Structural and Algorithmic Aspects of Chordal Graph Embeddings

In this thesis, we consider graph embeddings into the class of chordal graphs (so-called triangulations), by which, for greater classes of graphs, we obtain new structural results and efficient algorithms for computing various graph parameters. The minimal separators in a graph play an important role for graph triangulations. For this reason, we study different approaches to model the underlying structure that links the minimal separators in arbitrary graphs. In this context, we introduce the separator graph and the structure graph of graphs. One of our main results states that there is a 1-1 correspondence between the inclusion-minimal triangulations of a graph and the maximal cliques of its separator graph. Because of that, we can reduce some graph problems to weighted clique problems in the corresponding separator graph. This enables us to solve the treewidth and the minimum fill-in problem on a generalization of both the interval and the permutation graphs, namely the d-trapezoid g..