Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments

A digraph such that every proper induced subdigraph has a kernel is said to be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI for short) resp.) if the digraph has a kernel (does not have a kernel resp.). The unique CKI-tournament is $\overrightarrow{C}_3$ and the unique KP-tournaments are the transitive tournaments, however bipartite tournaments are KP. In this paper we characterize the CKI- and KP-digraphs for the following families of digraphs: locally in-/out-semicomplete, asymmetric arc-locally in-/out-semicomplete, asymmetric $3$-quasi-transitive and asymmetric $3$-anti-quasi-transitive $TT_3$-free and we state that the problem of determining whether a digraph of one of these families is CKI is polynomial, giving a solution to a problem closely related to the following conjecture posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for locally in-semicomplete digraphs.Comment: 13 pages and 5 figure

Combinatorial and graph theoretical aspects of two-edge connected reliability

Die Untersuchung von Zuverlässigkeitsnetzwerken geht bis zum frühen 20. Jahrhundert zurück. Diese Arbeit beschäftigt sich hauptsächlich mit der Zweifach-Kantenzusammenhangswahrscheinlichkeit. Zuerst werden einfache Algorithmen, die aber für allgemeine Graphen nicht effizient sind, gezeigt, zusammen mit Reduktionen. Weiterhin werden Charakterisierungen von Kanten bezogen auf Wegemengen gezeigt. Neue strukturelle Bedingungen für diese werden vorgestellt. Neue Ergebnisse liegen ebenfalls für Graphen hoher Dichte und Symmetrie vor, genauer für vollständige und vollständig bipartite Graphen. Naturgemäß sind Graphen von geringer Dichte hier einfacher in der Untersuchung. Die Arbeit zeigt Ergebnisse für Kreise, Räder und Leiterstrukturen. Graphen mit beschränkter Weg- beziehungsweise Baumweite haben polynomiale Algorithmen und in Spezialfällen einfache Formeln, die ebenfalls vorgestellt werden. Der abschließende Teil beschäftigt sich mit Schranken und Approximationen

Hybrid dynamics in large-scale logistics networks

We study stability properties of interconnected hybrid systems with application to large-scale logistics networks. Hybrid systems are dynamical systems that combine two types of dynamics: continuous and discrete. Such behaviour occurs in wide range of applications. Logistics networks are one of such applications, where the continuous dynamics occurs in the production and processing of material and the discrete one in the picking up and delivering of material. Stability of logistics networks characterizes their robustness to the changes occurring in the network. However, the hybrid dynamics and the large size of the network lead to complexity of the stability analysis. In this thesis we show how the behaviour of a logistics networks can be described by interconnected hybrid systems. Then we recall the small gain conditions used in the stability analysis of continuous and discrete systems and extend them to establish input-to-state stability (ISS) of interconnected hybrid systems. We give the mixed small gain condition in a matrix form, where one matrix describes the interconnection structure of the system and the other diagonal matrix takes into account whether ISS condition for a subsystem is formulated in the maximization or the summation sense. The small gain condition is sufficient for ISS of an interconnected hybrid system and can be applied to an interconnection of an arbitrary finite number of ISS subsystems. We also show an application of this condition to particular subclasses of hybrid systems: impulsive systems, comparison systems and the systems with stability of only a part of the state. Furthermore, we introduce an approach for structure-preserving model reduction for large-scale logistics networks. This approach supposes to aggregate typical interconnection patterns (motifs) of the network graph. Such reduction allows to decrease the number of computations needed to verify the small gain condition

Topology of complex networks: models and analysis

There is a large variety of real-world phenomena that can be modelled and analysed as networks. Part of this variety is reflected in the diversity of network classes that are used to model these phenomena. However, the differences between network classes are not always taken into account in their analysis. This thesis carefully addresses how to deal with distinct classes of networks in two different contexts. First, the well-known switching model has been used to randomise different classes of networks, and is typically referred to as the switching model. We argue that really we should be talking about a family of switching models. Ignoring the distinction between the switching model with respect to different network classes has lead to biased sampling. Given that the most common use of the switching model is as a null-model, it is critical that it samples without bias. We provide a comprehensive analysis of the switching model with respect to nine classes of networks and prove under which conditions sampling is unbiased for each class. Recently the Curveball algorithm was introduced as a faster approach to network randomisation. We prove that the Curveball algorithm samples without bias; a position that was previously implied, but unproven. Furthermore, we show that the Curveball algorithm provides a flexible framework for network randomisation by introducing five variations with respect to different network classes. We compare the switching models and Curveball algorithms to several other random network models. As a result of our findings, we recommend using the configuration model for multi-graphs with self-loops, the Curveball algorithm for networks without multiple edges or without self-loops and the ordered switching model for directed acyclic networks. Second, we extend the theory of motif analysis to directed acyclic networks. We establish experimentally that there is no difference in the motifs detected by existing motif analysis methods and our customised method. However, we show that there are differences in the detected anti-motifs. Hence, we recommend taking into account the acyclic nature of directed acyclic networks. Network science is a young and active field of research. Most existing network measures originate in statistical mechanics and focus on statistics of local network properties. Such statistics have proven very useful. However, they do not capture the complete structure of a network. In this thesis we present experimental results on two novel network analysis techniques. First, at the local level, we show that the neighbourhood of a node is highly distinctive and has the potential to match unidentified entities across networks. Our motivation is the identification of individuals across dark social networks hidden in recorded networks. Second, we present results of the application of persistent homology to network analysis. This recently introduced technique from topological data analysis offers a new perspective on networks: it describes the mesoscopic structure of a network. Finally, we used persistent homology for a classification problem in pharmaceutical science. This is a novel application of persistent homology. Our analysis shows that this is a promising approach for the classification of lipid formulations

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Global Constraint Catalog, 2nd Edition (revision a)

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms

Global Constraint Catalog, 2nd Edition

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms

Proceedings of the Fifth European Workshop on Probabilistic Graphical Models

Peer reviewe