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

Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs

In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs. Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber. Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses. Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture

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

Network-centric methods for heterogeneous multiagent systems

We present tools for a network topology based characterization of heterogeneity in multiagent systems, thereby providing a framework for the analysis and design of heterogeneous multiagent networks from a network structure view-point. In heterogeneous networks, agents with a diverse set of resources coordinate with each other. Coordination among different agents and the structure of the underlying network topology have significant impacts on the overall behavior and functionality of the system. Using constructs from graph theory, a qualitative as well as a quantitative analysis is performed to examine an inter-relationship between the network topology and the distribution of agents with various capabilities in heterogeneous networks. Our goal is to allow agents maximally exploit heterogeneous resources available within the network through local interactions, thus exploring a promise heterogeneous networks hold to accomplish complicated tasks by leveraging upon the assorted capabilities of agents. For a reliable operations of such systems, the issue of security against intrusions and malicious agents is also addressed. We provide a scheme to secure a network against a sequence of intruder attacks through a set of heterogeneous guards. Moreover, robustness of networked systems against noise corruption and structural changes in the underlying network topology is also examined.Ph.D

Doctor of Philosophy

dissertationFunctional magnetic resonance imaging (fMRI) measures the change of oxygen consumption level in the blood vessels of the human brain, hence indirectly detecting the neuronal activity. Resting-state fMRI (rs-fMRI) is used to identify the intrinsic functional patterns of the brain when there is no external stimulus. Accurate estimation of intrinsic activity is important for understanding the functional organization and dynamics of the brain, as well as differences in the functional networks of patients with mental disorders. This dissertation aims to robustly estimate the functional connectivities and networks of the human brain using rs-fMRI data of multiple subjects. We use Markov random field (MRF), an undirected graphical model to represent the statistical dependency among the functional network variables. Graphical models describe multivariate probability distributions that can be factorized and represented by a graph. By defining the nodes and the edges along with their weights according to our assumptions, we build soft constraints into the graph structure as prior information. We explore various approximate optimization methods including variational Bayesian, graph cuts, and Markov chain Monte Carlo sampling (MCMC). We develop the random field models to solve three related problems. In the first problem, the goal is to detect the pairwise connectivity between gray matter voxels in a rs-fMRI dataset of the single subject. We define a six-dimensional graph to represent our prior information that two voxels are more likely to be connected if their spatial neighbors are connected. The posterior mean of the connectivity variables are estimated by variational inference, also known as mean field theory in statistical physics. The proposed method proves to outperform the standard spatial smoothing and is able to detect finer patterns of brain activity. Our second work aims to identify multiple functional systems. We define a Potts model, a special case of MRF, on the network label variables, and define von Mises-Fisher distribution on the normalized fMRI signal. The inference is significantly more difficult than the binary classification in the previous problem. We use MCMC to draw samples from the posterior distribution of network labels. In the third application, we extend the graphical model to the multiple subject scenario. By building a graph including the network labels of both a group map and the subject label maps, we define a hierarchical model that has richer structure than the flat single-subject model, and captures the shared patterns as well as the variation among the subjects. All three solutions are data-driven Bayesian methods, which estimate model parameters from the data. The experiments show that by the regularization of MRF, the functional network maps we estimate are more accurate and more consistent across multiple sessions

Colouring on hereditary graph classes

The graph colouring problems ask if one can assign a colour from a palette of colour to every vertex of a graph so that any two adjacent vertices receive different colours. We call the resulting problem k-Colourability if the palette is of fixed size k, and Chromatic Number if the goal is to minimize the size of the palette. One of the earliest NP-completeness results states that 3-Colourability is NP-complete. Thereafter, numerous studies have been devoted to the graph colouring problems on special graph classes. For a fixed set of graphs H we denote F orb(H) by the set of graphs that exclude any graph H ∈ H as an induced subgraph. In this thesis, we explore the computational complexity of graph colouring problems on F orb(H) for different sets of H.In the first part of this thesis, we study k-Colourability on classes F orb(H) when H contains at most two graphs. We show that 4-Colourability and 5-Colourability are NPcomplete on F orb({P7}) and F orb({P6}), respectively, where Pt denotes a path of order t. These results leave open, for k ≥ 4, only the complexity of k-Colourability on F orb({Pt}) for k = 4 and t = 6. Secondly, we refine our NP-completeness results on k-Colourability to classes F orb({Cs, Pt}), where Cs denotes a cycle of length s. We prove new NP-completeness results for different combinations of values of k, s and t. Furthermore, we consider two common variants of the k-colouring problem, namely the list k-colouring problem and the pre-colouring extension of k-colouring problem. We show that in most cases these problems are also NP-complete on the class F orb({Cs, Pt}). Thirdly, we prove that the set of forbidden induced subgraph that characterizes the class of k-colourable (C4, P6)-free graphs is of finite size. For k ∈ {3, 4}, we obtain an explicit list of forbidden induced subgraphs and the first polynomial certifying algorithms for k-Colourability on F orb({C4, P6}).We also discuss one particular class F orb(H) when the size of H is infinite. We consider the intersection class of F orb({C4, C6, . . .}) and F orb(caps), where a cap is a graph obtained from an induced cycle by adding an additional vertex and making it adjacent to two adjacent vertices on the cycle. Our main result is a polynomial time 3/2-approximation algorithm for Chromatic Number on this class

Minimal circular-imperfect graphs of large clique number and large independence number

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A lightweight, graph-theoretic model of class-based similarity to support object-oriented code reuse.

The work presented in this thesis is principally concerned with the development of a method and set of tools designed to support the identification of class-based similarity in collections of object-oriented code. Attention is focused on enhancing the potential for software reuse in situations where a reuse process is either absent or informal, and the characteristics of the organisation are unsuitable, or resources unavailable, to promote and sustain a systematic approach to reuse. The approach builds on the definition of a formal, attributed, relational model that captures the inherent structure of class-based, object-oriented code. Based on code-level analysis, it relies solely on the structural characteristics of the code and the peculiarly object-oriented features of the class as an organising principle: classes, those entities comprising a class, and the intra and inter-class relationships existing between them, are significant factors in defining a two-phase similarity measure as a basis for the comparison process. Established graph-theoretic techniques are adapted and applied via this model to the problem of determining similarity between classes. This thesis illustrates a successful transfer of techniques from the domains of molecular chemistry and computer vision. Both domains provide an existing template for the analysis and comparison of structures as graphs. The inspiration for representing classes as attributed relational graphs, and the application of graph-theoretic techniques and algorithms to their comparison, arose out of a well-founded intuition that a common basis in graph-theory was sufficient to enable a reasonable transfer of these techniques to the problem of determining similarity in object-oriented code. The practical application of this work relates to the identification and indexing of instances of recurring, class-based, common structure present in established and evolving collections of object-oriented code. A classification so generated additionally provides a framework for class-based matching over an existing code-base, both from the perspective of newly introduced classes, and search "templates" provided by those incomplete, iteratively constructed and refined classes associated with current and on-going development. The tools and techniques developed here provide support for enabling and improving shared awareness of reuse opportunity, based on analysing structural similarity in past and ongoing development, tools and techniques that can in turn be seen as part of a process of domain analysis, capable of stimulating the evolution of a systematic reuse ethic