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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 oneof $m$ 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 arccolours, and the arc directions of edges and arcs incident with $v$. The groupof all allowed switches is $\Gamma$.  Let $k \geq 1$ be a fixed integer and $\Gamma$ a fixed permutation group. Weconsider the problem that takes as input an $(m,n)$-mixed graph $G$ and asks ifthere a sequence of switches at vertices of $G$ with respect to $\Gamma$ sothat the resulting $(m,n)$-mixed graph admits a homomorphism to an$(m,n)$-mixed graph on $k$ vertices. Our main result establishes this problemcan be solved in polynomial time for $k \leq 2$, and is NP-hard for $k \geq 3$.This provides a step towards a general dichotomy theorem for the$\Gamma$-switchable homomorphism decision problem.

The 2-colouring problem for $(m,n)$-mixed graphs with switching is polynomial