Delta-matroids as subsystems of sequences of Higgs lifts

Abstract

In [30], 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 deltamatroids 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.

Lattice Path Matroids and Quotients

“…are precisely flags of positroids. Example 7 Let M 1 be the positroid on [3] whose set of bases is B 1 = {1, 3} and let M 2 = U 2,3 be the uniform matroid of rank 2 on [3]. That is, the bases of M 2 are B 2 = {12, 13, 23}.…”

Lattice path matroids and quotients

“…Given two matroids M and M such that M ≤ q M , we say that a matroid M is the ith Higgs lift of M towards M if M is the maximal matroid (with respect to ≤ w ) such that r(M ) = r(M ) + i and M ≤ q M ≤ q M . See [12, Propositions 2.2, 2.6] and [8] for proof that the Higgs lift always exists. Notice that the ith Higgs lift of U 0,n towards M is simply the i-truncation of M , i.e., its base set is given by B i := {X ∈ [n] i | ∃B ∈ B : X ⊆ B }.…”

The excluded 3-minors for vf-safe delta-matroids

“…Proof. For part (1), note that a vertex minor of an unlooped simple graph is obtained by a sequence of local complementations and vertex deletions. The result follows from Proposition 5.4 and the fact that if v is a vertex of G then…”

“…The order in which elements are deleted or contracted can matter. See [1] for an example. However, for disjoint subsets X and Y of E, if some set in F is disjoint from X and contains Y , then the deletions and contractions in S\X/Y can be done in any order, and…”

“…(2) Let D be a twisted dual of a basic binary delta-matroid and let D be a 3-minor of D.Then there are graphs G and G such that D(G) and D(G ) are twisted duals of D and D respectively, and G is a vertex minor of G.Proof. For part(1), note that a vertex minor of an unlooped simple graph is obtained by a sequence of local complementations and vertex deletions. The result follows from Proposition 5.4 and the factthat if v is a vertex of G then D(G \ v) = D(G) \ v.For part (2), let F be a feasible set of D and letB = {e ∈ E(D) : {e} ∈ F(D * F )}.The remark following Corollary 5.2 implies that D * F is basic binary, so (D * F )+B is an even basic binary delta-matroid, so (D * F ) + B = D(G) for some unlooped simple graph G. It follows from Lemma 3.4 that there is a delta-matroid D that is a 3-minor of D(G) and a twisted dual of D .…”