Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the existence and non-existence of improper homomorphisms of oriented
  and $2$-edge-coloured graphs to reflexive targets   We consider non-trivial homomorphisms to reflexive oriented graphs in which
some pair of adjacent vertices have the same image. Using a notion of convexity
for oriented graphs, we study those oriented graphs that do not admit such
homomorphisms. We fully classify those oriented graphs with tree-width $2$ that
do not admit such homomorphisms and show that it is NP-complete to decide if a
graph admits an orientation that does not admit such homomorphisms. We prove
analogous results for $2$-edge-coloured graphs. We apply our results on
oriented graphs to provide a new tool in the study of chromatic number of
orientations of planar graphs -- a long-standing open problem.

3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom