#### Identifier

etd-07052014-001609

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

Matroid k-connectivity is typically defined in terms of a connectivity function. We can also say that a matroid is 2-connected if and only if for each pair of elements, there is a circuit containing both elements. Equivalently, a matroid is 2-connected if and only if each pair of elements is in a certain 2-element minor that is 2-connected. Similar results for higher connectivity had not been known. We determine a characterization of 3-connectivity that is based on the containment of small subsets in 3-connected minors from a given list of 3-connected matroids. Bixby’s Lemma is a well-known inductive tool in matroid theory that says that each element in a 3-connected matroid can be deleted or contracted to obtain a matroid that is 3-connected up to minimal 2-separations. We consider the binary matroids for which there is no element whose deletion and contraction are both 3-connected up to minimal 2-separations. In particular, we give a decomposition for such matroids to establish that any matroid of this type can be built from sequential matroids and matroids with many fans using a few natural operations. Wagner defined biconnectivity to translate connectivity in a bicircular matroid to certain connectivity conditions in its underlying graph. We extend a characterization of biconnectivity to higher connectivity. Using these graphic connectivity conditions, we call upon unavoidable minor results for graphs to find unavoidable minors for large 4-connected bicircular matroids.

#### Date

2014

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Moss, John Tyler, "Extremal Problems in Matroid Connectivity" (2014). *LSU Doctoral Dissertations*. 2234.

https://digitalcommons.lsu.edu/gradschool_dissertations/2234

#### Committee Chair

Oxley, James