Coloring half-planes and bottomless rectangles

We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover we give efficient coloring algorithms

Blocking the k-Holes of Point Sets in the Plane

Let P be a set of n points in the plane in general position. A subset H of P consisting of k elements that are the vertices of a convex polygon is called a k-hole of P, if there is no element of P in the interior of its convex hull. A set B of points in the plane blocks the k-holes of P if any k-hole of P contains at least one element of B in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of k-hole blocking sets, with emphasis in the case k=5

Survey on Decomposition of Multiple Coverings

The study of multiple coverings was initiated by Davenport and L. Fejes Tóth more than 50 years ago. In 1980 and 1986, the rst named author published the rst papers about decompos-ability of multiple coverings. It was discovered much later that, besides its theoretical interest, this area has practical applications to sensor networks. Now there is a lot of activity in this eld with several breakthrough results, although, many basic questions are still unsolved. In this survey, we outline the most important results, methods, and questions. 1 Cover-decomposability and the sensor cover problem Let P = { Pi | i ∈ I} be a collection of sets in Rd. We say that P is an m-fold covering if every point of Rd is contained in at least m members of P. The largest such m is called the thickness of the covering. A 1-fold covering is simply called a covering. To formulate the central question of this survey succinctly, we need a denition. Denition 1.1. A planar set P is said to be cover-decomposable if there exists a (minimal) constant m = m(P) such that every m-fold covering of the plane with translates of P can be decomposed into two coverings. Note that the above term is slightly misleading: we decompose (partition) not the set P, but a collection P of its translates. Such a partition is sometimes regarded a coloring of the members of P

Blocking the k-holes of point sets in the plane

Let P be a set of n points in the plane in general position. A subset H of P consisting of k elements that are the vertices of a convex polygon is called a k-hole of P, if there is no element of P in the interior of its convex hull. A set B of points in the plane blocks the k-holes of P if any k-hole of P contains at least one element of B in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of k-hole blocking sets, with emphasis in the case k=5Peer ReviewedPostprint (author's final draft

Decomposition of Geometric Set Systems and Graphs

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.Comment: This is my PhD thesi

Flip cycles in plabic graphs

Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian $\text{Gr}^{\geq 0}(n,k)$. Any two plabic graphs for the same positroid cell can be related by a sequence of certain moves. The flip graph has plabic graphs as vertices and has edges connecting the plabic graphs which are related by a single move. A recent result of Galashin shows that plabic graphs can be seen as cross-sections of zonotopal tilings for the cyclic zonotope $Z(n,3)$. Taking this perspective, we show that the fundamental group of the flip graph is generated by cycles of length 4, 5, and 10, and use this result to prove a related conjecture of Dylan Thurston about triple crossing diagrams. We also apply our result to make progress on an instance of the generalized Baues problem.Comment: 26 pages, 7 figures. Journal versio

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Selected topics in algorithmic geometry

Let P be a set of n points on the plane with no three points on a line. A crossing-free structure on P is a straight-edge plane graph whose vertex set is P. In this thesis we consider problems of two different topics in the area of algorithmic geometry: Geometry using Steiner points, and counting algorithms. These topics have certain crossing-free structures on P as our primary objects of study. Our results can roughly be described as follows: i) Given a k-coloring of P, with k >= 3 colors, we will show how to construct a set of Steiner points S = S(P) such that a k-colored quadrangulation can always be constructed on (P U S). The bound we show of |S| significantly improves on previously known results. ii) We also show how to construct a se S = S(P) of Steiner points such that a triangulation of (P U S) having all its vertices of even (odd) degree can always be constructed. We show that |S| <= n/3 + c, where c is a constant. We also look at other variants of this problem. iii) With respect to counting algorithms, we show new algorithms for counting triangulations, pseudo-triangulations, crossing-free matchings and crossing-free spanning cycles on P. Our algorithms are simple and allow good analysis of their running times. These algorithms significantly improve over previously known results. We also show an algorithm that counts triangulations approximately, and a hardness result of a particular instance of the problem of counting triangulations exactly. iv) We show experiments comparing our algorithms for counting triangulations with another well-known algorithm that is supposed to be very fast in practice.Sei P eine Menge von n Punkte in der Ebene, so dass keine drei Punkten auf einer Geraden liegen. Eine kreuzungsfreie Struktur von P ist ein geradliniger ebener Graph, der P als Knotenmenge hat. In dieser Dissertation behandeln wir zwei verschiedene Problemkreise auf dem Gebiet der algorithmischen Geometrie: Geometrie mit Steinerpunkten und Anzahl bestimmende Algorithmen auf P und auf gewissen kreuzungsfreien Strukturen von P. Unsere Resultate können wie folgt beschrieben werden: i) Gegeben sei eine k-Färbung von P, mit k >= 3 Farben. Es wird gezeigt, wie eine Menge S = S(P) von Steiner Punkten konstruiert werden kann, die die Konstruktion einer k-gefärbten Quadrangulierung von (P U S) ermöglicht. Die von uns gezeigte Schranke für |S| verbessert die bisher bekannte Schranke. ii) Gezeigt wird auch die Konstruktion einer Menge S = S(P) von Steiner Punkten, so dass eine Triangulierung von (P U S) konstruiert werden kann, bei der der Grad aller Knoten gerade (ungerade) ist. Wir zeigen, dass |S| <= n/3 + c möglich ist, wobei c eine Konstante ist. Wir betrachten auch andere Varianten dieses Problems. iii) Was die Anzahl bestimmenden Algorithmen betrifft, zeigen wir neue Algorithmen, um Triangulierungen, Pseudotriangulierungen, kreuzungsfreie Matchings und kreuzungsfreie aufspannende Zyklen von P zu zählen. Unsere Algorithmen sind einfach und lassen eine gute Analyse der Laufzeiten zu. Diese neuen Algorithmen verbessern wesentlich die bisherigen Ergebnisse. Weiter zeigen wir einen Algorithmus, der Triangulierungen approximativ zählt, und bestimmen die Komplexitätsklasse einer bestimmten Variante des Problems des exakten Zählens von Triangulierungen. iv) Wir zeigen Experimente, die unsere triangulierungszählenden Algorithmen mit einem anderen bekannten Algorithmus vergleichen, der in der Praxis als besonders schnell bekannt ist