On Extremal Connectivity Properties of Unavoidable Matroids

Ding, Oporowski, Oxley, and Vertigan proved that, for alln⩾3, there is an integerN(n) such that every 3-connected matroid with at leastN(n) elements has a minor isomorphic to a wheel or whirl of rankn, M(K3, n) or its dual,U2, n+2or its dual, or a rank-nspike. This paper characterizes each of these classes of unavoidable matroids in terms of an extremal connectivity condition. In particular, it is proved that ifMis a 3-connected matroid of at least rank 7 for which every single-element deletion or contraction is 3-connected but no 2-element deletion or contraction is, thenMis a spike with its tip deleted. It is further proved that ifMis a 3-connected matroid of at least rank 4 for which every single-element deletion is 3-connected but no 1-element contraction or 2-element deletion is, thenM≅M*(K3, n).