Extremal and Ramsey Type Questions for Graphs and Ordered Graphs

In this thesis we study graphs and ordered graphs from an extremal point of view. In the first part of the thesis we prove that there are ordered forests H and ordered graphs of arbitrarily large chromatic number not containing such H as an ordered subgraph. In the second part we study pairs of graphs that have the same set of Ramsey graphs. We support a negative answer to the question whether there are pairs of non-isomorphic connected graphs that have this property. Finally we initiate the study of minimal ordered Ramsey graphs. For large families of ordered graphs we determine whether their members have finitely or infinitely many minimal ordered Ramsey graphs

Coloring and covering problems on graphs

The \emph{separation dimension} of a graph $G$, written $\pi(G)$, is the minimum number of linear orderings of $V(G)$ such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} $\pi_f(G)$, which is the minimum of $a/b$ such that some $a$ linear orderings (repetition allowed) separate every two nonincident edges at least $b$ times. In contrast to separation dimension, we show fractional separation dimension is bounded: always $\pi_f(G)\le 3$, with equality if and only if $G$ contains $K_4$. There is no stronger bound even for bipartite graphs, since $\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}$. We also compute $\pi_f(G)$ for cycles and some complete tripartite graphs. We show that $\pi_f(G)<\sqrt{2}$ when $G$ is a tree and present a sequence of trees on which the value tends to $4/3$. We conjecture that when $n=3m$ the $K_4$-free $n$-vertex graph maximizing $\pi_f(G)$ is $K_{m,m,m}$. We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let $\pi^\circ(G)$ be the number of circular orderings needed to separate all pairs, and let $\pi_f^\circ(G)$ be the fractional version. Among our results: (1) $\pi^\circ(G)=1$ if and only $G$ is outerplanar. (2) $\pi^\circ(G)\le2$ when $G$ is bipartite. (3) $\pi^\circ(K_n)\ge\log_2\log_3(n-1)$. (4) $\pi_f^\circ(G)\le\frac{3}{2}$, with equality if and only if $K_4\subseteq G$. (5) $\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}$. A \emph{star $k$-coloring} is a proper $k$-coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to prove that every graph with maximum average degree less than $\frac{5}{2}$ has an \Ifp, which is sharp and improves the result of Bu, Cranston, Montassier, Raspaud, and Wang (2009). As a corollary, we gain that every planar graph with girth at least 10 has a star 4-coloring. A proper vertex coloring of a graph $G$ is \emph{$r$-dynamic} if for each $v\in V(G)$, at least $\min\{r,d(v)\}$ colors appear in $N_G(v)$. We investigate $3$-dynamic versions of coloring and list coloring. We prove that planar and toroidal graphs are 3-dynamically 10-choosable, and this bound is sharp for toroidal graphs. Given a proper total $k$-coloring $c$ of a graph $G$, we define the \emph{sum value} of a vertex $v$ to be $c(v) + \sum_{uv \in E(G)} c(uv)$. The smallest integer $k$ such that $G$ has a proper total $k$-coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} $\chi''_{\Sigma}(G)$. Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that $\chi''_{\Sigma}(G)\leq \Delta(G)+3$ for any simple graph with maximum degree $\Delta(G)$. We prove this bound to be asymptotically correct by showing that $\chi''_{\Sigma}(G)\leq \Delta(G)(1+o(1))$. The main idea of our argument relies on Przyby{\l}o's proof (2014) for neighbor sum distinguishing edge-coloring

Entropic Characterization and Time Evolution of Complex Networks

In this thesis, we address problems encountered in complex network analysis using graph theoretic methods. The thesis specifically centers on the challenge of how to characterize the structural properties and time evolution of graphs. We commence by providing a brief roadmap for our research in Chapter 1, followed by a review of the relevant research literature in Chapter 2. The remainder of the thesis is structured as follows. In Chapter 3, we focus on the graph entropic characterizations and explore whether the von Neumann entropy recently defined only on undirected graphs, can be extended to the domain of directed graphs. The substantial contribution involves a simplified form of the entropy which can be expressed in terms of simple graph statistics, such as graph size and vertex in-degree and out-degree. Chapter 4 further investigates the uses and applications of the von Neumann entropy in order to solve a number of network analysis and machine learning problems. The contribution in this chapter includes an entropic edge assortativity measure and an entropic graph embedding method, which are developed for both undirected and directed graphs. The next part of the thesis analyzes the time-evolving complex networks using physical and information theoretic approaches. In particular, Chapter 5 provides a thermodynamic framework for handling dynamic graphs using ideas from algebraic graph theory and statistical mechanics. This allows us to derive expressions for a number of thermodynamic functions, including energy, entropy and temperature, which are shown to be efficient in identifying abrupt structural changes and phase transitions in real-world dynamical systems. Chapter 6 develops a novel method for constructing a generative model to analyze the structure of labeled data, which provides a number of novel directions to the study of graph time-series. Finally, in Chapter 7, we provide concluding remarks and discuss the limitations of our methodologies, and point out possible future research directions

