Structural and Algorithmic Aspects of Chordal Graph Embeddings

In this thesis, we consider graph embeddings into the class o f chordal graphs (so-calledtriangulations), by which, for greater classes of graphs, we obtain new structural results and efficient algorithms for computing v arious graph parameters. The minimal separators in a graph play an important role for g raph triangulations. For this reason, we study different approaches to m odel the underlying structure that links the minimal separators in arbitrary gr aphs. In this context, we introduce theseparator graphand thestructure graphof graphs. One of our main results states that there is a 1-1 correspondence between th inclusion-minimal triangulations of a graph and the maximal cliques of its separat or graph. Because of that, we can reduce some graph problems to weighted clique pr oblems in the corresponding separator graph. This enables us to solve the treew idth and the minimum fill-in problem on a generalization of both the interval and t he permutation graphs, namely thed-trapezoid graphs. The structure graph models the lattice s ructure of minimal separators, which is well-known for fixed pairs of ve rtices. This approach leads to new characterizations and a polynomial time algori thm for computing the treewidth of graphs without asteroidal triples. Subsequen tly, we study the bandwidth and the ranking problem on subclasses of graphs withou t asteroidal triples. Eventually, for a generalization of the tolerance graphs, w e show the relation of the treewidth and pathwidth and solve the minimum fill-in proble m.