LATIN 1998: LATIN'98: Theoretical Informatics pp 226-238

# Circuit covers in series-parallel mixed graphs

• Orlando Lee
• Yoshiko Wakabayashi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)

## 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.

## References

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