidence structure as a quasiline arrangement.

Planar Matchings for Weighted Straight Skeletons

We introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.

Chambers of wiring diagrams

Given a wiring diagram (pseudo-line arrangement) of a permutation w∈Sn, the chambers can be labeled with subsets of [n] and they are called chamber sets. In this short note, we show that two wiring diagrams (of same w), can be mutated from one to another via the usual moves (coming from nil and braid relations), while freezing the chamber sets they have in common.

On the Pseudolinear Crossing Number

A drawing of a graph is {\em pseudolinear} if there is a pseudoline arrangement such that each pseudoline contains exactly one edge of the drawing. The {\em pseudolinear crossing number} of a graph $G$ is the minimum number of pairwise crossings of edges in a pseudolinear drawing of $G$. We establish several facts on the pseudolinear crossing number, including its computational complexity and its relationship to the usual crossing number and to the rectilinear crossing number. This investigation was motivated by open questions and issues raised by Marcus Schaefer in his comprehensive survey of the many variants of the crossing number of a graph.

$$\hbox {H}_\infty $$ H ∞ and $$\upmu $$ μ synthesis control of virtual structure satellites formation flying

Recent developments in the space industry and the great tendency to define efficient, precise and low cost missions are the main reasons for the growing interest in satellites formation flight. Regarding the aforementioned, the study of multiple satellites control methods is the subject of the present paper. In this paper, two linear robust control strategies are applied to position control system of certain satellites in the virtual structure formation flight. The proposed controllers are designed considering different parametric uncertainties such as semimajor axis and eccentricity changes as well as some disturbances like second term of gravitational acceleration function (\(j_{2}\)) and drag acceleration which are resulted in asymmetrical shape of the Earth and low density atmosphere in high altitude. In the presence of all parametric and un-modeled dynamic uncertainties, the robust stability and performance of the nonlinear system are guaranteed and good tracking, disturbance and noise rejection characteristics are achieved. Computer simulations show that these approaches have very promising performance when position controllers are applied with PWPF modulators while they are working in their pseudo linear range.