Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results

Propriétés géométriques du nombre chromatique : polyèdres, structures et algorithmes

Computing the chromatic number and finding an optimal coloring of a perfect graph can be done efficiently, whereas it is an NP-hard problem in general. Furthermore, testing perfection can be carried- out in polynomial-time. Perfect graphs are characterized by a minimal structure of their sta- ble set polytope: the non-trivial facets are defined by clique-inequalities only. Conversely, does a similar facet-structure for the stable set polytope imply nice combinatorial and algorithmic properties of the graph ? A graph is h-perfect if its stable set polytope is completely de- scribed by non-negativity, clique and odd-circuit inequalities. Statements analogous to the results on perfection are far from being understood for h-perfection, and negative results are missing. For ex- ample, testing h-perfection and determining the chromatic number of an h-perfect graph are unsolved. Besides, no upper bound is known on the gap between the chromatic and clique numbers of an h-perfect graph. Our first main result states that the operations of t-minors keep h- perfection (this is a non-trivial extension of a result of Gerards and Shepherd on t-perfect graphs). We show that it also keeps the Integer Decomposition Property of the stable set polytope, and use this to answer a question of Shepherd on 3-colorable h-perfect graphs in the negative. The study of minimally h-imperfect graphs with respect to t-minors may yield a combinatorial co-NP characterization of h-perfection. We review the currently known examples of such graphs, study their stable set polytope and state several conjectures on their structure. On the other hand, we show that the (weighted) chromatic number of certain h-perfect graphs can be obtained efficiently by rounding-up its fractional relaxation. This is related to conjectures of Goldberg and Seymour on edge-colorings. Finally, we introduce a new parameter on the complexity of the matching polytope and use it to give an efficient and elementary al- gorithm for testing h-perfection in line-graphs.Le calcul du nombre chromatique et la détermination d'une colo- ration optimale des sommets d'un graphe sont des problèmes NP- difficiles en général. Ils peuvent cependant être résolus en temps po- lynomial dans les graphes parfaits. Par ailleurs, la perfection d'un graphe peut être décidée efficacement. Les graphes parfaits sont caractérisés par la structure de leur poly- tope des stables : les facettes non-triviales sont définies exclusivement par des inégalités de cliques. Réciproquement, une structure similaire des facettes du polytope des stables détermine-t-elle des propriétés combinatoires et algorithmiques intéressantes? Un graphe est h-parfait si les facettes non-triviales de son polytope des stables sont définies par des inégalités de cliques et de circuits impairs. On ne connaît que peu de résultats analogues au cas des graphes parfaits pour la h-perfection, et on ne sait pas si les problèmes sont NP-difficiles. Par exemple, les complexités algorithmiques de la re- connaissance des graphes h-parfaits et du calcul de leur nombre chro- matique sont toujours ouvertes. Par ailleurs, on ne dispose pas de borne sur la différence entre le nombre chromatique et la taille maxi- mum d'une clique d'un graphe h-parfait. Dans cette thèse, nous montrons tout d'abord que les opérations de t-mineurs conservent la h-perfection (ce qui fournit une extension non triviale d'un résultat de Gerards et Shepherd pour la t-perfection). De plus, nous prouvons qu'elles préservent la propriété de décompo- sition entière du polytope des stables. Nous utilisons ce résultat pour répondre négativement à une question de Shepherd sur les graphes h-parfaits 3-colorables. L'étude des graphes minimalement h-imparfaits (relativement aux t-mineurs) est liée à la recherche d'une caractérisation co-NP com- binatoire de la h-perfection. Nous faisons l'inventaire des exemples connus de tels graphes, donnons une description de leur polytope des stables et énonçons plusieurs conjectures à leur propos. D'autre part, nous montrons que le nombre chromatique (pondéré) de certains graphes h-parfaits peut être obtenu efficacement en ar- rondissant sa relaxation fractionnaire à l'entier supérieur. Ce résultat implique notamment un nouveau cas d'une conjecture de Goldberg et Seymour sur la coloration d'arêtes. Enfin, nous présentons un nouveau paramètre de graphe associé aux facettes du polytope des couplages et l'utilisons pour donner un algorithme simple et efficace de reconnaissance des graphes h- parfaits dans la classe des graphes adjoints

