JaJa, Joseph F.Kosaraju, S. RaoWe present new efficient parallel algorithms for planar graph isomorphism and several related problems. Our main parallel model is the SQRT n x SQRT n mesh of processors, where n is the length of the input. Our results include O(SQRT n) time algorithms for finding a good separating cycle and the triconnected components of a planar graph, and for solving the single function coarsest partitioning problem.en-USTechnical Report