06401 Abstracts Collection -- Complexity of Constraints

From 01.10.06 to 06.10.06, the Dagstuhl Seminar 06401 ``Complexity of Constraints\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Grafos com poucos cruzamentos e o número de cruzamentos do Kp,q em superfícies topológicas

Orientador: Orlando LeeTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O número de cruzamentos de um grafo G em uma superfície ? é o menor número de cruzamentos de arestas dentre todos os possíveis desenhos de G em ?. Esta tese aborda dois problemas distintos envolvendo número de cruzamentos de grafos: caracterização de grafos com número de cruzamentos igual a um e determinação do número de cruzamentos do Kp,q em superfícies topológicas. Para grafos com número de cruzamentos um, apresentamos uma completa caracterização estrutural. Também desenvolvemos um algoritmo "prático" para reconhecer estes grafos. Em relação ao número de cruzamentos do Kp,q em superfícies, mostramos que para um inteiro positivo p e uma superfície ? fixos, existe um conjunto finito D(p,?) de desenhos "bons" de grafos bipartidos completos Kp,r (possivelmente variando o r) tal que, para todo inteiro q e todo desenho D de Kp,q, existe um desenho bom D' de Kp,q obtido através de duplicação de vértices de um desenho D'' em D(p,?) tal que o número de cruzamentos de D' é menor ou igual ao número de cruzamentos de D. Em particular, para todo q suficientemente grande, existe algum desenho do Kp,q com o menor número de cruzamentos possível que é obtido a partir de algum desenho de D(p,?) através da duplicação de vértices do mesmo. Esse resultado é uma extensão de outro obtido por Cristian et. al. para esferaAbstract: The crossing number of a graph G in a surface ? is the least amount of edge crossings among all possible drawings of G in ?. This thesis deals with two problems on crossing number of graphs: characterization of graphs with crossing number one and determining the crossing number of Kp,q in topological surfaces. For graphs with crossing number one, we present a complete structural characterization. We also show a "practical" algorithm for recognition of such graphs. For the crossing number of Kp,q in surfaces, we show that for a fixed positive integer p and a fixed surface ?, there is a finite set D(p,?) of good drawings of complete bipartite graphs Kp,r (with distinct values of r) such that, for every positive integer q and every good drawing D of Kp,q, there is a good drawing D' of Kp,q obtained from a drawing D'' of D(p,?) by duplicating vertices of D'' and such that the crossing number of D' is at most the crossing number of D. In particular, for any large enough q, there exists some drawing of Kp,q with fewest crossings which can be obtained from a drawing of D(p,?) by duplicating vertices. This extends a result of Christian et. al. for the sphereDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação2014/14375-9FAPES

On Geometric Drawings of Graphs

This thesis is about geometric drawings of graphs and their topological generalizations. First, we study pseudolinear drawings of graphs in the plane. A pseudolinear drawing is one in which every edge can be extended into an infinite simple arc in the plane, homeomorphic to $\mathbb{R}$, and such that every two extending arcs cross exactly once. This is a natural generalization of the well-studied class of rectilinear drawings, where edges are straight-line segments. Although, the problem of deciding whether a drawing is homeomorphic to a rectilinear drawing is NP-hard, in this work we characterize the minimal forbidden subdrawings for pseudolinear drawings and we also provide a polynomial-time algorithm for recognizing this family of drawings. Second, we consider the problem of transforming a topological drawing into a similar rectilinear drawing preserving the set of crossing pairs of edges. We show that, under some circumstances, pseudolinearity is a necessary and sufficient condition for the existence of such transformation. For this, we prove a generalization of Tutte's Spring Theorem for drawings with crossings placed in a particular way. Lastly, we study drawings of $K_n$ in the sphere whose edges can be extended to an arrangement of pseudocircles. An arrangement of pseudocircles is a set of simple closed curves in the sphere such that every two intersect at most twice. We show that (i) there is drawing of $K_{10}$ that cannot be extended into an arrangement of pseudocircles; and (ii) there is a drawing of $K_9$ that can be extended to an arrangement of pseudocircles, but no extension satisfies that every two pseudocircles intersects exactly twice. We also introduce the notion pseudospherical drawings of $K_n$, a generalization of spherical drawings in which each edge is a minor arc of a great circle. We show that these drawings are characterized by a simple local property. We also show that every pseudospherical drawing has an extension into an arrangement of pseudocircles where the ``at most twice'' condition is replaced by ``exactly twice''