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.
Highlights
A result widely used in the study of 3-connected matroids is due to Bixby [1]: if e is an element of a 3-connected matroid M, either M \e or M/e has no non-minimal 2-separations, in which case, co(M \e), the cosimplification of M, or si(M/e), the simplification of M, is 3-connected
All wheels and whirls of rank at least four have no elastic elements. The reason for this is that every element of such a matroid is in a 4-element fan and the way, geometrically, every 4-element fan is positioned in relation to the rest of the elements of the matroid
Provided X ∪ {e} is a maximal fan, the instance illustrated in Fig. 1(i) is essentially the only way in which X does not contain two elastic elements
Summary
A result widely used in the study of 3-connected matroids is due to Bixby [1]: if e is an element of a 3-connected matroid M , either M \e or M/e has no non-minimal 2-separations, in which case, co(M \e), the cosimplification of M , or si(M/e), the simplification of M , is 3-connected. Provided X ∪ {e} is a maximal fan, the instance illustrated in Fig. 1(i) is essentially the only way in which X does not contain two elastic elements. Brettell and Semple [2] establish a Splitter Theorem counterpart to this last result where, again, 3-connectivity is preserved up to simplification and cosimplification. Throughout the paper, the notation and terminology follows [3]
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callSimilar Papers
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