List-coloring and sum-list-coloring problems on graphs

Graph coloring is a well-known and well-studied area of graph theory that has many applications. In this dissertation, we look at two generalizations of graph coloring known as list-coloring and sum-list-coloring. In both of these types of colorings, one seeks to first assign palettes of colors to vertices and then choose a color from the corresponding palette for each vertex so that a proper coloring is obtained. A celebrated result of Thomassen states that every planar graph can be properly colored from any arbitrarily assigned palettes of five colors. This result is known as 5-list-colorability of planar graphs. Albertson asked whether Thomassen\u27s theorem can be extended by precoloring some vertices which are at a large enough distance apart. Hutchinson asked whether Thomassen\u27s theorem can be extended by allowing certain vertices to have palettes of size less than five assigned to them. In this dissertation, we explore both of these questions and answer them in the affirmative for various classes of graphs. We also provide a catalog of small configurations with palettes of different prescribed sizes and determine whether or not they can always be colored from palettes of such sizes. These small configurations can be useful in reducing certain planar graphs to obtain more information about their structure. Additionally, we look at the newer notion of sum-list-coloring where the sum choice number is the parameter of interest. In sum-list-coloring, we seek to minimize the sum of varying sizes of palettes of colors assigned the vertices of a graph. We compute the sum choice number for all graphs on at most five vertices, present some general results about sum-list-coloring, and determine the sum choice number for certain graphs made up of cycles

Boundary properties of graphs

A set of graphs may acquire various desirable properties, if we apply suitable restrictions on the set. We investigate the following two questions: How far, exactly, must one restrict the structure of a graph to obtain a certain interesting property? What kind of tools are helpful to classify sets of graphs into those which satisfy a property and those that do not? Equipped with a containment relation, a graph class is a special example of a partially ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set satisfies a desirable property or not. This tool can give a complete characterisation of lower ideals defined by a finite forbidden set, into those that satisfy the given property and to those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph relation is known as a hereditary graph class. We study three interrelated types of properties for hereditary graph classes: the existence of an efficient solution to an algorithmic graph problem, the boundedness of the graph parameter known as clique-width, and well-quasi-orderability by the induced subgraph relation. It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness of clique-width immediately implies an efficient solution to a wide range of algorithmic problems. This serves as one of the motivations to study clique-width. As for well-quasiorderability, we conjecture that every hereditary graph class that is well-quasi-ordered by the induced subgraph relation also has bounded clique-width. We discover the first boundary classes for several algorithmic graph problems, including the Hamiltonian cycle problem. We also give polynomial-time algorithms for the dominating induced matching problem, for some restricted graph classes. After discussing the special importance of bipartite graphs in the study of clique-width, we describe a general framework for constructing bipartite graphs of large clique-width. As a consequence, we find a new minimal class of unbounded clique-width. We prove numerous positive and negative results regarding the well-quasi-orderability of classes of bipartite graphs. This completes a characterisation of the well-quasi-orderability of all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also make considerable progress in characterising general graph classes defined by two forbidden induced subgraphs, reducing the task to a small finite number of open cases. Finally, we show that, in general, for hereditary graph classes defined by a forbidden set of bounded finite size, a similar reduction is not usually possible, but the number of boundary classes to determine well-quasi-orderability is nevertheless finite. Our results, together with the notion of boundary ideals, are also relevant for the study of other partially ordered sets in mathematics, such as permutations ordered by the pattern containment relation

Boundary properties of graphs

A set of graphs may acquire various desirable properties, if we apply suitable restrictions on the set. We investigate the following two questions: How far, exactly, must one restrict the structure of a graph to obtain a certain interesting property? What kind of tools are helpful to classify sets of graphs into those which satisfy a property and those that do not? Equipped with a containment relation, a graph class is a special example of a partially ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set satisfies a desirable property or not. This tool can give a complete characterisation of lower ideals defined by a finite forbidden set, into those that satisfy the given property and to those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph relation is known as a hereditary graph class. We study three interrelated types of properties for hereditary graph classes: the existence of an efficient solution to an algorithmic graph problem, the boundedness of the graph parameter known as clique-width, and well-quasi-orderability by the induced subgraph relation. It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness of clique-width immediately implies an efficient solution to a wide range of algorithmic problems. This serves as one of the motivations to study clique-width. As for well-quasiorderability, we conjecture that every hereditary graph class that is well-quasi-ordered by the induced subgraph relation also has bounded clique-width. We discover the first boundary classes for several algorithmic graph problems, including the Hamiltonian cycle problem. We also give polynomial-time algorithms for the dominating induced matching problem, for some restricted graph classes. After discussing the special importance of bipartite graphs in the study of clique-width, we describe a general framework for constructing bipartite graphs of large clique-width. As a consequence, we find a new minimal class of unbounded clique-width. We prove numerous positive and negative results regarding the well-quasi-orderability of classes of bipartite graphs. This completes a characterisation of the well-quasi-orderability of all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also make considerable progress in characterising general graph classes defined by two forbidden induced subgraphs, reducing the task to a small finite number of open cases. Finally, we show that, in general, for hereditary graph classes defined by a forbidden set of bounded finite size, a similar reduction is not usually possible, but the number of boundary classes to determine well-quasi-orderability is nevertheless finite. Our results, together with the notion of boundary ideals, are also relevant for the study of other partially ordered sets in mathematics, such as permutations ordered by the pattern containment relation.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research Council (EPSRC)University of Warwick. Centre for Discrete Mathematics and its Applications (DIMAP)GBUnited Kingdo