Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

Abstract

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^7/6). No known input causes our algorithm to use area Ω(n^1+ϵ) for any ϵ > 0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.