Minimal Separators in Intersection Graphs

In this report, we present the intersection graphs and describe the role they play in computations of treewidth. We introduce minimal triangulations, minimal separators, potential maximal cliques and blocks as the main tools for exact computation and approximation of that important graph parameter. The first chapter is an introduction to the intersection graph theory. We give all the necessary definitions and present in detail our approach to the known results, that are important for understanding intersection graphs and treewidth computations. The second chapter presents the notion of treewidth of a graph and gives the details of many structural properties related to this graph parameter. The most important is the result of Parra, giving a characterization of minimal triangulations, using minimal separators of a graph. The third chapter focuses on the notions of potential maximal clique and block of a graph. They are characterized in detail. Later on, they are put at work in exact computations and approximations of treewidth. The main results of this chapter come from Bouchitté and Todinca. The fourth chapter brings discussion on the possibilities of further research. We focus on minimal separators and analyze the cases, in which their number is or is not polynomially bounded in the number of vertices of the considered graph. These investigations take place on the ground of two intersection graph classes: the polygon-circle graphs and the interval-filament graphs.