Delta-matroids as subsystems of sequences of Higgs lifts

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.

Figures (10)

• Figure 1: The graphs whose edges give the proper, nonempty feasible sets of $U_3$, $U_4$, $U_5$, $U_6$, and $U_7$, respectively.
• Figure 2: Examples of (a) the region of interest, (b) the lattice path matroid it gives, which is the transversal matroid that has the presentation $\{\{1,2,3\},\{2,\ldots,6\},\{5,\ldots,8\},\{8,9\}\}$, (c) a quotient of that matroid, and (d) a region that yields the quotient.
• Figure 3: Above, a typical region of interest. Below, the lattice path representations of the two associated lattice path matroids, $M(\mathcal{R}_{\max})$ and $M(\mathcal{R}_{\min})$.
• Figure 4: A sketch of how to get the path $P_{B'}$ (dashed) from $P_B$ (in gray) in the proof of Proposition \ref{['prop:genlpquot']}.
• Figure 5: Exchanging $f$ for a smaller element $e$ diverts the solid path around the shaded region to the left, as the dashed path in the first part shows. Exchanging $f$ for a larger element $e$ diverts the path around the shaded region to the right, as the dashed path in the second part shows.

Theorems & Definitions (45)

• Proposition 1.1
• Lemma 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Proposition 3.1
• proof
• Lemma 3.2
• Corollary 3.3