There are four prominent product graphs in graph theory: Cartesian, strong, direct, and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted $G \circ H$, for some digraphs $G$ and $H$. The paper concludes by proving a cancellation property for the lexicographic product of digraphs $G$, $H$, $A$, and $B$: if $G \circ H \cong A \circ B$ and $|V(G)| = |V(A)|$, then $G \cong A$. It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments.

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This paper concerns a specific pursuit-evasion problem with a node-located evader which we call the monk problem. First, we propose a way of verifying a strategy using a new kind of recursive systems, called EL-systems. We show how an EL-system representing a graph-instance of the problem can be represented using matrices, and we give an example of how this can be used to efficiently implement a verifier. In the later parts we propose heuristics to construct a strategy, based on a greedy algorithm. Our main focus is to minimise the number of pursuers needed, called the search number. The heuristics rely on properties of minimal stable components. We show that the minimal stable components are equivalent to the strongly connected components of a graph, and prove that the search number is equal to the maximum search number of its strongly connected components. We also establish lower and upper bounds for the search number to narrow the search space. / Denna rapport avhandlar ett specifikt pursuit-evasion problem med en hörnplacerad flykting, som vi kallar för munkproblemet. Först föreslår vi ett sätt att verifiera en strategi med en ny typ av rekursivt system, kallat EL-system. Vi visar hur ett EL-system som representerar en grafinstans av munkproblemet kan representeras med matriser, och vi ger ett exempel på hur detta kan användas för att effektivt implementera en verifikator. I de senare delarna föreslår vi heuristiker för att konstruera en strategi, baserad på giriga algoritmer. Vårt huvudfokus är att minimera antalet förföljare som krävs för att dekontaminera grafen, det så kallade söktalet. Vår heuristik förlitar sig på egenskaper för minimala stabila komponenter. Vi visar att minimala stabila komponenter är ekvivalenta med de starka komponenterna i en graf, och härleder att söktalet är lika med det maximala söktalet för grafens starka komponenter. Vi etablerar också undre och övre gränser för söktalet i syfte att minska sökintervallet.

The decisions taken during the aircraft conceptual design stage are of paramount importance since these commit up to eighty percent of the product life cycle costs. Thus in order to obtain a sound baseline which can then be passed on to the subsequent design phases, various studies ought to be carried out during this stage. These include trade-off analysis and multidisciplinary optimisation performed on computational processes assembled from hundreds of relatively simple mathematical models describing the underlying physics and other relevant characteristics of the aircraft. However, the growing complexity of aircraft design in recent years has prompted engineers to substitute the conventional algebraic equations with compiled software programs (referred to as models in this thesis) which still retain the mathematical models, but allow for a controlled expansion and manipulation of the computational system. This tendency has posed the research question of how to dynamically assemble and solve a system of non-linear models. In this context, the objective of the present research has been to develop methods which significantly increase the flexibility and efficiency with which the designer is able to operate on large scale computational multidisciplinary systems at the conceptual design stage. In order to achieve this objective a novel computational process modelling method has been developed for generating computational plans for a system of non-linear models. The computational process modelling was subdivided into variable flow modelling, decomposition and sequencing. A novel method named Incidence Matrix Method (IMM) was developed for variable flow modelling, which is the process of identifying the data flow between the models based on a given set of input variables. This method has the advantage of rapidly producing feasible variable flow models, for a system of models with multiple outputs. In addition, criteria were derived for choosing the optimal variable flow model which would lead to faster convergence of the system. Cont/d.

