Proportional contact representations of planar graphs

Abstract

We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a point-contact or a side-contact between the corresponding polygons. Specifically, we consider proportional contact representations, where given vertex weights are represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe optimal (with respect to complexity) constructive algorithms for proportional contact representations for general planar graphs and planar 2-segment graphs, which include maximal outerplanar graphs and partial 2-trees. Specifically, we show that: (a) 4-sided polygons are necessary and sufficient for a point-contact proportional representation for any planar graph; (b) triangles are necessary and sufficient for point-contact proportional representation of partial 2-trees; (c) trapezoids are necessary and sufficient for side-contact proportional representation of partial 2-trees; (d) convex quadrilaterals are necessary and sufficient for hole-free side-contact proportional representation for maximal outer-planar graphs.