Induced-minor-closed classes of matroids

Author

James Dylan Douthitt, Louisiana State University and Agricultural and Mechanical CollegeFollow

Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of simple GF(q)-representable matroids that can be built from projective geometries over GF(q) by repeated generalized parallel connections across projective geometries. We show that this class of matroids is closed under induced minors and characterize the class by its forbidden induced minors, noting that the case when q=2 is distinctive. Additionally, we show that the class of GF(2)-chordal matroids coincides with the class of binary matroids that have none of M(K₄), M^∗(K_3,3), or M(C_n) for n ≥ 4 as a flat. We also show that GF(q)-chordal matroids can be characterized by an analogous result to Rose’s 1970 characterization of chordal graphs as those that have a perfect elimination ordering of vertices. We then describe the classes of binary matroids with pairs from the set {M(C₄),M(K₄\e),M(K₄), F₇} as excluded induced minors. Additionally, we prove structural lemmas toward characterizing the class of binary matroids that do not contain M(K₄) as an induced minor.

Date

3-9-2025

Recommended Citation

Douthitt, James Dylan, "Induced-minor-closed classes of matroids" (2025). LSU Doctoral Dissertations. 6688.
https://repository.lsu.edu/gradschool_dissertations/6688

Committee Chair

Oxley, James