A combinatorial algorithm for the planar multiflow problem with demands located on three holes

We consider an undirected multi(commodity)flow demand problem in which a supply graph is planar, each source-sink pair is located on one of three specified faces of the graph, and the capacities and demands are integer-valued and Eulerian. It is known that such a problem has a solution if the cut and (2,3)-metric conditions hold, and that the solvability implies the existence of an integer solution. We develop a purely combinatorial strongly polynomial solution algorithm.