GraphQL is a database handling API created by Facebook, that provides an effective al-ternative to REST-style architectures. GraphQL provides the ability for a client to spec-ify exactly what data it wishes to receive. A problem with GraphQL is that the freedomof creating customized requests allows data to be included several times in the response,growing the response’s size exponentially. The thesis contributes to the field of GraphQLanalysis by studying the prevalence of simple cycles in GraphQL schemas. We have im-plemented a locally-run tool and webtool using Tarjan’s and Johnson’s algorithms, thatparses the schemas, creates a directed graph and enumerates all simple cycles in the graph.A collection of schemas was analysed with the tool to collect empirical data. It was foundthat 39.73 % of the total 2094 schemas contained at least one simple cycle, with the averagenumber of cycles per schema being 4. The runtime was found to be on average 11 mil-liseconds, most of which consisted of the time for parsing the schemas. It was found that44 out of the considered schemas could not be enumerated due to containing a staggeringamount of simple cycles. It can be concluded that it is possible to test schemas for cyclicityand enumerate all simple cycles in a given schema efficiently.

The decisions taken during the aircraft conceptual design stage are of paramount importance since these commit up to eighty percent of the product life cycle costs. Thus in order to obtain a sound baseline which can then be passed on to the subsequent design phases, various studies ought to be carried out during this stage. These include trade-off analysis and multidisciplinary optimisation performed on computational processes assembled from hundreds of relatively simple mathematical models describing the underlying physics and other relevant characteristics of the aircraft. However, the growing complexity of aircraft design in recent years has prompted engineers to substitute the conventional algebraic equations with compiled software programs (referred to as models in this thesis) which still retain the mathematical models, but allow for a controlled expansion and manipulation of the computational system. This tendency has posed the research question of how to dynamically assemble and solve a system of non-linear models. In this context, the objective of the present research has been to develop methods which significantly increase the flexibility and efficiency with which the designer is able to operate on large scale computational multidisciplinary systems at the conceptual design stage. In order to achieve this objective a novel computational process modelling method has been developed for generating computational plans for a system of non-linear models. The computational process modelling was subdivided into variable flow modelling, decomposition and sequencing. A novel method named Incidence Matrix Method (IMM) was developed for variable flow modelling, which is the process of identifying the data flow between the models based on a given set of input variables. This method has the advantage of rapidly producing feasible variable flow models, for a system of models with multiple outputs. In addition, criteria were derived for choosing the optimal variable flow model which would lead to faster convergence of the system. Cont/d.

The IEEE 802.11a protocol is part of the IEEE 802 protocols for implementing WLAN Wi- Fi computer communications in various frequencies. These protocols find applications worldwide, covering a wide range of devices like mobile phones, computers, laptops, household appliances, etc. Since wireless communication is being used, data that is transmitted is susceptible to noise. As a means to recover from noise, the data transmitted is encoded using convolutional encoding and correspondingly decoded on the receiver side. The decoder used is the Viterbi decoder, in the PHY layer of the protocol. This thesis investigates soft-decision Viterbi decoder implementations that meet the requirements of the IEEE 802.11a protocol. It aims to implement a rate-flexible design as a coarse grain re-configurable architecture using the SiLago framework. SiLago is a modular approach towards ASIC design. Components are designed as hardened blocks, which means they are synthesised and pre-verified. Each block is also abuttable like LEGO blocks, which allows users to connect compatible blocks and make designs specific to their requirements, while getting performance similar to that of traditional ASICs. This approach significantly reduces the design costs, as verification is a one-time task. The thesis discusses the strongly connected trellis Viterbi decoding algorithm and proposes a design for a soft decision Viterbi decoder. The proposed design meets the throughput requirements of the communication protocol and it can be reconfigured to work for 45 different code rates, with programmable soft decision width and parallelism. The algorithm used is compared against MATLAB for its BER performance. Results from RTL simulations, advantages and disadvantages of the proposed design are discussed. Recommendations for future improvements are also made. / IEEE 802.11a-protokollet är en del av IEEE 802-protokollen för att implementera WLAN Wi-Fi-datorkommunikation i olika frekvenser. Dessa protokoll används i applikationer över hela världen som täcker ett brett spektrum av produkter som mobiltelefoner, datorer, bärbara datorer, hushållsapparater etc. Eftersom trådlös kommunikation används är data som överförs känslig för brus. Som ett medel för att återhämta sig från brus kodas överförd data med hjälp av faltningskodning och avkodas på motsvarande sätt på mottagarsidan. Den avkodare som används är Viterbi-avkodaren, i PHY-skiktet i protokollet. Denna avhandling undersöker mjuka beslut Viterbi avkodarimplementeringar som uppfyller kraven i IEEE 802.11a protokollet. Det syftar till att implementera en hastighetsflexibel design som en grovkornig konfigurerbar arkitektur som använder SiLago ramverket. SiLago är ett modulärt synsätt på ASIC design. Komponenterna är utformade som härda block, vilket innebär att de är syntetiserade och förverifierade. Varje block kan också kopplas ihop, som LEGO block, vilket gör det möjligt för användare att ansluta kompatibla block och göra designer som är specifika för deras krav, samtidigt som de får prestanda som liknar traditionella ASICs. Detta tillvägagångssätt minskar designkostnaderna avsevärt, eftersom verifiering är en engångsuppgift. Avhandlingen diskuterar den starkt kopplade trellis Viterbi-avkodningsalgoritmen och föreslår en design för en mjuk Viterbi-avkodare. Den föreslagna designen uppfyller kommunikationsprotokollets genomströmningskrav och den kan konfigureras om för att fungera för 45 olika kodhastigheter, med programmerbar mjuk beslutsbredd och parallellitet. Algoritmen som används jämförs mot MATLAB för dess BER-prestanda. Resultat från RTL-simuleringar, fördelar och nackdelar med den föreslagna designen diskuteras. Rekommendationer för framtida förbättringar görs också.

