# Vertically N-contractible elements in 3-connected matroids

@article{Costalonga2012VerticallyNE, title={Vertically N-contractible elements in 3-connected matroids}, author={Jo{\~a}o Paulo Costalonga}, journal={arXiv: Combinatorics}, year={2012} }

In this paper we establish a variation of the Splitter Theorem. Let $M$ and $N$ be simple 3-connected matroids. We say that $x\in E(M)$ is vertically $N$-contractible if $si(M/x)$ is a 3-connected matroid with an $N$-minor. Whittle (for $k=1,2$) and Costalonga(for $k=3$) proved that, if $r(M)- r(N)\ge k$, then $M$ has a $k$-independent set $I$ of vertically $N$-contractible elements. Costalonga also characterized an obstruction for the existence of such a 4-independent set $I$ in the binary…

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