On the integral plane two-commodity flow problem

The authors consider in this note the maximum integral two-commodity flow problem in augmented planar graphs, that is with both source-sink edges added. They provide an O(nlogn) simple algorithm to determine the minimal increase in the weight function required to ensure that the value of a maximum plane integral two-commodity flow equals the value of the maximum plane two-commodity flow with both values equal to the original maximum two-commodity flow value.