Flows in One-Crossing-Minor-Free Graphs

We study the maximum flow problem in directed H-minor-free graphs where H can be drawn in the plane with one crossing. If a structural decomposition of the graph as a clique-sum of planar graphs and graphs of constant complexity is given, we show that a maximum flow can be computed in O(nlogn) time. In particular, maximum flows in directed K 3,3-minor-free graphs and directed K 5-minor-free graphs can be computed in O(nlogn) time without additional assumptions.

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