The alternating path problem revisited

It is well known that, given n red points and n blue points on a circle, it is not always possible to find a plane geometric Hamiltonian alternating path. In this work we prove that if we relax the constraint on the path from being plane to being 1-plane, then the problem always has a solution, and even a Hamiltonian alternating cycle can be obtained on all instances. We also extend this kind of result to other configurations and provide remarks on similar problems.Ministerio de Economía y CompetitividadGeneralitat de CatalunyaEuropean Science FoundationMinisterio de Ciencia e InnovaciónJunta de Andalucía (Consejería de Innovación, Ciencia y Empresa

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

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

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

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

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

On the barrier-resilience of arrangements of ray-sensors

Given an arrangement A of n sensors and two points s and t in the plane, the barrier resilience of A with respect to s and t is the minimum number of sensors whose removal permits a path from s to t such that the path does not intersect the coverage region of any sensor in A. When the surveillance domain is the entire plane and sensor coverage regions are unit line segments, even with restricted orientations, the problem of determining the barrier resilience is known to be NP-hard. On the other hand, if sensor coverage regions are arbitrary lines, the problem has a trivial linear time solution. In this paper, we give an O(n2m) time algorithm for computing the barrier resilience when each sensor coverage region is an arbitrary ray, where m is the number of sensor intersections.Natural Sciences and Engineering Research Council of Canad