Homomorphisms of (n,m)-graphs with respect to generalised switch

The study of homomorphisms of $(n,m)$-graphs, that is, adjacency preserving vertex mappings of graphs with $n$ types of arcs and $m$ types of edges was initiated by Ne\v{s}et\v{r}il and Raspaud [Journal of Combinatorial Theory, Series B 2000]. Later, some attempts were made to generalize the switch operation that is popularly used in the study of signed graphs, and study its effect on the above mentioned homomorphism. In this article, we too provide a generalization of the switch operation on $(n,m)$-graphs, which to the best of our knowledge, encapsulates all the previously known generalizations as special cases. We approach to study the homomorphism with respect to the switch operation axiomatically. We prove some fundamental results, which, we believe, will be essential tools in the further study of this topic. We also prove the existence of a categorical product for $(n,m)$-graphs with respect to a particular class of generalized switch. We also provide a way to calculate the product explicitly, and prove general properties of the product. Also, in the process of proving the fundamental results, we have provided yet another solution to an open problem posed by Klostermeyer and MacGillivray [Discrete Mathematics 2004].