• Corpus ID: 117358727

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…
1 Citations

Figures from this paper

On $K_5$ and $K_{3,3}$-minors of graphs and regular matroids
In this paper we prove two main results about obstruction to graph planarity. One is that, if $G$ is a 3-connected graph with a $K_5$-minor and $T$ is a triangle of $G$, then $G$ has a $K_5$-minor

References

SHOWING 1-10 OF 17 REFERENCES
On 3-connected minors of 3-connected matroids and graphs
It is shown that, in the graphic case, with the extra assumption that r(M)-r(N)>=6, the authors can guarantee the existence of a 4-independent set of M with such a property.
On chains of 3-connected matroids
• Mathematics, Computer Science
Discret. Appl. Math.
• 1986
It is proved that for M and N 3-connected and N a minor of M, if the authors do not allow the insertion of an isomorphic copy for N, then there is always a 3-chain from N to M of gap at most 3.
On contractible and vertically contractible elements in 3-connected matroids and graphs
• Haidong Wu
• Computer Science, Mathematics
Discret. Math.
• 1998
The contractible and vertically contractible elements in 3-connected matroids are studied and the best-possible lower bounds for the number of vertically contractable elements in3-connected and minimally 3- connected matroIDS are obtained.
On the Structure of 3-connected Matroids and Graphs
• Computer Science, Mathematics
Eur. J. Comb.
• 2000
It is proved that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both and if an essential element e of M is in more than one fan, then that fan has three or five elements.
Stabilizers of Classes of Representable Matroids
• G. Whittle
• Computer Science, Mathematics
J. Comb. Theory, Ser. B
• 1999
One of the main theorems of this paper proves that if M is minor-closed and closed under duals, and N is 3- connected, then to show that N is a stabilizer it suffices to check 3-connected matroids in M that are single-element extensions or coextensions of N, or are obtained by a single- element extension followed by asingle-element coextension.
Finding a small 3-connected minor maintaining a fixed minor and a fixed element
• Mathematics, Computer Science
Comb.
• 1987
This result generalizes a theorem of Truemper and can be used to prove Seymour’s 2-roundedness theorem, as well as a result of Oxley on triples in nonbinary matroids.
On nonbinary 3-connected matroids
It is well known that a matroid is binary if and only if it has no minor isomorphic to U2,4, the 4-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid
Minors of 3-Connected Matroids
• P. Seymour
• Computer Science, Mathematics
Eur. J. Comb.
• 1985
It is shown that to test whether U24 (or any other 3-connected matroid N) has the property described above, it is only necessary to test that it works for those matroids M with 5 (or more generally, |E (N)| + 1) elements.
On minors of non-binary matroids
It is proved that for every two elements of a 3-connected non-binary matroid, there is aU42 minor using them both.
Extensions of Tutte's wheels-and-whirls theorem
• Computer Science, Mathematics
J. Comb. Theory, Ser. B
• 1992
Tutte's wheels-and-whirls theorem is proved and some extensions of this theorem are proved, one of which states that if M is 3-connected and has both a wheel and a whirl minor, then either M has only seven elements or there is some element the deletion or contraction of which maintains 3-connectivity and leaves a matroid with both aWheel and a Whirl minor.