Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rank connectivity and pivot-minors of graphs   The cut-rank of a set $X$ in a graph $G$ is the rank of the $X\times
(V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a
partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$
is less than $2$ and both $X$ and $Y$ have at least two vertices. A graph is
prime (with respect to the split decomposition) if it is connected and has no
splits. A graph $G$ is $k^{+\ell}$-rank-connected if for every set $X$ of
vertices with the cut-rank less than $k$, $\lvert X\rvert$ or $\lvert
V(G)-X\rvert $ is less than $k+\ell$. We prove that every prime
$3^{+2}$-rank-connected graph $G$ with at least $10$ vertices has a prime
$3^{+3}$-rank-connected pivot-minor $H$ such that $\lvert V(H)\rvert =\lvert
V(G)\rvert -1$. As a corollary, we show that every excluded pivot-minor for the
class of graphs of rank-width at most $k$ has at most $(3.5 \cdot 6^{k}-1)/5$
vertices for $k\ge 2$. We also show that the excluded pivot-minors for the
class of graphs of rank-width at most $2$ have at most $16$ vertices.

9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat