On chains of 3-connected matroids
A sequence \(N_ 0,N_ 1,...,N_ m\) of 3-connected matroids is called a chain from \(N_ 0\) to \(N_ m\) if each of \(N_ 0,N_ 1,...,N_{m-1}\) is a minor of its successor. The gap of such a chain is \(\max \{| E(N_ i)| -| E(N_{i-1})|:\) \(i=1,2,...,m\}\). A result of \textit{K. Truemper} [Eur. J. Comb. 5, 377-394 (1984; Zbl 0561.05018)] shows that if N is a 3-connected minor of a 3-connected matroid M, then there is always a chain from N to M of gap at most three and this paper proves several extensions of Truemper's result. In particular, the authors show that the chain of gap at most three from N to M may be chosen so that each term is obtained from its successor by a single-element contraction followed by a simplification, or by a single-element deletion followed by a cosimplification. In addition, they prove that, provided N has no circuits or cocircuits of size less than four, one may find a chain whose gap is at most two.
scientific article; zbMATH DE number 3985241