Algorithms for orthogonal drawings (abstract)

Abstract

The Andreev-Thurston circle packing theorem is generalized and improved in three ways. First, to arbitrary maps on closed surfaces. Second, we get the simultaneous circle packing of the map and its dual map so that, in the corresponding straight-line representations of the map and the dual, any two edges dual to each other cross at the right angle. The necessary and sufficient condition for a map to have such a primal-dual circle packing representation is that its universal cover is 3-connected (the map has no "planar" 2-separations). Finally, a polynomial time algorithm is obtained that given a map M and a rational number ε > 0 finds an ε-approximation for the primal-dual circle packing representation of M. In particular, we get a polynomial time algorithm for geodesic convex representation of reduced maps on arbitrary surfaces.