The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Perfect Graphs

This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement

Partitioning a graph into disjoint cliques and a triangle-free graph

A graph G=(V,E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques (i.e., G[A] is P_3-free) and B induces a triangle-free graph (i.e., G[B] is K_3-free). In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K_4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomassé, Trotignon and Vušković. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs

Simplicial decompositions of graphs: a survey of applications

AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects

More on discrete convexity

In several recent papers some concepts of convex analysis were extended to discrete sets. This paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form

Algoritmos para teste de perfeição de grafos

Orientador : Prof. André Luiz Pires GuedesDissertação (mestrado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa: Curitiba, 26/08/2004Inclui referências : f. 65-67Resumo: Esta dissertação apresenta e discute os dois recentemente descobertos algoritmos de teste de perfeição de grafos. A parte central dos dois algoritmos e a mesma. Este núcleo que os dois algoritmos compartilham, que certamente e a parte mais complexa dos mesmos, foi discutido detalhadamente e implementado. Ate o momento, o autor desta dissertação não tem notícias de outras implementações destes algoritmos. A apresentação do algoritmo foi dividida em três partes distintas. A primeira parte agrupa vários algoritmos que testam pela presença de subgrafos específicos. A segunda parte estuda em detalhes o núcleo que os dois algoritmos compartilham. A terceira parte apresenta os dois algoritmos de teste de perfeição de grafos propriamente ditos. Adicionalmente, nesta dissertação foram definidos quatro parâmetros que podem ser associados a um grafo para exprimir seu grau de imperfeição. Estes parâmetros foram denotados p 1, p2, p3 e p4. O autor relacionou estes parâmetros com algumas operações que podem ser aplicadas a um grafo imperfeito para torna-lo perfeito. As operações utilizadas para definir estes parâmetros de foram a remoção de arestas do grafo (pi), a inversão de arestas no grafo (p2), a execução de remoção e inserção de arestas no grafo (p 3) e, finalmente, a remoção de vértices do grafo (p4). Mostrou-se que para qualquer grafo temos p4 < p3 < p1 e p4 < p3 < p2. Alem disso foram apresentados exemplos de grafos em que cada uma destas desigualdades pode ser estrita. O autor apresentou também alguns limitantes inferiores e superiores para estes parâmetros. Finalmente, utilizando um dos limitantes inferiores para p4, mostrou-se que existem grafos que são "bastante imperfeitos" . Mais especificamente, foi demonstrado que existem grafos com n vértices para os quais o número de vértices que deve ser removido n para tornás-lo perfeitos é pelo menos --;- - - lg (2n ). lg (2n) Palavras-chave: teoria dos grafos, algoritmos, teoria algorítmica dos grafos, grafos perfeitos, otimização combinatória.Abstract: This dissertation presents and discusses two recently discovered algorithms th a t test if a graph is perfect. The core shared by the two algorithms is discussed in details and the results of its implementation are presented. It is worthwhile to mention th a t no other similar implementation is known so far. The presentation of the algorithms is divided into three parts. The first part presents several algorithms th a t test some particular subgraphs. The second part reviews the core of the algorithms and the third part presents the two algorithms for perfectness. Additionally, in this work it is defined four parameters th a t can measure how imperfect a graph is. These parameters are denoted p1, p2, p3 and p4. The defined parameters are related to some operations th a t can be applied to a graph to make it perfect. The following operations are considered: edge deletion (pi), edge insertion (p2), both deletion and insertion of edges (p 3) and, finally, vertex deletion (p4). It is shown th a t for any graph it holds th a t p4 < p3 < p 1 and p4 < p3 < p2. It is also shown examples of graphs where such inequalities are strict. Finally, some lower bounds and upper bounds for these paramenters are shown. As a consequence of a lower bound for p4 , the author shows th a t there are "highly" imperfect graphs. More precisely, there are graphs with n vertices where n , . . Keywords: graph theory, algorithms, algorithmic graph theory, perfect graphs, combinatorial optimization