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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 aK   3  ,  3minor 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 themain result of this paper, it is shown that the Ford-Fulkerson algorithm naturally extends to this more general class of graphs.

international journal of combinatorics

Graphs with no <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mn mathvariant="bold">3</mml:mn><mml:mtext>,</mml:mtext><mml:mn mathvariant="bold">3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> Minor Containing a Fixed Edge

Graphs with no K3,3 Minor Containing a Fixed Edge