On Minors Avoiding Elements in Matroids》.pdfVIP

Europ. J. Combinatorics (1991) 12, 531-539 On Minors Avoiding Elements in Matroids JAMES G. OXLEY Let· fF be a collection of 3-connected matroids, none a proper minor of another, such that if M is a 3-connected matroid having a proper fF-minor and e is an element of M, then M has an fF-minor avoiding e. This paper proves that there are precisely two collections fF with this property: {U ,4} and {U ,4, M(K4)} Several extensions of this result and some similar results 2 2 for 2-connected matroids are also established. 1. INTRODUCTION The problem of relating the minors of a matroid to particular elements of the matroid has received considerable research attention recently (see, for example, [1-3,5-10,12-16,18-21]). This paper addresses a new aspect of this problem. Let fff be a collection of matroids. A matroid M is an fff-minor of the matroid M if M is a minor of M that is isomorphic to some member of fff. If X is a set of elements of M, then M uses X if X!;;; E(M). Let k and I be integers with k ~ 2 and I ~ 1. Then fff is (k, I)-rounded [3] if every member of fff is k-connected [22] having at least four elements and the following condition holds: (1.1) If M is a k-connected matroid having an fff-minor and X is a subset of E(M) with at most I elements, then M has an fff-minor using X. Bixby [2] and Seymour [18], respectively, proved that {U ,4} is (2,1)- and 2 (3,2)-rounded. Moreover, numerous other collections of matro