60 research outputs found
Decomposing Berge graphs and detecting balanced skew partitions
We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an O(n9)-time algorithm for the same problem restricted to Berge graphs. Our algorithm is not constructive : it certifies that a graph has a balanced skew partition if it has one. It relies on a new decomposition theorem for Berge graphs, that is more precise than the previously known theorems and implies them easily. Our theorem also implies that every Berge graph can be decomposed in a first step by using only balanced skew partitions, and in a second step by using only 2-joins. Our proof of this new theorem uses at an essential step one of the decomposition theorems of Chudnovsky.Perfect graph, Berge graph, 2-join, balanced skew partition, decomposition, detection, recognition.
Clique-Stable Set separation in perfect graphs with no balanced skew-partitions
Inspired by a question of Yannakakis on the Vertex Packing polytope of
perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary
subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates
a clique K and a stable set S if and . A
Clique-Stable Set Separator is a family of cuts such that for every clique K,
and for every stable set S disjoint from K, there exists a cut in the family
that separates K and S. Given a class of graphs, the question is to know
whether every graph of the class admits a Clique-Stable Set Separator
containing only polynomially many cuts. It is open for the class of all graphs,
and also for perfect graphs, which was Yannakakis' original question. Here we
investigate on perfect graphs with no balanced skew-partition; the balanced
skew-partition was introduced in the proof of the Strong Perfect Graph Theorem.
Recently, Chudnovsky, Trotignon, Trunck and Vuskovic proved that forbidding
this unfriendly decomposition permits to recursively decompose Berge graphs
using 2-join and complement 2-join until reaching a basic graph, and they found
an efficient combinatorial algorithm to color those graphs. We apply their
decomposition result to prove that perfect graphs with no balanced
skew-partition admit a quadratic-size Clique-Stable Set Separator, by taking
advantage of the good behavior of 2-join with respect to this property. We then
generalize this result and prove that the Strong Erdos-Hajnal property holds in
this class, which means that every such graph has a linear-size biclique or
complement biclique. This property does not hold for all perfect graphs (Fox
2006), and moreover when the Strong Erdos-Hajnal property holds in a hereditary
class of graphs, then both the Erdos-Hajnal property and the polynomial
Clique-Stable Set Separation hold.Comment: arXiv admin note: text overlap with arXiv:1308.644
Detecting 2-joins faster
2-joins are edge cutsets that naturally appear in the decomposition of
several classes of graphs closed under taking induced subgraphs, such as
balanced bipartite graphs, even-hole-free graphs, perfect graphs and claw-free
graphs. Their detection is needed in several algorithms, and is the slowest
step for some of them. The classical method to detect a 2-join takes
time where is the number of vertices of the input graph and the number
of its edges. To detect \emph{non-path} 2-joins (special kinds of 2-joins that
are needed in all of the known algorithms that use 2-joins), the fastest known
method takes time . Here, we give an -time algorithm for both
of these problems. A consequence is a speed up of several known algorithms
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
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
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
Three-in-a-Tree in Near Linear Time
The three-in-a-tree problem is to determine if a simple undirected graph
contains an induced subgraph which is a tree connecting three given vertices.
Based on a beautiful characterization that is proved in more than twenty pages,
Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known
polynomial-time algorithm, running in time, to solve the
three-in-a-tree problem on an -vertex -edge graph. Their three-in-a-tree
algorithm has become a critical subroutine in several state-of-the-art graph
recognition and detection algorithms.
In this paper we solve the three-in-a-tree problem in time,
leading to improved algorithms for recognizing perfect graphs and detecting
thetas, pyramids, beetles, and odd and even holes. Our result is based on a new
and more constructive characterization than that of Chudnovsky and Seymour. Our
new characterization is stronger than the original, and our proof implies a new
simpler proof for the original characterization. The improved characterization
gains the first factor in speed. The remaining improvement is based on
dynamic graph algorithms.Comment: 46 pages, 12 figures, accepted to STOC 202
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