On removable circuits in graphs and matroids

Abstract

Mader proved that every 2-connected simple graph G with minimum degree d exceeding three has a cycle C, the deletion of whose edges leaves a 2-connected graph. Jackson extended this by showing that C may be chosen to avoid any 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 matroid dual of Mader's theorem fails. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 51–66, 1999