Vertically N-contractible elements in 3-connected matroids

Abstract

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 case, provided $r(M)-r(N)\ge 5$, and improved this result when $r(M)-r(N)\ge 6$, and in the graphic case. In this paper we generalize the results of Costalonga to the non-binary case. Moreover, we apply our results to the study of properties similar to 3-roundedness in classes of matroids.

arXiv commentThis paper has been withdrawn by the author because it's results are obsolete (the more general results are on a more recent work:arXiv:1405.6454) and still has many minor, being not worth to correct due to obsolescence

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