Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complexity results for two kinds of colored disconnections of graphs   The concept of rainbow disconnection number of graphs was introduced by
Chartrand et al. in 2018. Inspired by this concept, we put forward the concepts
of rainbow vertex-disconnection and proper disconnection in graphs. In this
paper, we first show that it is $NP$-complete to decide whether a given
edge-colored graph $G$ with maximum degree $\Delta(G)=4$ is proper
disconnected. Then, for a graph $G$ with $\Delta(G)\leq 3$ we show that
$pd(G)\leq 2$ and determine the graphs with $pd(G)=1$ and $2$, respectively.
Furthermore, we show that for a general graph $G$, deciding whether $pd(G)=1$
is $NP$-complete, even if $G$ is bipartite. We also show that it is
$NP$-complete to decide whether a given vertex-colored graph $G$ is rainbow
vertex-disconnected, even though the graph $G$ has $\Delta(G)=3$ or is
bipartite.

6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat