Planar Graph Isomorphism Is in Log-Space

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best-known lower and upper bounds on its complexity. The gap between the known upper and lower bounds continues to be very significant for many subclasses of graphs as well.
We bridge the gap for a natural and important class of graphs, namely, planar graphs, by presenting a log-space upper bound that matches the known log-space hardness. In fact, we show a stronger result that planar graph canonization is in log-space.