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. Furthermore, such a rectilinear cartogram can be constructed in O ( n log n ) time where n = | V | .