Linear - Time Algorithms for Proportional Contact Graph Representations ?

In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12-sided rectilinear polygons and takes O(n log n) time. We describe a new algorithm that guarantees 10-sided rectilinear polygons and runs in O(n) time. We also describe a linear-time algorithm for proportional contact representation of planar 3-trees with 8-sided rectilinear polygons and show that this optimal, as there exist planar 3-trees that requires 8-sided polygons. Finally, we show that a maximal outer-planar graph admits a proportional contact representation with 6-sided rectilinear polygons when the outer-boundary is a rectangle and with 4 sides otherwise.