It is a well known fact in graph theory that in a connected graph any two longest paths must have a vertex in common. In this paper we will explore what happens when we look at k - connected graphs, leading us to make a conjecture about the intersection of any two longest paths. We then look at cycles and look at what would be needed to improve on a result by Chen, Faudree and Gould about the intersection of two longest cycles.

Rectilinear crossing number of a graph is the number of crossing edges in a drawing with all straight line edges. The problem of drawing an n-vertex complete graph such that its rectilinear crossing number is minimum is known to be an NP-Hard problem. In this thesis, we present a heuristic that attempts to achieve the theoretical lower bound value of the rectilinear crossing number of a n+1 vertex complete graph from that of n vertices. Our algorithm accepts an optimal or near-optimal rectilinear drawing of Kn graph as input and tries to place a new node such that the crossing number is minimized. Based on prior optimal drawings of Kn, we make an empirical observation that the optimal drawings are triangular in shape. The proposed heuristic has three steps: (1) Given the optimal or near-optimal drawing of Kn, the outer triangle is determined; (2) A set of candidate positions for the (n+1)th node is determined by ensuring none of them are collinear with two or more nodes in the graph; and (3) the best drawing with least rectilinear crossing number is chosen based on the drawings corresponding to the candidate position. A loose bound on the worst-case time complexity of the proposed algorithm is O(n7). The heuristic is not guaranteed to yield optimal solution as the search space is constrained by the input graph. In our experimental results, we obtained optimal results for complete graphs of up to n=27.

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在這篇論文中，我們根據局部連通和局部補連通性質將圖分類，計算在 Harary 圖裡大小為 2k - 1 和 2k 邊切集的個數，和證明當圖形有最大的最小邊度數和最小點度數差，一些關於度數為 1 的點個數性質。 / In this thesis, we classify some graphs into locally coconnected graphs or locally connected graphs, compute the number of its edge cuts of size 2k - 1 and 2k in a Harary graph, and show some properties of the number of vertices of degree 1 when the graph has the maximum difference of minimum edge degree and minimum vertex degree.

Relational data is characterised by the rich structure it encodes in the dependencies between the individual entities of a given domain. Statistical Relational Learning (SRL) combines first-order logic and probability to learn and reason over relational domains by creating parametric probability distributions over relational structures. SRL models can succinctly represent the complex dependencies in relational data and admit learning and inference under uncertainty. However, these models are significantly limited when it comes to the tractability of learning and inference. This limitation emerges from the intractability of Weighted First Order Model Counting (WFOMC), as both learning and inference in SRL models can be reduced to instances of WFOMC. Hence, fragments of first-order logic that admit tractable WFOMC, widely known as domain-liftable, can significantly advance the practicality and efficiency of SRL models. Recent works have uncovered another limitation of SRL models, i.e., they lead to unintuitive behaviours when used across varying domain sizes, violating fundamental consistency conditions expected of sound probabilistic models. Such inconsistencies also mean that conventional machine learning techniques, like training with batched data, cannot be soundly used for SRL models. In this thesis, we contribute to both the tractability and consistency of probabilistic inference in SRL models. We first expand the class of domain-liftable fragments with counting quantifiers and cardinality constraints. Unlike the algorithmic approaches proposed in the literature, we present a uniform combinatorial approach, admitting analytical combinatorial formulas for WFOMC. Our approach motivates a new family of weight functions allowing us to express a larger class of probability distributions without losing domain-liftability. We further expand the class of domain-liftable fragments with constraints inexpressible in first-order logic, namely acyclicity and connectivity constraints. Finally, we present a complete characterization for a statistically consistent (a.k.a projective) models in the two-variable fragment of a widely used class of SRL models, namely Markov Logic Networks.

Pour faire face aux mutations du paysage énergétique, les réseaux de distribution d'électricité sont soumis à des exigences de fonctionnement avec des indices de fiabilité à garantir. Dans les années à venir, de grands investissements sont prévus pour la construction des réseaux électriques flexibles, cohérents et efficaces, basés sur de nouvelles architectures et des solutions techniques innovantes, adaptatifs à l'essor des énergies renouvelables. En prenant en compte ces besoins industriels sur le développement des réseaux de distribution du futur, nous proposons, dans cette thèse, une approche reposant sur la théorie des graphes et l'optimisation combinatoire pour la conception de nouvelles architectures pour les réseaux de distribution. Notre démarche consiste à étudier le problème général de recherche d'une architecture optimale qui respecte l'ensemble de contraintes topologiques (redondance) et électrotechniques (courant maximal, plan de tension) selon des critères d'optimisation bien précis : minimisation du coût d'exploitation (OPEX) et minimisation de l'investissement (CAPEX). Ainsi donc, les deux familles des problèmes combinatoires (et leurs relaxations) ont été explorées pour proposer des résolutions efficaces (exactes ou approchées) du problème de planification des réseaux de distribution en utilisant une formulation adaptée. Nous nous sommes intéressés particulièrement aux graphes 2-connexes et au problème de flot arborescent avec pertes quadratiques minimales. Les résultats comparatifs de tests sur les instances de réseaux (fictifs et réels) pour les méthodes proposées ont été présentés. / To cope with the changes in the energy landscape, electrical distribution networks are submitted to operational requirements in order to guarantee reliability indices. In the coming years, big investments are planned for the construction of flexible, consistent and effective electrical networks, based on the new architectures, innovative technical solutions and in response to the development of renewable energy. Taking into account the industrial needs of the development of future distribution networks, we propose in this thesis an approach based on the graph theory and combinatorial optimization for the design of new architectures for distribution networks. Our approach is to study the general problem of finding an optimal architecture which respects a set of topological (redundancy) and electrical (maximum current, voltage plan) constraints according to precise optimization criteria: minimization of operating cost (OPEX) and minimization of investment (CAPEX). Thus, the two families of combinatorial problems (and their relaxations) were explored to propose effective resolutions (exact or approximate) of the distribution network planning problem using an adapted formulation. We are particularly interested in 2-connected graphs and the arborescent flow problem with minimum quadratic losses. The comparative results of tests on the network instances (fictional and real) for the proposed methods were presented.