On Rectilinear Duals for Vertex-Weighted Plane Graphs

Let G = (V,E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of G is a partition of a rectangle into |V | simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge in E. A rectilinear dual is called a cartogram if the area of each region is equal to the weight of the corresponding vertex. We show that every vertex-weighted plane triangulated graph G admits a cartogram of constant complexity, that is, a cartogram where the number of vertices of each region is constant.