Doctor of Philosophy (PhD)
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.
Document Availability at the Time of Submission
Release the entire work immediately for access worldwide.
Moss, John Tyler, "Extremal Problems in Matroid Connectivity" (2014). LSU Doctoral Dissertations. 2234.