Genus Distributions of Graphs Constructed Through Amalgamations

Graphs are commonly represented as points in space connected by lines. The points in space are the vertices of the graph, and the lines joining them are the edges of the graph. A general definition of a graph is considered here, where multiple edges are allowed between two vertices and an edge is permitted to connect a vertex to itself. It is assumed that graphs are connected, i.e., any vertex in the graph is reachable from another distinct vertex either directly through an edge connecting them or by a path consisting of intermediate vertices and connecting edges. Under this visual representation, graphs can be drawn on various surfaces. The focus of my research is restricted to a class of surfaces that are characterized as compact connected orientable 2-manifolds. The drawings of graphs on surfaces that are of primary interest follow certain prescribed rules. These are called 2-cellular graph embeddings, or simply embeddings. A well-known closed formula makes it easy to enumerate the total number of 2-cellular embeddings for a given graph over all surfaces. A much harder task is to give a surface-wise breakdown of this number as a sequence of numbers that count the number of 2-cellular embeddings of a graph for each orientable surface. This sequence of numbers for a graph is known as the genus distribution of a graph. Prior research on genus distributions of graphs has primarily focused on making calculations of genus distributions for specific families of graphs. These families of graphs have often been contrived, and the methods used for finding their genus distributions have not been general enough to extend to other graph families. The research I have undertaken aims at developing and using a general method that frames the problem of calculating genus distributions of large graphs in terms of a partitioning of the genus distributions of smaller graphs. To this end, I use various operations such as edge-amalgamation, self-edge-amalgamation, and vertex-amalgamation to construct large graphs out of smaller graphs, by coupling their vertices and edges together in certain consistent ways. This method assumes that the partitioned genus distribution of the smaller graphs is known or is easily calculable by computer, for instance, by using the famous Heffter-Edmonds algorithm. As an outcome of the techniques used, I obtain general recurrences and closed-formulas that give genus distributions for infinitely many recursively specifiable graph families. I also give an easily understood method for finding non-trivial examples of distinct graphs having the same genus distribution. In addition to this, I describe an algorithm that computes the genus distributions for a family of graphs known as the 4-regular outerplanar graphs

Comparing graphs

Graphs are a well-studied mathematical concept, which has become ubiquitous to represent structured data in many application domains like computer vision, social network analysis or chem- and bioinformatics. The ever-increasing amount of data in these domains requires to efficiently organize and extract information from large graph data sets. In this context techniques for comparing graphs are fundamental, e.g., in order to obtain meaningful similarity measures between graphs. These are a prerequisite for the application of a variety of data mining algorithms to the domain of graphs. Hence, various approaches to graph comparison evolved and are wide-spread in practice. This thesis is dedicated to two different strategies for comparing graphs: maximum common subgraph problems and graph kernels. We study maximum common subgraph problems, which are based on classical graph-theoretical concepts for graph comparison and are NP-hard in the general case. We consider variants of the maximum common subgraph problem in restricted graph classes, which are highly relevant for applications in cheminformatics. We develop a polynomial-time algorithm, which allows to compute a maximum common subgraph under block and bridge preserving isomorphism in series-parallel graphs. This generalizes the problem of computing maximum common biconnected subgraphs in series-parallel graphs. We show that previous approaches to this problem, which are based on the separators represented by standard graph decompositions, fail. We introduce the concept of potential separators to overcome this issue and use them algorithmically to solve the problem in series-parallel graphs. We present algorithms with improved bounds on running time for the subclass of outerplanar graphs. Finally, we establish a sufficient condition for maximum common subgraph variants to allow derivation of graph distance metrics. This leads to polynomial-time computable graph distance metrics in restricted graph classes. This progress constitutes a step towards solving practically relevant maximum common subgraph problems in polynomial time. The second contribution of this thesis is to graph kernels, which have their origin in specific data mining algorithms. A key property of graph kernels is that they allow to consider a large (possibly infinite) number of features and can support graphs with arbitrary annotation, while being efficiently computable. The main contributions of this part of the thesis are (i) the development of novel graph kernels, which are especially designed for attributed graphs with arbitrary annotations and (ii) the systematic study of implicit and explicit mapping into a feature space for computation of graph kernels w.r.t. its impact on the running time and the ability to consider arbitrary annotations. We propose graph kernels based on bijections between subgraphs and walks of fixed length. In an experimental study we show that these approaches provide a viable alternative to known techniques, in particular for graphs with complex annotations

Asymptotic study of regular planar graphs

The central topic of this dissertation is the study of some families of regular planar graphs and maps. We are in particular interested in their asymptotic enumeration in order to understand of the associated uniform random model. In a first part, we give both an exact and an asymptotic enumeration of labelled cubic planar graphs, multigraphs and simple maps, via a recursive scheme following the iterative decompositon of a graph in smaller components of higher connecttivity. In the second part, we apply those results to the study a the uniform random labelled cubic planar graph. We compute for instance the probability of connectivity, and prove that some significant parameters are distributed following a Gaussian limit law: the numbers of cut-vertices, isthmuses, blocks, cherries, near-bricks, and triangles. In the third and last part, we develop the first recursive combinatorial scheme to enumerate 4-regular labelled planar graphs. This scheme is based on a decomposition in terms of connectivity, similar to that of cubic planar graphs, which leads to the exact enumeration of 4-regular planar graphs and simple maps.Das zentrale Thema dieser Dissertation sind Familien von regulären planaren Graphen und Karten. Insbesondere sind wir an daran interessiert, diese zu zählen und die Zusammenhänge zu deren zufälligen Gegenstücken zu erforschen. Im ersten Teil geben wir sowohl eine rekursive als auch eine asymptotische Abzählung von kubischen, planaren Graphen, Multigraphen und einfachen Karten, durch eine Dekomposition entlang deren Komponenten. Im zweiten Teil wenden wir diese Resultate auf zufällige kubische planare Graphen an. Insbesondere berechnen wir die Wahrscheinlichkeit von Zusammenhängigkeit, und beweisen das einige bedeutende Parameter normalverteilt sind: die Anzahl der cut-vertices, isthmuses, Blöcke, cherries, near-bricks und Dreiecke. Im dritten und letzten Teil entwickeln wir das erste kombinatorisches Schema, basierend auf einem Dekompositionsschema das ähnlich zu dem im Kontext von kubischen planaren Graphen ist, das zur rekursiven Abzählung von 4-regulären planaren Graphen und einfachen Karten führt

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum