Stabbing simplices of point sets with k-flats

Let S be a set of n points in Rd in general position. A set H of k-flats is called an mk-stabber of S if the relative interior of any m-simplex with vertices in S is intersected by at least one element of H. In this paper we give lower and upper bounds on the size of minimum mk-stabbers of point sets in Rd. We study mainly mk-stabbers in the plane and in R3.Consejo Nacional de Ciencia y Tecnología (México)Ministerio de Economía y CompetitividadGeneralitat de CatalunyaEuropean Science FoundationMinisterio de Ciencia e Innovació

Improved enumeration of simple topological graphs

A simple topological graph T = (V (T ), E(T )) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Tóth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2 O(n2log(m/n)), and at most 2O(mn1/2 log n) if m ≤ n 3/2. As a consequence we obtain a new upper bound 2 O(n3/2 log n) on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2n2 ·α(n) O(1), using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2 m2+O(mn) and at least 2 Ω(m2) for graphs with m > (6 + ε)n.Graph Drawings and Representations, EuroGIGA ProjectCentre Interfacultaire Bernoull

Phase transitions in the Ramsey-Turán theory

Let f(n) be a function and L be a graph. Denote by RT(n, L, f(n)) the maximum number of edges of an L-free graph on n vertices with independence number less than f(n). Erdos and Sós asked if RT (n, K5, c√ n) = o (n2) for some constant c. We answer this question by proving the stronger RT(n, K5, o (√n log n)) = o(n2). It is known that RT (n, K5, c√n log n )= n2/4 + o (n2) for c > 1, so one can say that K5 has a Ramsey-Turán-phase transition at c√n log n. We extend this result to several other Kp's and functions f(n), determining many more phase transitions. We shall formulate several open problems, in particular, whether variants of the Bollobás-Erdos graph, which is a geometric construction, exist to give good lower bounds on RT (n, Kp, f(n)) for various pairs of p and f(n). These problems are studied in depth by Balogh-HuSimonovits, where among others, the Szemerédi's Regularity Lemma and the Hypergraph Dependent Random Choice Lemma are used.National Science Foundatio

Three location tapas calling for CG sauce

Based on some recent modelling considerations in location theory we call for study of three CG constructs of Voronoi type that seem not to have been studied much before

Abstract Voronoi diagrams

Abstract Voronoi diagrams are a unifying framework that covers many types of concrete Voronoi diagrams. This talk reports on the state of the art, including recent progress.European Science Foundatio

Continuous surveillance of points by rotating floodlights

Let P and F be sets of n ≥ 2 and m ≥ 2 points in the plane, respectively, so that P∪F is in general position. We study the problem of finding the minimum angle α ∈ [2π/m, 2π] such that one can install at each point of F a stationary rotating floodlight with illumination angle α, initially oriented in a suitable direction, in such a way that, at all times, every target point of P is illuminated by at least one light. All floodlights rotate at unit speed and clockwise. We give an upper bound for the 1-dimensional problem and present results for some instances of the general problem. Specifically, we solve the problem for the case in which we have two floodlights and many points, and give an upper bound for the case in which there are many floodlights and only two target points.Ministerio de Educación y CienciaEuropean Science FoundationMinisterio de Ciencia e InnovaciónComisión Nacional de Investigación Científica y Tecnológica (Chile)Fondo Nacional de Desarrollo Científico y Tecnológico (Chile

On the nonexistence of k-reptile simplices in R3 and R4

A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled without overlaps by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d=2, triangular k-reptiles exist for many values of k and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d≥3, have k=m d, where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d=3, k-reptile tetrahedra can exist only for k=m 3. We also prove a weaker analogue of this result for d=4 by showing that four-dimensional k-reptile simplices can exist only for k=m 2.Czech Science FoundationCentre Interfacultaire BernoulliSwiss National Science Foundatio

Recent developments on the crossing number of the complete graph

Ministerio de Economía y CompetitividadEuropean Science Foundatio

Guarding the vertices of thin orthogonal polygons is NP-hard

An orthogonal polygon of P is called “thin” if the dual graph of the partition obtained by extending all edges of P towards its interior until they hit the boundary is a tree. We show that the problem of computing a minimum guard set for either a thin orthogonal polygon or only its vertices is NP-hard, indeed APX-hard, either for guards lying on the boundary or on vertices of the polygon.Fondo Europeo de Desarrollo RegionalFundação para a Ciência e a Tecnologi

On the enumeration of permutominoes

Although the exact counting and enumeration of polyominoes remain challenging open problems, several positive results were achieved for special classes of polyominoes. We give an algorithm for direct enumeration of permutominoes by size, or, equivalently, for the enumeration of grid orthogonal polygons. We show how the construction technique allows us to derive a simple characterization of the class of convex permutominoes, which has been extensively investigated. The approach extends to other classes, such as the row convex and the directed convex permutominoes.Fondo Europeo de Desarrollo RegionalFundação para a Ciência e a Tecnologi