The 2colouring problem for $(m,n)$mixed graphs with switching is polynomial
Abstract
A mixed graph is a set of vertices together with an edge set and an arc set. An $(m,n)$mixed graph $G$ is a mixed graph whose edges are each assigned one of $m$ colours, and whose arcs are each assigned one of $n$ colours. A \emph{switch} at a vertex $v$ of $G$ permutes the edge colours, the arc colours, and the arc directions of edges and arcs incident with $v$. The group of all allowed switches is $\Gamma$. Let $k \geq 1$ be a fixed integer and $\Gamma$ a fixed permutation group. We consider the problem that takes as input an $(m,n)$mixed graph $G$ and asks if there a sequence of switches at vertices of $G$ with respect to $\Gamma$ so that the resulting $(m,n)$mixed graph admits a homomorphism to an $(m,n)$mixed graph on $k$ vertices. Our main result establishes this problem can be solved in polynomial time for $k \leq 2$, and is NPhard for $k \geq 3$. This provides a step towards a general dichotomy theorem for the $\Gamma$switchable homomorphism decision problem.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.08070
 arXiv:
 arXiv:2203.08070
 Bibcode:
 2022arXiv220308070B
 Keywords:

 Mathematics  Combinatorics;
 05C15 (Primary) 68R10 (Secondary)
 EPrint:
 Accepted version Discrete Mathematics and Theoretical Computing Science. 13 page, 1 figure,