In [12] (CCC 2013), the authors presented an algorithm for the reachability problem over directed planar graphs that runs in polynomial-time and uses O ( n 1 2+ ) space. A critical ingredient of their algorithm is a polynomial-time, \tldO ( n ) -space algorithm to compute a separator of a planar graph. The conference version provided a sketch of the algorithm and many nontrivial details were left unexplained. In this work, we provide a detailed construction of their algorithm.