# Circuit covers in series-parallel mixed graphs

## Abstract

A mixed graph is a graph that contains both edges and arcs. Given a nonnegative integer weight function *p* on the edges and arcs of a mixed graph *M*, we wish to decide whether *(M,p)* has a circuit cover, that is, if there is a list of circuits in *M* such that every edge (arc) *e* is contained in exactly *p(e)* circuits in the list. When *M* is a directed graph or an undirected graph with no Petersen graph as a minor, good necessary and sufficient conditions are known for the existence of a circuit cover. For general mixed graphs this problem is known to be NP-complete. We provide necessary and sufficient conditions for the existence of a circuit cover of (*M, p*) when *M* is a *series-parallel* mixed graph, that is, the underlying graph of *M* does not have *Ka* as a minor. We also describe a polynomial-time algorithm to find such a circuit cover, when it exists. Further, we show that *p* can be written as a nonnegative integer linear combination of at most *m* incidence vectors of circuits of *M*, where *m* is the number of edges and arcs. We also present a polynomial-time algorithm to find a minimum circuit in a series-parallel mixed graph with arbitrary weights. Other results on the fractional circuit cover and the circuit double cover problem are discussed.

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