An Excluded Minor Characterization of Seymour Graphs

Abstract

A graph G is said to be aSeymour 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 ∩ F| ≤ |C ∖ 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 Ageev, Kostochka and Szigeti [1], the recognition problem has been solved in a particular case by Gerards [2], 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.