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AbstractA regular edge labeling of an irreducible triangulation G uniquely defines a rectangular dual of G. Rectangular duals find applications in various areas: as floor plans of electronic chips, in architectural designs, as rectangular cartograms, or as treemaps. An irreducible triangulation can have many regular edge labelings and hence many rectangular duals. Depending on the specific application different duals might be desirable. In this thesis we consider optimization problems on regular edge labelings and show how to find optimal or near-optimal ones for various quality criteria. We evaluate our optimization methods by applying them to generate high quality rectangular cartograms. Furthermore, we show how to efficiently enumerate the regular edge labelings of an irreducible triangulation. Since the running time of the enumeration algorithm depends on the number of regular edge labelings, we also consider the maximal number of regular edge labelings an irreducible triangulation can have. We show that every irreducible triangulation with n vertices has less than O(4.6807 n ) regular edge labelings and that there are irreducible triangulations with Ω(3.0426 n ) regular edge labelings.i
