Elastic Elements in 3-Connected Matroids

Abstract

It follows by Bixby's Lemma that if $e$ is an element of a $3$-connected matroid $M$, then either $\textrm{co}(M\backslash e)$, the cosimplification of $M\backslash e$, or $\textrm{si}(M/e)$, the simplification of $M/e$, is $3$-connected. A natural question to ask is whether $M$ has an element $e$ such that both $\textrm{co}(M\backslash e)$ and $\textrm{si}(M/e)$ are $3$-connected. Calling such an element "elastic", in this paper we show that if $|E(M)|\ge 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans.

A splitter theorem for elastic elements in $3$-connected matroids

“…The following theorem is the above mentioned analogue of Tutte's Wheelsand-Whirls Theorem for elastic elements established in [3]. The main result of this paper is the next theorem.…”

“…The third lemma is [3, Theorem 1]. The fourth lemma follows by combining [3,Lemma 18] and [3,Lemma 19]. Lemma 2.9.…”

“…Recall the definition of a Θ-separator given in the introduction. The next four lemmas are extracted from [3]. The first lemma follows from the proofs of [3, Lemma 13] and [3, Lemma 16], while the second lemma is an immediate consequence of the first.…”

“…For example, Oxley et al [7] showed that if B is a given basis of a 3-connected matroid M with no 4element fans, and N is a 3-connected minor of M , then, provided a certain necessary condition holds, there is either an element b ∈ B such that M/b is 3-connected with an N -minor or an element b * ∈ E(M ) − B such that M \b * is 3-connected with an N -minor. More recently, Drummond et al [3] proved a variant of Tutte's Wheels-and-Whirls Theorem for elastic elements. An element e of a 3-connected matroid M is elastic if si(M/e), the simplification of M/e, and co(M \e), the cosimplification of M \e, are both 3-connected.…”

“…An element e of a 3-connected matroid M is elastic if si(M/e), the simplification of M/e, and co(M \e), the cosimplification of M \e, are both 3-connected. It is shown in [3] that if |E(M )| ≥ 4, and M has no 4-element fans and no member of a particular family of 3-separators, generically referred to as Θ-separators, then M has at least four elastic elements. As a comparison, Bixby's Lemma [1] says that if e is an element of M , then either si(M/e) or co(M \e) is 3-connected.…”