Flows in One-Crossing-Minor-Free Graphs

Abstract

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 K₅-minor-free graphs can be computed in O(nlogn) time without additional assumptions.