If you are interested in max-flow computations in planar graphs, below is the talk I have given at WORKER 2013. It gives an overview of how to use shortest paths and Monge property to get the following three results:
- O(n log log n) time algorithm for undirected max-flow in planar graphs , shop
- almost linear time algorithm for all-source all-sink max-flow in undirected planar graphs, i.e., it computes Gumory-Hu tree ,
- almost linear time algorithm for single-source all-sink max-flow in directed planar graphs .
 Giuseppe F. Italiano, Yahav Nussbaum, Piotr Sankowski, Christian Wulff-Nilsen: Improved algorithms for min cut and max flow in undirected planar graphs. STOC 2011: 313-322.
 Glencora Borradaile, Piotr Sankowski, Christian Wulff-Nilsen: Min st-cut Oracle for Planar Graphs with Near-Linear Preprocessing Time. FOCS 2010: 601-610.
 Jakub Lacki, Yahav Nussbaum, Piotr Sankowski, Christian Wulff-Nilsen: Single Source - All Sinks Max Flows in Planar Digraphs. FOCS 2012: 599-608.