an:05912590
Zbl 1298.05295
Ageev, Alexander; Benchetrit, Yohann; Seb??, Andr??s; Szigeti, Zolt??n
An excluded minor characterization of Seymour graphs
EN
G??nl??k, Oktay (ed.) et al., Integer programming and combinatoral optimization. 15th international conference, IPCO 2011, New York, NY, USA, June 15--17, 2011. Proceedings. Berlin: Springer (ISBN 978-3-642-20806-5/pbk). Lecture Notes in Computer Science 6655, 1-13 (2011).
2011
a
05C83 05C75
Summary: A graph \(G\) is said to be a Seymour graph if for any edge set \(F\) there exist \(|F|\) pairwise disjoint cuts each containing exactly one element of \(F\), provided for every circuit \(C\) of \(G\) the necessary condition \(|C \cap F| \leq |C \setminus F|\) is satisfied. Seymour graphs behave well with respect to some integer programs including multiflow problems, or more generally odd cut packings, and are closely related to matching theory.
A first coNP characterization of Seymour graphs has been shown by \textit{A. A. Ageev} et al. [J. Graph Theory 24, No. 4, 357--364 (1997; Zbl 0869.05051)], the recognition problem has been solved in a particular case by \textit{A. M. H. Gerards} [J. Comb. Theory, Ser. B 55, No. 1, 73--82 (1992; Zbl 0810.05056)], and the related cut packing problem has been solved in the corresponding special cases. In this article we show a new, minor-producing operation that keeps this property, and prove excluded minor characterizations of Seymour graphs: the operation is the contraction of full stars, or of odd circuits. This sharpens the previous results, providing at the same time a simpler and self-contained algorithmic proof of the existing characterizations as well, still using methods of matching theory and its generalizations.
For the entire collection see [Zbl 1216.90002].
Zbl 0869.05051; Zbl 0810.05056