Bielefeld, University, Diss., 2001.

Thesis (Ph. D.)--Ohio State University, 2008. / Title from first page of PDF file. Includes bibliographical references (p. 96-97).

Indiana University-Purdue University Indianapolis (IUPUI) / A Multigraph is a set of graphs with a common set of nodes but different sets of edges. Multigraph visualization has not received much attention so far. In this thesis, I will introduce an interactive application in brain network data analysis that has a strong need for multigraph visualization. For this application, multigraph was used to represent brain connectome networks of multiple human subjects. A volumetric data set was constructed from the matrix representation of the multigraph. A volume visualization tool was then developed to assist the user to interactively and iteratively detect network features that may contribute to certain neurological conditions. I applied this technique to a brain connectome dataset for feature detection in the classification of Alzheimer's Disease (AD) patients. Preliminary results showed significant improvements when interactively selected features are used.

Konstanz, Univ., Diplomarb., 2004.

Les problèmes de tournées de véhicules (VRPs) ont fait l’objet de plusieurs travaux de recherche depuis maintenant plus de 50 ans. La plupart des approches trouvées dans la littérature s’appuient sur un graphe complet ou un nœud est introduit pour tout point d’intérêt du réseau routier (typiquement les clients et le dépôt). Cette modélisation est, implicitement, basée sur l’hypothèse que le meilleur chemin entre toute paire de points du réseau routier est bien défini. Cependant, cette hypothèse n’est pas toujours valide dans de nombreuses situations. Souvent, plus d’informations sont nécessaires pour modéliser et résoudre correctement le problème. Nous commençons par examiner ces situations et définir les limites de la modélisation basée sur un graphe complet. Nous proposons un état de l’art des travaux qui examinent ces limites et qui traitent des VRPs en considérant plus d’informations issues du réseau routier. Nous décrivons les approches alternatives proposées, à savoir la modélisation utilisant un multi-graphe et celle utilisant la résolution directe sur un graph représentant le réseau routier. Dans une seconde étude, nous nous intéressons à l’approche basée sur la construction d’un multi-graphe. Nous proposons, d’abord, un algorithme qui permet de calculer d’une manière efficace la représentation par multi-graph du réseau routier. Puis, nous présentons une analyse empirique sur l’impact de cette modélisation sur la qualité de la solution. Pour ce faire, nous considérons le problème classique VRPTW comme un problème de pilote. Par la suite, nous développons une méthode heuristique efficace afin de résoudre le VRPTW basée sur une représentation par un multi-graphe.Dans une troisième étape, nous nous concentrons sur l’approche basée sur la résolution directe du problème sur un graphe représentant le réseau routier. Nous développons un algorithme de type branch-and-price pour la résolution de cette variante du problème. Une étude expérimentale est, ensuite, menée afin d’évaluer l’efficacité relative des deux approches. Enfin, nous étudions les problèmes de tournées de véhicules dans lesquels les temps de parcours varient au cours de la journée. Nous proposons un algorithme de type branch-and-price afin de résoudre le problème avec des fenêtres de temps directement sur le graphe représentant le réseau routier. Une analyse empirique sur l’impact de l’approche proposée sur la qualité de la solution est proposée. / Vehicle routing problems (VRPs) have drawn many researchers’ attention for more than fifty years. Most approaches found in the literature are, implicitly, based on the key assumption that the best path between each two points of interest in the road network (customers, depot, etc.) can be easily defined. Thus, the problem is tackled using the so-called customer-based graph, a complete graph representation of the road network. In many situations, such a graph may fail to accurately represent the original road network and more information are needed to address correctly the routing problem.We first examine these situations and point out the limits of the traditional customer-based graph. We propose a survey on works investigating vehicle routing problems by considering more information from the road network. We outline the proposed alternative approaches, namely the multigraph representation and the road network approach.Then, we are interested in the multigraph approach. We propose an algorithm that efficiently compute the multigraph representation for large sized road networks. We present an empirical analysis on the impact of the multigraph representation on the solution quality for the VPR with time windows (VRPTW) when several attributes are defined on road segments. Then, we develop an efficient heuristic method for the multigraph-based VRPTW.Next, we investigate the road network approach. We develop a complete branch-and-price algorithm that can solve the VRPTW directly on the original road network. We evaluate the relative efficiency of the two approaches through an extensive computational study.Finally, we are interested in problems where travel times vary over the time of the day, called time dependent vehicle routing problems (TDVRPs). We develop a branch-and-price algorithm that solves the TDVRP with time windows directly on the road network and we analyze the impact of the proposed approach on the solution quality.