Modelos para sequenciação de padrões em problemas de corte de stock

Tese de doutoramento em Engenharia Industrial e de SistemasIn this thesis, we address an optimization problem that appears in cutting stock operations research called the minimization of the maximum number of open stacks (MOSP) and we put forward a new integer programming formulation for the MOSP. By associating the duration of each stack with an interval of time, it is possible to use the rich theory that exists in interval graphs in order to create a model based on the completion of a graph with edges. The structure of this type of graphs admits a linear ordering of the vertices that de nes an ordering of the stacks, and consequently decides a sequence for the cutting patterns. The polytope de ned by this formulation is full-dimensional and the main inequalities in the model are proved to be facets. Additional inequalities are derived based on the properties of chordal graphs and comparability graphs. The maximum number of open stacks is related with the chromatic number of the solution graph; thus the formulation is strengthened by adding the representatives formulation for the vertex coloring problem. The model is applied to the minimization of open stacks, and also to the minimum interval graph completion problem and other pattern sequencing problems such as the minimization of the order spread (MORP) and the minimization of the number of tool switches (MTSP). Computational tests of the model are presented.Nesta tese e abordado um problema de optimização que surge em operações de corte de stock chamado minimização do número máximo de pilhas abertas (MOSP) e e proposta uma nova formulação de programação inteira. Associando a duração de cada pilha a um intervalo de tempo, e possível usar a teoria rica que existe em grafos de intervalos para criar um modelo baseado no completamento de um grafo por arcos. A estrutura deste tipo de grafos admite uma ordenação linear dos vértices que define uma ordenação linear das pilhas e, por sua vez, determina a sequência dos padrões de corte. O politopo definido por esta formulação tem dimensão completa e prova-se que as principais desigualdades do modelo são facetas. São derivadas desigualdades adicionais baseadas nas propriedades de grafos cordais e de grafos de comparabilidades. O número máximo de pilhas abertas está relacionado com o número cromático do grafo solução, pelo que o modelo e reforçado com a formulação por representativos para o problema de coloração de vértices. O modelo e aplicado a minimização de pilhas abertas, e também ao problema de completamento mínimo de um grafo de intervalos e a outros problemas de sequenciação de padrões, tais como a minimização da dispersão de encomendas (MORP) e a minimização do número de trocas de ferramentas (MTSP). São apresentados testes computacionais do modelo.Fundação para a Ciência e a Tecnologia (FCT), programa de financiamento QREN-POPH-Tipologia 4.1-Formação Avançada comparticipado pelo Fundo Social Europeu e por fundos do MCTES (Bolsa individual com a refer^encia SFRH/BD/32151/2006) entre 2006 e 2009, e pela Escola Superior de Estudos Industriais e de Gest~ao do Instituto Polit ecnico do Porto (Bolsa PROTEC com a refer^encia SFRH/BD/49914/2009) entre 2009 e 2010

COMBINATORIAL ALGORITHMS FOR GRAPH DISCOVERY AND EXPERIMENTAL DESIGN

In this thesis, we study the design and analysis of algorithms for discovering the structure and properties of an unknown graph, with applications in two different domains: causal inference and sublinear graph algorithms. In both these domains, graph discovery is possible using restricted forms of experiments, and our objective is to design low-cost experiments. First, we describe efficient experimental approaches to the causal discovery problem, which in its simplest form, asks us to identify the causal relations (edges of the unknown graph) between variables (vertices of the unknown graph) of a given system. For causal discovery, we study algorithms for the problem of learning the causal relationships between a set of observed variables in the presence of hidden or unobserved variables while minimizing a suitable cost of interventions on the observed variables. An intervention on a set of variables helps learn the presence of causal relations adjacent to them. Under various cost models for interventions, we design combinatorial algorithms for causal discovery by identifying new connections between discrete optimization, graph property testing, and efficient intervention design. Next, we investigate query-efficient experimental approaches for estimating various graph properties, such as the number of edges and graph connectivity. The access to the graph, or equivalently performing an experiment, is via a Bipartite Independent Set (BIS) oracle. The BIS oracle is related to the interventional access model used in our work for causal graph discovery, with other applications in group testing and fine-grained complexity. In this setting, we develop non-adaptive algorithms that lead to efficient implementations in highly parallelized and low-memory streaming settings

Algorithms for nonuniform networks

In this thesis, observations on structural properties of natural networks are taken as a starting point for developing efficient algorithms for natural instances of different graph problems. The key areas discussed are sampling, clustering, routing, and pattern mining for large, nonuniform graphs. The results include observations on structural effects together with algorithms that aim to reveal structural properties or exploit their presence in solving an interesting graph problem. Traditionally networks were modeled with uniform random graphs, assuming that each vertex was equally important and each edge equally likely to be present. Within the last decade, the approach has drastically changed due to the numerous observations on structural complexity in natural networks, many of which proved the uniform model to be inadequate for some contexts. This quickly lead to various models and measures that aim to characterize topological properties of different kinds of real-world networks also beyond the uniform networks. The goal of this thesis is to utilize such observations in algorithm design, in addition to empowering the process of network analysis. Knowing that a graph exhibits certain characteristics allows for more efficient storage, processing, analysis, and feature extraction. Our emphasis is on local methods that avoid resorting to information of the graph structure that is not relevant to the answer sought. For example, when seeking for the cluster of a single vertex, we compute it without using any global knowledge of the graph, iteratively examining the vicinity of the seed vertex. Similarly we propose methods for sampling and spanning-tree construction according to certain criteria on the outcome without requiring knowledge of the graph as a whole. Our motivation for concentrating on local methods is two-fold: one driving factor is the ever-increasing size of real-world problems, but an equally important fact is the nonuniformity present in many natural graph instances; properties that hold for the entire graph are often lost when only a small subgraph is examined.reviewe