International Journal of Combinatorics
Volume 2013 (2013), Article ID 783710, 7 pages
Research Article

Graphs with no Minor Containing a Fixed Edge

Mathematical, Computer, and Information Sciences Division, Office of Naval Research, Arlington, VA 22203, USA

Received 30 November 2012; Accepted 6 February 2013

Academic Editor: Chính T. Hoang

Copyright © 2013 Donald K. Wagner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 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.