Abstract

We address the problem of drawing a 3-connected planar graph as a convex polyhedron in R³. We give an efficient algorithm for producing such a realization using O(n) volume under the vertex-resolution rule. Each vertex in the drawing resulting from this method is guaranteed to need no more than O(n log n) bits to represent (as a pair of rational numbers). This solves an open problem of Cohen, Eades, Lin, and Ruskey. We also show that under the angularresolution rule drawing a 3-connected planar graph as a convex polyhedron in R³ requires at least exponential volume in the worst case.

Citations