On chains of 3-connected matroids

Abstract A sequence of k-connected matroids N0, N1,…,Nm is called a k-chain, from N0 to Nm, if Ni−1 is a minor of Ni(i=1,…,m); this chain is said to have gap t = max {|E(Ni)| − |E(Ni−1)|: i = 1,…,m}. Chains of gap 1 are said to be dense. If M and N are 3-connected matroids, M is not a wheel or whirl, |E(N)| ⩾ 4 and N is a minor of M, then there is a dense 3-chain from N′ to M where N′ is isomorphic to N. For graphs this is a theorem of Negami, and for matroids a theorem of Seymour and Tan. Truemper has proved that for M and N 3-connected and N a minor of M, if we do not allow the insertion of an isomorphic copy for N, then there is always a 3-chain from N to M of gap at most 3. We investigate the structure of these chains, and show that if N has no circuits or cocircuits with 3 or fewer elements, then there is a 3-chain of gap at most 2.