Graphs with no \(K_{3,3}\) minor containing a fixed edge (Q1953665)

Summary: It is well known that every cycle of a graph must intersect every cut in an even number of edges. For planar graphs, Ford and Fulkerson proved that, for any edge \(e\), there exists a cycle containing \(e\) that intersects every minimal cut containing \(e\) in exactly two edges. The main result of this paper generalizes this result to any nonplanar graph \(G\) provided \(G\) does not have a \(K_{3,3}\) minor containing the given edge \(e\). Ford and Fulkerson used their result to provide an efficient algorithm for solving the maximum-flow problem on planar graphs. As a corollary to the main result of this paper, it is shown that the Ford-Fulkerson algorithm naturally extends to this more general class of graphs.