Planar Embedding of Planar Graphs

Abstract

Planar embedding with minimal area of graphs on an integer grid is an interesting problem in VLSI theory. Valiant [V] gave an algorithm to construct a planar embedding for trees in linear area; he also proved that there are planar graphs that require quadratic area. We fill in a spectrum between Valiant's results by showing that a N-node planar graph has a planar embedding with area O(NF), where F is a bound on the path length from any node to the exterior face. In particular, an outerplanar graph can be embedded without crossovers in linear area. This bound is tight, up to constant factors: for any N and F, there exist graphs requiring Ω(NF) area for planar embedding. Also, finding a minimal embedding area is shown to be NP-complete for forests, and hence for more general types of graphs.