Proportional Contact Representations of Planar Graphs - Alam, Md. Jawaherul; Biedl, Therese; Felsner, Stefan; Kaufmann, Michael; Kobourov, Stephen (Journal of Graph Algorithms and Applications, 2012)
We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by point-contacts or side-contacts between the corresponding polygons. Specifically, we consider proportional contact representations, where pre-specified vertex weights must be represented by the areas of the corresponding polygons. Natural optimization goals for such representations include minimizing the complexity of the polygons, and the unused area. We describe algorithms for proportional contact representations with optimal polygonal complexity for general planar graphs and planar 2-segment graphs, which include maximal outer-planar graphs and partial 2-trees.