Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The rainbow vertex-disconnection in graphs   Let $G$ be a nontrivial connected and vertex-colored graph. A subset $X$ of
the vertex set of $G$ is called rainbow if any two vertices in $X$ have
distinct colors. The graph $G$ is called \emph{rainbow vertex-disconnected} if
for any two vertices $x$ and $y$ of $G$, there exists a vertex subset $S$ of
$G$ such that when $x$ and $y$ are nonadjacent, $S$ is rainbow and $x$ and $y$
belong to different components of $G-S$; whereas when $x$ and $y$ are adjacent,
$S+x$ or $S+y$ is rainbow and $x$ and $y$ belong to different components of
$(G-xy)-S$. For a connected graph $G$, the \emph{rainbow vertex-disconnection
number} of $G$, denoted by $rvd(G)$, is the minimum number of colors that are
needed to make $G$ rainbow vertex-disconnected.
  In this paper, we characterize all graphs of order $n$ with rainbow
vertex-disconnection number $k$ for $k\in\{1,2,n\}$, and determine the rainbow
vertex-disconnection numbers of some special graphs. Moreover, we study the
extremal problems on the number of edges of a connected graph $G$ with order
$n$ and $rvd(G)=k$ for given integers $k$ and $n$ with $1\leq k\leq n$.

36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph