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 K4 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.