2018/01/21 by Tara Fife, Fife, Tara, James Oxley +1 · 1 citation
Mathematics · #05B35 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B35
paper · pdf · doi:10.48550/arxiv.1801.06882
12 pages
arxiv created 2018/01/21 · arxiv updated 2018/01/23
Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid M is Hamiltonian if it has a spanning circuit. A matroid M is nested if and only if its Hamiltonian flats form a chain under inclusion; M is laminar if and only if, for every 1-element independent set X, the Hamiltonian flats of M containing X form a chain under inclusion. We generalize these notions to define the classes of k-closure-laminar and k-laminar matroids. This paper focuses on structural properties of these classes noting that, while the second class is always minor-closed, the first is if and only if k ≤ 3. The main results are excluded-minor characterizations for the classes of 2-laminar and 2-closure-laminar matroids.