The circuit cover problem for mixed graphs (those containing edges and/or arcs) is defined
as follows. Given a mixed graph M with a nonnegative integer weight function p on its
edges and arcs, decide whether (M, p) has a circuit cover, that is, a list of circuits in M
such that every edge (arc) e is contained in exactly p(e) circuits of the list. In the special
case when M is a directed graph (contains only arcs), the problem is easy, but when M is
an undirected graph not many results are known. For general mixed graphs this problem
was shown to be NP-complete by Arkin and Papadimitriou in 1986. We prove that this
problem remains NP-complete for planar mixed graphs. Furthermore, we present a good
characterization for the existence of a circuit cover when M is series-parallel (a similar
result holds for the fractional version). We also describe a polynomial algorithm to find
such a circuit cover, when it exists. This is an ellipsoid-based algorithm whose separation
problem is the minimum circuit problem on series-parallel mixed graphs, which we show to
be polynomially solvable. Results on two well-known combinatorial problems, the problem
of detecting negative circuits and the problem of finding shortest paths, are also presented.
We prove that both problems are NP-hard for planar mixed graphs.