Abstract
Analogous pairs of theorems are investigated concerning coverings of directed and odd cuts. One such pair of results is the Lucchesi-Younger theorem on directed cuts and Seymour’s theorem on odd cuts. Here we strengthen these results (incidently providing a simple proof of Seymour’s theorem). For example, the minimum cardinality of a T-join in a graph G=(V,E) is proved to equal the maximum of Σq T(V ι)/2 over all partitions of V where q T (X) is the number of T-odd components of V-X. Moreover, if G is bipartite, there is an optimal partition arising from a partition of the two parts. Secondly some orientation problems of undirected graphs are discussed. The results also emphasize the analogy between strong connectivity and parity conditions.
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References
F.D.J. Dunstan, “Matroids and submodular functions”, Quarterly Journal of Mathematics 27 (1976) 339–347.
A. Frank, “Combinatorial algorithms, algorithmic proofs” (in Hungarian), Ph.D. Thesis, Eötvös L. University (Budapest, 1975).
A. Frank, “How to make a digraph strongly connected”, Combinatorica 1 (1981) 145–153.
A. Frank and A. Gyárfás, “How to orient the edges of a graph”, in: A. Hajnal and V.T. Sós, eds., Colloquia Mathematica Societatis János Bolyai 18, Combinatorics (North-Holland, Amsterdam, 1978) pp. 353–364.
L. Lovász, “2-matchings and 2-covers of hypergraphs”, Acta Mathematica Academiae Scientiarum Hungaricae 26 (1975), 433–444.
L. Lovász, “Flats in matroids and geometric graphs”, in: P.J. Cameron, ed., Combinatorial Surveys: Proceedings of the sixth British Combinatorial Conference (Academic Press, New York, 1979) pp. 45–86.
L. Lovász, Combinatorial Problems and Exercises (Akadémiai Kiadó, Budapest, 1979).
C.L. Lucchesi and D.H. Younger, “A minimax relation for directed graphs”, Journal of the London Mathematical Society (2) 17 (1978) 369–374.
W. Mader, “Ueber die Maximalzahl kantendisjunkter A-wege”, Archiv der Mathemaik 30 (1978) 325–336.
H.E. Robbins, “A theorem on graphs with an application to a problem of traffic control”, American Mathematical Monthly 46 (1939) 281–283.
A. Schrijver, “Total dual integrality directed graphs, crossing families and sub-and supermodular functions”, in: Proceedings of the Silver Jubilee Conference on Combinatorics (held in Waterloo, June 1982, Academic Press).
A. Sebő, “On the structure of odd-joins”, in preparation.
P.D. Seymour, “On odd cuts and plane multicommodity flows”, Proceedings of the London Mathematical Society 42 (1981) 178–192.
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© 1984 The Mathematical Programming Society, Inc.
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Frank, A., Tardos, é., Sebő, A. (1984). Covering directed and odd cuts. In: Korte, B., Ritter, K. (eds) Mathematical Programming at Oberwolfach II. Mathematical Programming Studies, vol 22. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121011
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DOI: https://doi.org/10.1007/BFb0121011
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