Before revision:  First we replace an iterative procedure of Wang that uses k invocations of a maximum flow algorithm in a planar graph with k apices [Borradaile et al., FOCS 2012, SICOMP 2017], by an alternative procedure that only makes one invocation of the algorithm of Borradaile et al.
After revision:  First we replace an iterative procedure of Wang that uses O(k) invocations of an O(k^3 n \log^3 n)-time algorithm for maximum flow algorithm in a planar graph with k apices [Borradaile et al., FOCS 2012, SICOMP 2017], by an alternative procedure that only makes one invocation of the algorithm of Borradaile et al.