Cette thèse présente une méthodologie d’ordonnancement cyclique robuste appliquée à la gestion des conteneurs dans les ports maritimes de taille moyenne. Ces derniers sont sujet constamment à des variations des conditions des terminaux, la visibilité réduite sur des évènements futurs ne permet pas de proposer une planification précise des tâches à accomplir. L’ordonnancement cyclique robuste peut jouer un rôle primordial. Il permettra non seulement de proposer un ordonnancement prédictif pour le transport des conteneurs, mais aussi, il proposera également une planification robuste permettant d’éliminer les perturbations éventuelles en temps réel. Dans ce travail nous utilisons les Véhicules Intelligents Automatisés (AIV) pour transporter les conteneurs et nous modélisons les procédures de transit de ces derniers par des graphes d’évènements P-temporels fortement connexes (PTSCEG). Avant l’arrivée d’un porte conteneur au port, un plan (planning) de transport des conteneurs est proposé en un temps court par la programmation linéaire mixte (MIP). Des algorithmes polynomiaux de calcul de robustesse permettent de calculer sur les différents nœuds du système les marges de robustesse. Une fois le navire à quai, l’ordonnancement cyclique robuste est appliqué. Lorsqu’une perturbation est observée (localisée) dans le système, une comparaison avec la marge de robustesse connue est effectuée. Si cette perturbation est incluse dans la marge de robustesse, l’algorithme robuste est utilisé pour éliminer ces perturbations en quelques cycles. Dans le cas où la perturbation est trop importante, la méthode MIP est utilisée pour calculer un nouvel ordonnancement cyclique en un temps réduit / This PhD thesis is dedicated to propose a robust cyclic scheduling methodology applied to container management of medium sized seaport which faces ever changing terminal conditions and the limited predictability of future events and their timing. The robust cyclic scheduling can be seen not just a predictable scheduling to compute a container transportation schedule, but also a reactive scheduling to eliminate the disturbances in real time. In this work, the automated intelligent vehicles (AIV) are used to transport the containers, and the P-time strongly connected event graph (PTSCEG) is used as a graphical tool to model the container transit procedures. Before the arrival of the container vessel, a cyclic container transit schedule can be given by the mixed integer programming (MIP) method in short time. The robustness margins on the nodes of the system can be computed by robustness algorithms in polynomial computing time. After the stevedoring begins, this robust cyclic schedule is used. When a disturbance is observed in system, it should be compared with the known robustness margin. If the disturbance belongs to the robustness margin, the robustness algorithm is used to eliminate the disturbance in a few cycle times. If not, the MIP method is used to compute a new cyclic schedule in short time