The paper studies directed embeddings of \(2\)-regular, diplanar, digraphs on surfaces. The key definitions here are the following. A directed embedding of a digraph on a surface is a \(2\)-cell embedding with the property that each face is bounded by a directed cycle. A digraph is \(2\)-regular if each vertex has both indegree and outdegree two, and is diplanar if it admits a directed embedding in the sphere.
There are two main results in this paper:
1.) It gives a shorter and simpler proof of an extension of Whitney’s theorem to directed embeddings first obtained by D. Archdeacon et al. [Australas. J. Comb. 67, Part 2, 159–165 (2017; Zbl 1375.05064)]. This result states that every strongly \(2\)-edge-connected \(2\)-regular diplanar digraph admits a unique (up to homeomorphisms) directed embedding on the sphere.
2.) It gives a structural description and characterisation of the possible directed embeddings of any strongly \(2\)-edge-connected \(2\)-regular diplanar digraphs on the projective plane, on the torus, and on the Klein bottle. In particular, this allows for an estimation of the number of different such embeddings for each fixed digraph with the above properties.