# 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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## References

- 1.B. Alspach, L. Goddyn, and C.Q. Zhang. Graphs with the circuit cover property.
*Trans. Am. Math. Soc.*, 344(1):131–154, 1994.CrossRefMathSciNetGoogle Scholar - 2.R.K. Ahuja, T.L. Magnanti, and J.B. Orlin.
*Network Flows: Theory, Algorithms and Applications*. Prentice Hall, 1993.Google Scholar - 3.E.M. Arkin and C.H. Papadimitriou. On the complexity of circulations.
*J. Algorithms*, 7:134–145, 1986.CrossRefMathSciNetGoogle Scholar - 4.E.M. Arkin.
*Complexity of Cycle and Path Problems in Graphs*. PhD thesis, Stanford University, 1986.Google Scholar - 5.J.C. Bermond, B. Jackson, and F. Jaeger. Shortest coverings of graphs with cycles.
*J. Comb. Theory Ser. B*, 35:297–308, 1983.CrossRefMathSciNetGoogle Scholar - 6.J.A. Bondy and U.S.R. Murty.
*Graph Theory with Applications*. MacMillan Press, 1976.Google Scholar - 7.V. Batagelj and T. Pisanski. On partially directed eulerian multigraphs.
*Publ. de l'Inst. Math.*, Nouvelle série 25(39):16–24, 1979.MathSciNetGoogle Scholar - 8.G.A. Dirac. A property of 4-chromatic graphs and some remarks on critical graphs.
*J. London Math. Soc.*, 27:85–92, 1952.zbMATHMathSciNetGoogle Scholar - 9.G. Fan. Integer flows and cycle covers.
*J. Comb. Theory Ser. B*, 54:113–122, 1992.zbMATHCrossRefGoogle Scholar - 10.L.R. Ford and D.R. Fulkerson.
*Flows in Networks*. Princeton U. Press, Princeton, 1973.Google Scholar - 11.M. Grötschel and L. Lovász and A. Schrijver.
*Geometric Algorithms and Combinatorial Optimization*. Springer-Verlag, 1988.Google Scholar - 12.A.J. Hoffman. Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In
*Proc. Symp. Appl. Math*, volume 10, 1960.Google Scholar - 13.F. Jaeger. Flows and generalized coloring theorems in graphs.
*J. Comb. Theory Ser. B*, 26:205–216, 1979.zbMATHCrossRefMathSciNetGoogle Scholar - 14.A. Schrijver.
*Theory of Linear and Integer Programming*. Wiley, 1986.Google Scholar - 15.A. Sebö. Hilbert bases, Carathéodory's theorem and combinatorial optimization. In R. Kannan and W.R. Pulleyblank, editors,
*Integer Programming and Combinatorial Optimization Proceedings*, pages 431–456, Waterloo, 1990. University of Waterloo Press.Google Scholar - 16.P.D. Seymour. Sum of circuits. In
*Graph Theory and Related Topics*, pages 341–355. Academic Press, N. York, 1979.Google Scholar