Linear-Time Algorithms for Rectilinear Hole-free Proportional Contact Representations

A proportional contact representation of a planar graph is one where each vertex is represented by a simple polygon with area proportional to a given weight and adjacencies between polygons represent edges between the corresponding pairs of vertices. In this paper we study proportional contact representations that use only rectilinear polygons and contain no unused area or hole. There is an algorithm that gives a hole-free proportional contact representation of a maximal planar graph with 12-sided rectilinear polygons inO(n logn) time. We improve this result by giving a linear-time algorithm that produces a hole-free proportional contact representation of a maximal planar graph with a 10-sided rectilinear polygons. For a planar 3-tree we give a linear-time algorithm for a hole-free proportional contact representation with 8-sided rectilinear polygons. Furthermore, there exist a planar 3-tree that requires 8-sided polygons in any hole-free contact representation with rectilinear polygons. A maximal outerplanar graph admits a hole-free proportional contact representation with rectangles.