The crossing number of a graph 𝐺 is the minimum number of crossings in any drawing of 𝐺 in the plane. The rectilinear crossing number of 𝐺 is the minimum number of crossings in any straight-line drawing of 𝐺. The Fáry-Wagner theorem states that planar graphs have rectilinear crossing number zero. By Wagner’s theorem, that is equivalent to stating that every graph that excludes 𝐾₅ and 𝐾₃,₃ as minors has rectilinear crossing number 0. We are interested in discovering other proper minor-closed families of graphs which admit strong upper bounds on their rectilinear crossing numbers. Unfortunately, it is known that the crossing number of 𝐾₃,ₙ with 𝑛 ≥ 1, which excludes 𝐾₅ as a minor, is quadratic in 𝑛, more specifically Ω(𝑛²). Since every 𝑛-vertex graph in a proper minor closed family has O(𝑛) edges, the rectilinear crossing number of all such graphs is trivially O(𝑛²). In fact, it is not hard to argue that O(𝑛) bound on the crossing number is the best one can hope for general enough proper minor-closed families of graphs and that to achieve O(𝑛) bounds, one has to both exclude a minor and bound the maximum degree of the graphs in the family. In this thesis, we do that for bounded degree graphs that exclude a single-crossing graph as a minor. A single-crossing graph is a graph whose crossing number is at most one. The main result of this thesis states that every graph 𝐺 that does not contain a single-crossing graph as a minor has a rectilinear crossing number O(∆𝑛), where 𝐺 has 𝑛 vertices and maximum degree ∆. This dependence on 𝑛 and ∆ is best possible. Note that each planar graph is a single-crossing graph, as is the complete graph 𝐾₅ and the complete bipartite graph 𝐾₃,₃. Thus, the result applies to 𝐾₅-minor-free graphs, 𝐾₃,₃-minor free graphs, as well as to bounded treewidth graphs. In the case of bounded treewidth graphs, the result improves on the previous best known bound of O(∆² · 𝑛) by Wood and Telle [New York Journal of Mathematics, 2007]. In the case of 𝐾₃,₃-minor free graphs, our result generalizes the result of Dujmovic, Kawarabayashi, Mohar and Wood [SCG 2008].

No description available.

This thesis is concerned with multigraphs and their complexity which is defined and quantified by the distribution of edge multiplicities. Two random multigraph models are considered.  The first model is random stub matching (RSM) where the edges are formed by randomly coupling pairs of stubs according to a fixed stub multiplicity sequence. The second model is obtained by independent edge assignments (IEA) according to a common probability distribution over the edge sites. Two different methods for obtaining an approximate IEA model from an RSM model are also presented. In Paper I, multigraphs are analyzed with respect to structure and complexity by using entropy and joint information. The main results include formulae for numbers of graphs of different kinds and their complexity. The local and global structure of multigraphs under RSM are analyzed in Paper II. The distribution of multigraphs under RSM is shown to depend on a single complexity statistic. The distributions under RSM and IEA are used for calculations of moments and entropies, and for comparisons by information divergence. The main results include new formulae for local edge probabilities and probability approximation for simplicity of an RSM multigraph. In Paper III, statistical tests of a simple or composite IEA hypothesis are performed using goodness-of-fit measures. The results indicate that even for very small number of edges, the null distributions of the test statistics under IEA have distributions that are  well approximated by their asymptotic χ2-distributions. Paper IV contains the multigraph algorithms that are used for numerical calculations in Papers I-III.

No description available.