On Finding Some New Excluded Minors for Gammoids

Abstract

The class of gammoids is the smallest class of matroids closed under duality and minors that contains the class of transversal matroids. Ingleton showed that the class of gammoids does not allow for a finite characterization in terms of excluded minors [Ingleton, A., “Transversal Matroids and Related Structures,” Springer Netherlands, Dordrecht, 1977 pp. 117–131], and Mayhew showed that every gammoid may be extended to a matroid that is an excluded minor for the class of gammoids [Mayhew, D., The antichain of excluded minors for the class of gammoids is maximal, ArXiv e-prints (2016)]. The properties of the antichain of excluded minors has yet to be examined thoroughly. The natural approach to this would be to start with some non-gammoid and then examine its minors for the minimal minors that are not gammoids. Although it is comparatively easy to check whether a given matroid is either a strict gammoid or a transversal matroid, we still have to deal with minors that are neither strict gammoids nor transversal matroids. We introduce an easily implemented and automated sufficient condition that a given matroid is not a gammoid, which greatly reduces the handwork needed in this investigation.