Biochemical network matching and composition

This paper looks at biochemical network matching and compositio

On the linear extension complexity of stable set polytopes for perfect graphs

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-joins and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behavior of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph G depending linearly on the size of G and involving the depth of a decomposition tree of G in terms of basic perfect graphs

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

Structure and algorithms for (cap, even hole)-free graphs

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most [View the MathML source], and hence [View the MathML source], where ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set, where n is the number of vertices and m is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G)−1 and clique-width at most 48

Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family

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

Interactions entre les Cliques et les Stables dans un Graphe

This thesis is concerned with different types of interactions between cliques and stable sets, two very important objects in graph theory, as well as with the connections between these interactions. At first, we study the classical problem of graph coloring, which can be stated in terms of partioning the vertices of the graph into stable sets. We present a coloring result for graphs with no triangle and no induced cycle of even length at least six. Secondly, we study the Erdös-Hajnal property, which asserts that the maximum size of a clique or a stable set is polynomial (instead of logarithmic in random graphs). We prove that the property holds for graphs with no induced path on k vertices and its complement.Then, we study the Clique-Stable Set Separation, which is a less known problem. The question is about the order of magnitude of the number of cuts needed to separate all the cliques from all the stable sets. This notion was introduced by Yannakakis when he studied extended formulations of the stable set polytope in perfect graphs. He proved that a quasi-polynomial number of cuts is always enough, and he asked if a polynomial number of cuts could suffice. Göös has just given a negative answer, but the question is open for restricted classes of graphs, in particular for perfect graphs. We prove that a polynomial number of cuts is enough for random graphs, and in several hereditary classes. To this end, some tools developed in the study of the Erdös-Hajnal property appear to be very helpful. We also establish the equivalence between the Clique-Stable set Separation problem and two other statements: the generalized Alon-Saks-Seymour conjecture and the Stubborn Problem, a Constraint Satisfaction Problem.Cette thèse s'intéresse à différents types d'interactions entre les cliques et les stables, deux objets très importants en théorie des graphes, ainsi qu'aux relations entre ces différentes interactions. En premier lieu, nous nous intéressons au problème classique de coloration de graphes, qui peut s'exprimer comme une partition des sommets du graphe en stables. Nous présentons un résultat de coloration pour les graphes sans triangles ni cycles pairs de longueur au moins 6. Dans un deuxième temps, nous prouvons la propriété d'Erdös-Hajnal, qui affirme que la taille maximale d'une clique ou d'un stable devient polynomiale (contre logarithmique dans les graphes aléatoires) dans le cas des graphes sans chemin induit à k sommets ni son complémentaire, quel que soit k.Enfin, un problème moins connu est la Clique-Stable séparation, où l'on cherche un ensemble de coupes permettant de séparer toute clique de tout stable. Cette notion a été introduite par Yannakakis lors de l’étude des formulations étendues du polytope des stables dans un graphe parfait. Il prouve qu’il existe toujours un séparateur Clique-Stable de taille quasi-polynomiale, et se demande si l'on peut se limiter à une taille polynomiale. Göös a récemment fourni une réponse négative, mais la question se pose encore pour des classes de graphes restreintes, en particulier pour les graphes parfaits. Nous prouvons une borne polynomiale pour la Clique-Stable séparation dans les graphes aléatoires et dans plusieurs classes héréditaires, en utilisant notamment des outils communs à l'étude de la conjecture d'Erdös-Hajnal. Nous décrivons également une équivalence entre la Clique-Stable séparation et deux autres problèmes  : la conjecture d'Alon-Saks-Seymour généralisée et le Problème Têtu, un problème de Satisfaction de Contraintes

Structural Characterisations of Hereditary Graph Classes and Algorithmic Consequences

A hole is a chordless cycle of length at least four, and is even or odd depending onthe parity of its length. Many interesting classes of graphs are defined by excluding (possibly among other graphs) holes of certain lengths. Most famously perhaps is the class of Berge graphs, which are the graphs that contain no odd hole and no complement of an odd hole. A graph is perfect if the chromatic number of each of its induced subgraphs is equal to the size of a maximum clique in that subgraph. It was conjectured in the 1960’s by Claude Berge that Berge graphs and perfect graphs are equivalent, that is, a graph is perfect if and only if it is Berge. This conjecture was finally resolved by Chudnovsky, Robertson, Seymour and Thomas in 2002, and it is now called the strong perfect graph theorem. Graphs that do not contain even holes are structurally similar to Berge graphs, and for this reason Conforti, Cornuéjols, Kapoor and Vušković initiated the study of even-hole-free graphs. One of their main results was a decomposition theorem and a recognition algorithm for even-hole-free graphs, and many techniques developed in the pursuit of a decomposition theorem for even-hole-free graphs proved useful in the study of perfect graphs. Indeed, the proof of the strong perfect graph theorem relied on decomposition, and many interesting graph classes have since then been understood from the viewpoint of decomposition. In this thesis we study several classes of graphs that relate to even-hole-free graphs. First, we focus on β-perfect graphs, which form a subclass of even-hole-free graphs. While it is unknown whether even-hole-free graphs can be coloured in polynomial time, β-perfect graphs can be coloured optimally in polynomial time using the greedy colouring algorithm. The class of β-perfect graphs was introduced in 1996 by Markossian, Gasparian and Reed, and since then several classes of β-perfect graphs have been identified but no forbidden induced subgraph characterisation is known. In this thesis we identify a new class of β-perfect graphs, and we present forbidden induced subgraph characterisations for the class of β-perfect hyperholes and for the class of claw-free β-perfect graphs. We use these characterisations to decide in polynomial time whether a given hyperhole, or more generally a claw-free graph, is β-perfect. A graph is l-holed (for an integer l ≥ 4) if every one of its holes is of length l. Another focus of the thesis is the class of l-holed graphs. When l is odd, the l-holed graphs form a subclass of even-hole-free graphs. Together with Preissmann, Robin, Sintiari, Trotignon and Vušković we obtained a structure theorem for l-holed graphs where l ≥ 7. Working independently, Cook and Seymour obtained a structure theorem for the same class of graphs. In this thesis we establish that these two structure theorems are equivalent. Furthermore, we present two recognition algorithms for l-holed graphs for odd l ≥ 7. The firs uses the structure theorem of Preissmann, Robin, Sintiari, Trotignon, Vušković and the present author, and relies on decomposition by a new variant of a 2-join called a special 2-join, and the second uses the structure theorem of Cook and Seymour, and relies only on a process of clique cutset decomposition. We also give algorithms that solve in polynomial time the maximum clique and maximum stable set problems for l-holed graphs for odd l ≥ 7. Finally, we focus on circular-arc graphs. It is a long standing open problem to characterise in terms of forbidden induced subgraphs the class of circular-arc graphs, and even the class of chordal circular-arc graphs. Motivated by a result of Cameron, Chap-lick and Hoàng stating that even-hole-free graphs that are pan-free can be decomposed by clique cutsets into circular-arc graphs, we investigate the class of even-hole-free circular-arc graphs. We present a partial characterisation for the class of even-hole-free circular-arc graphs that are not chordal