In this paper we present an O(nlog n) time 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 maximum flow 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 graphs, an algorithm with the same time complexity was recently claimed, but we show that it has a flaw. We also apply our technique to obtain a linear‐time algorithm to convert a flow to an acyclic flow, and a linear‐time algorithm to find a largest set of vertex‐disjoint s-t paths, in a directed planar graph.
