Flows in One-Crossing-Minor-Free Graphs

TL;DR

This paper presents an efficient algorithm for computing maximum flows in directed graphs that exclude certain minors, specifically those that can be drawn with one crossing, achieving O(n log n) time complexity.

Contribution

It introduces a method to compute maximum flows in directed H-minor-free graphs with a given structural decomposition, extending efficient flow algorithms to new classes of graphs.

Findings

Maximum flow in directed K_{3,3}-minor-free graphs computed in O(n log n) time.

Maximum flow in directed K_5-minor-free graphs computed in O(n log n) time.

Efficient algorithms depend on structural decompositions of the 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(n log n) 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(n log n) time without additional assumptions.