The role of twins in computing planar supports of hypergraphs

A support or realization of a hypergraph $H$ is a graph $G$ on the same vertex as $H$ such that for each hyperedge of $H$ it holds that its vertices induce a connected subgraph of $G$. The NP-hard problem of finding a planar} support has applications in hypergraph drawing and network design. Previous algorithms for the problem assume that twins}---pairs of vertices that are in precisely the same hyperedges---can safely be removed from the input hypergraph. We prove that this assumption is generally wrong, yet that the number of twins necessary for a hypergraph to have a planar support only depends on its number of hyperedges. We give an explicit upper bound on the number of twins necessary for a hypergraph with $m$ hyperedges to have an $r$-outerplanar support, which depends only on $r$ and $m$. Since all additional twins can be safely removed, we obtain a linear-time algorithm for computing $r$-outerplanar supports for hypergraphs with $m$ hyperedges if $m$ and $r$ are constant; in other words, the problem is fixed-parameter linear-time solvable with respect to the parameters $m$ and $r$

Combinatorics of embeddings

We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family. In higher dimensions n>2, we observe that in order to characterize those compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited minors", it suffices to establish finiteness of the list of all (n-1)-connected n-dimensional finite cell complexes that do not embed in S^{2n} yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n+1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i+2)-simplices) at least ten non-simplicial complexes.Comment: 49 pages, 1 figure. Minor improvements in v2 (subsection 4.C on transforms of dichotomial spheres reworked to include more details; subsection 2.D "Algorithmic issues" added, etc

Detecting Weakly Simple Polygons

A closed curve in the plane is weakly simple if it is the limit (in the Fr\'echet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201

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

Finding topological subgraphs is fixed-parameter tractable

We show that for every fixed undirected graph $H$, there is a $O(|V(G)|^3)$ time algorithm that tests, given a graph $G$, if $G$ contains $H$ as a topological subgraph (that is, a subdivision of $H$ is subgraph of $G$). This shows that topological subgraph testing is fixed-parameter tractable, resolving a longstanding open question of Downey and Fellows from 1992. As a corollary, for every $H$ we obtain an $O(|V(G)|^3)$ time algorithm that tests if there is an immersion of $H$ into a given graph $G$. This answers another open question raised by Downey and Fellows in 1992

A New Parallel Algorithm for Planarity Testing

Determining whether a graph is planar is both theoretically and practically interesting. Although several sequential algorithms have been introduced which accomplish planarity testing in O(V ) time for graphs with V vertices, very few of these have been parallelized. In a recent comparison of sequential planarity testing algorithms, the newest algorithms were found to be fastest; however, these are the ones which have not been parallelized. The goal of this thesis is to introduce a method for parallelizing one of the newest planarity testing algorithms

Efficient processing of hierarchical graphs

The standard representation of a graph is a list of its vertices and edges. However, graphs encountered in some areas have structural regularities that allow them to be represented using considerably less space. Typically, such a succinct encoding consists of a list of basic parts and a set of instructions for assembling the graph from the parts. Several models for succinctly representing graphs and other structures have been studied in the past. These models are capable of representing a graph using space polynomial in the logarithm of the size of the graph. Because of the potentially large difference in size between a graph and its description, it is natural to ask whether there are any problems that can be solved in time polynomial in the size of the succinct description, rather than in the size of the graph. It is known that even simple graph problems become NP-hard or worse under most models for succinct representation. Two exceptions are dynamic graphs and, the model we study, hierarchical graphs;A hierarchical graph [gamma] is a list of graphs and a set of rules describing how to attach the graphs together to form X([gamma]), the graph [gamma] represents. A hierarchical algorithm is an algorithm that, given [gamma], solves some problem defined on X([gamma]). Not every polynomially-solvable graph problem has a polynomial-time hierarchical algorithm. However, polynomial-time hierarchical algorithms for many problems have been developed using a framework called the bottom-up method;We develop a generalization of the bottom-up method that we use to construct and analyze hierarchical algorithms. Our method provides a uniform setting in which to present our algorithms, as well as many developed by other researchers. The problems we study belong to three classes: connectivity augmentation, subgraph homeomorphism, and matroid optimization. The P-connectivity augmentation problem, where P is a connectivity property, is to determine the number of edges that must be added to a graph to satisfy P. We present polynomial-time hierarchical algorithms for bridge-connectivity, biconnectivity, and strong-connectivity augmentation. Series-parallel and outer-planar graphs can be characterized by sets of forbidden graphs closed under homeomorphism. We present linear-time hierarchical algorithms that determine if X([gamma]) is series-parallel or outer-planar, and a polynomial-space hierarchical algorithm that generates a forbidden subgraph of X([gamma]) when one exists. The matroid optimization problem we consider is that of computing costs of optimum bases of matroids defined on graphs. We identify two infinite families of matroids for which polynomial-time hierarchical algorithms for this problem exist. We also develop polynomial-space hierarchical algorithms that generate optimum bases

Drawings of Complete Multipartite Graphs up to Triangle Flips

For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph Kn with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on n vertices is bounded by O(n16). The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the following sense: For the complete bipartite graph Km, n minus two edges and Km, n plus one edge for any m, n ≥ 4, as well as Kn minus a 4-cycle for any n ≥ 5, there exist two simple drawings with the same ERS that cannot be transformed into each other using triangle flips. So having the same ERS does not remain sufficient when removing or adding very few edges