The excluded minors for the intersection of bicircular and lattice path matroids

Abstract

The classes of bicircular matroids and lattice path matroids are closed under minors. The complete list of excluded minors for the class of lattice path matroids is known, and it has been recently shown that the analogous list for the class of bicircular matroids is finite. In this paper, we establish the complete list of excluded minors for the class of matroids that is the intersection of these two classes. This resolves a recently posed open problem.

Figures (10)

• Figure 1: Excluded minors for the class of bicircular and lattice path matroids that are bicircular.
• Figure 2: Excluded minors for the class of bicircular and lattice path matroids that are neither bicircular nor lattice path. Note that $r(B_{3,2})=3$ and $r(E_4) = 4$.
• Figure 3: An example lattice path presentation and the bipartite graph of its standard presentation. Path $R$ represents the basis $\{2,5,6,8,9\}$.
• Figure 4: Excluded minors for the class of lattice path matroids that are not in infinite families.
• Figure 5: Examples of excluded minors for the class of lattice path matroids from infinite families.

Theorems & Definitions (36)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1
• Lemma 3.2