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

Subgraph densities in a surface

Given a fixed graph $H$ that embeds in a surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $\Sigma$? We show that the answer is $\Theta(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers.Comment: v4: referee's comments implemented. v3: proof of the main theorem fully rewritten, fixes a serious error in the previous version found by Kevin Hendre

Deterministic Subgraph Detection in Broadcast CONGEST

We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: - For any constant k, detecting k-paths and trees on k nodes can be done in O(1) rounds. - For any constant k, detecting k-cycles and pseudotrees on k nodes can be done in O(n) rounds. - On d-degenerate graphs, cliques and 4-cycles can be enumerated in O(d + log n) rounds, and 5-cycles in O(d2 + log n) rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for d-degenerate graphs can be improved to O(d/logn) and O(d2/logn), respect- ively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique

Deterministic subgraph detection in broadcast CONGEST

We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: For any constant k, detecting k-paths and trees on k nodes can be done in O(1) rounds. For any constant k, detecting k-cycles and pseudotrees on k nodes can be done in O(n) rounds. On d-degenerate graphs, cliques and 4-cycles can be enumerated in O(d+log n) rounds, and 5-cycles in O(d2 + log n) rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for d-degenerate graphs can be improved to O(d/ log n) and O(d2/log n), respectively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique. © 2017 Janne H. Korhonen and Joel Rybicki.Peer reviewe

Clique minors in graphs with a forbidden subgraph

The classical Hadwiger conjecture dating back to 1940's states that any graph of chromatic number at least $r$ has the clique of order $r$ as a minor. Hadwiger's conjecture is an example of a well studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on $n$ vertices of independence number $\alpha(G)$ at most $r$. If true Hadwiger's conjecture would imply the existence of a clique minor of order $n/\alpha(G)$. Results of Kuhn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that $G$ is $H$-free for some bipartite graph $H$ then one can find a polynomially larger clique minor. This has recently been extended to triangle free graphs by Dvo\v{r}\'ak and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph $H$, answering a question of Dvo\v{r}\'ak and Yepremyan. In particular, we show that any $K_s$-free graph has a clique minor of order $c_s(n/\alpha(G))^{1+\frac{1}{10(s-2) }}$, for some constant $c_s$ depending only on $s$. The exponent in this result is tight up to a constant factor in front of the $\frac{1}{s-2}$ term.Comment: 11 pages, 1 figur

A hierarchy of randomness for graphs

AbstractIn this paper we formulate four families of problems with which we aim at distinguishing different levels of randomness.The first one is completely non-random, being the ordinary Ramsey–Turán problem and in the subsequent three problems we formulate some randomized variations of it. As we will show, these four levels form a hierarchy. In a continuation of this paper we shall prove some further theorems and discuss some further, related problems

The Widths of Strict Outerconfluent Graphs

Strict outerconfluent drawing is a style of graph drawing in which vertices are drawn on the boundary of a disk, adjacencies are indicated by the existence of smooth curves through a system of tracks within the disk, and no two adjacent vertices are connected by more than one of these smooth tracks. We investigate graph width parameters on the graphs that have drawings in this style. We prove that the clique-width of these graphs is unbounded, but their twin-width is bounded.Comment: 15 pages, 2 figure

Biplanar Crossing Numbers of Bipartite Graphs

The goal of this thesis is to compute upper and lower bounds on the biplanar crossing numbers of complete bipartite graphs. The concept of a biplanar crossing number was first introduced by Owens (Owens 1971) as an optimization problem in circuit design. To prove upper bounds, we follow a method used by Czabarka et. al. (Czabarka et. al. 2006), in which they start from an optimal drawing of a small bipartite graph and use it to generate drawings of larger bipartite graphs. We explore several possibilities for computing lower bounds. One is using Ramsey theory, via the Bipartite Ramsey Number and the Connected Bipartite Ramsey Number. We prove that these numbers are equal for complete bipartite graphs, except in a few trivial cases. The other method we use is a heavily computer-aided derivation, based on the counting method, of lower bounds for small complete bipartite graphs. This is the method used in Shavali and Zarrabi-Zadeh (Shavali and Zarrabi-Zadeh 2019). We present a slight improvement over their results