Drawing planar graphs with prescribed face areas

Abstract

This thesis deals with planar drawings of planar graphs such that each interior face has a prescribed area. Our work is divided into two main sections. The rst one deals with straight-line drawings and the second one with orthogonal drawings. For straight-line drawings, it was known that such drawings exist for all planar graphs with maximum degree 3. We show here that such drawings exist for all planar partial 3-trees, i.e., subgraphs of a triangulated planar graph obtained by repeatedly inserting a vertex in one triangle and connecting it to all vertices of the triangle. Moreover, vertices have rational coordinates if the face areas are rational, and we can bound the resolution. For orthogonal drawings, we give an algorithm to draw triconnected planar graphs with maximum degree 3. This algorithm produces a drawing with at most 8 bends per face and 4 bends per edge, which improves the previous known result of 34 bends per face. Both vertices and bends have rational coordinates if the face areas are rational.