Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $N$-detachable pairs in 3-connected matroids II: life in $X$   Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$.
A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids
$M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an
$N$-minor. This is the second in a series of three papers where we describe the
structures that arise when it is not possible to find an $N$-detachable pair in
$M$. In the first paper in the series, we showed that, under mild assumptions,
either $M$ has an $N$-detachable pair, $M$ has one of three particular
3-separators that can appear in a matroid with no N-detachable pairs, or there
is a 3-separating set $X$ with certain strong structural properties. In this
paper, we analyse matroids with such a structured set $X$, and prove that they
have either an $N$-detachable pair, or one of five particular 3-separators that
can appear in a matroid with no $N$-detachable pairs.

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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom