Planarity testing revisited

Abstract

Planarity Testing is the problem of determining whether a given graph is planar while planar embedding is the corresponding construction problem. The bounded space complexity of these problems has been determined to be exactly deterministic logarithmic space by Allender and Mahajan [AM00] with the aid of Reingold's result [Rei08]. Unfortunately, the algorithm is quite daunting and generalizing it to, say the bounded genus case, seems a tall order. We present a simple planar embedding algorithm running in logspace. The algorithm uses the unique embedding of 3-connected planar graphs, a variant of Tutte's criterion on the conflict graphs of cycles and an explicit change of basis for the cycle space. We also present a logspace algorithm to find an obstacle to planarity, viz. a Kuratowski minor, for non-planar graphs. To the best of our knowledge this is the first logspace algorithm for this problem. © 2011 Springer-Verlag.