Delta-matroids as subsystems of sequences of Higgs lifts

TL;DR

This paper characterizes classes of delta-matroids derived from Higgs lifts, providing excluded-minor characterizations and exploring their structural properties within the broader context of set systems and matroid theory.

Contribution

It offers the first excluded-minor characterizations of full Higgs lift delta-matroids and related classes, advancing the structural understanding of delta-matroids and their construction from matroids.

Findings

Excluded-minor characterization of full Higgs lift delta-matroids.

Identification of classes arising from lattice paths and their properties.

Connections between delta-matroids and unions of matroid bases with specific properties.

Abstract

In her paper "Generalized matroids and supermodular colourings", Tardos studied special delta-matroids obtained from sequences of Higgs lifts; these are the full Higgs lift delta-matroids that we treat and around which all of our results revolve. We give an excluded-minor characterization of the class of full Higgs lift delta-matroids within the class of all delta-matroids, and we give similar characterizations of two other minor-closed classes of delta-matroids that we define using Higgs lifts. We introduce a minor-closed, dual-closed class of Higgs lift delta-matroids that arise from lattice paths. It follows from results of Bouchet that all delta-matroids can be obtained from full Higgs lift delta-matroids by removing certain feasible sets; to address which feasible sets can be removed, we give an excluded-minor characterization of delta-matroids within the more general structure of set systems. Many of these excluded minors occur again when we characterize the delta-matroids in which the collection of feasible sets is the union of the collections of bases of matroids of different ranks, and yet again when we require those matroids to have special properties, such as being paving.