I'm working on a digraph problem in which bidirectional edges need to be treated separately. As such, we could consider them as undirected edges. Clearly, if I replace bidirectional edges with undirected edges I get a ``mixed graph'' (one with directed and undirected edges). But can a mixed graph have bidirectional edges in addition to undirected edges? Can a mixed graph have a directed edge and an undirected edge between two vertices?
The Wikipedia definition is not crystal clear on this. A paper, Enumeration of Mixed Graphs indicates bidirectional and parallel mixed edges are disallowed. The definition in this article claims that the ``complete mixed graph'' has a undirected and bidirectional edge between both vertices.
What are ``mixed graphs'' typically defined as?