On removable circuits in graphs and matroids

Mader proved that every 2-connected simple graph G with minimum degree d ex ceeding three has a cycle C, the deletion of whose edges leaves a 2-connect ed graph. Jackson extended this by showing that C may be chosen to avoid an y nominated edge of G and to have length at least d-1. This article proves an extension of Jackson's theorem. In addition, a conjecture of Goddyn, van den Heuvel, and McGuinness is disproved when it is shown that a natural ma troid dual of Mader's theorem fails. (C) 1999 John Wiley & Sons, Inc.