K3,3 minors and the maximum-flow problem

Let G be a graph, and let e be an edge of G. The main result of this paper is that any instance of the maximum-flow problem on G having e as the ‘return’ edge can be solved in linear time provided G does not have a K3,3 minor containing e. The primary tool in proving this result is a new graph decomposition. In particular, it is shown that if G is 3-connected and does not have a K3,3 minor containing e, then it can be decomposed into planar graphs, ‘almost’ planar graphs, and copies of K5.