  In this paper we present an O(n log n) algorithm for finding a maximum flow
in a directed planar graph, where the vertices are subject to capacity
constraints, in addition to the arcs. If the source and the sink are on the
same face, then our algorithm can be implemented in O(n) time.
  For general (not planar) graphs, vertex capacities do not make the problem
more difficult, as there is a simple reduction that eliminates vertex
capacities. However, this reduction does not preserve the planarity of the
graph. The essence of our algorithm is a different reduction that does preserve
the planarity, and can be implemented in linear time. For the special case of
undirected planar graph, an algorithm with the same time complexity was
recently claimed, but we show that it has a flaw.
