Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

Connectivity is a central theme in both graph theory and matroid theory. This dissertation investigates extremal connectivity in graphs and matroids, with emphasis on unavoidable structures and minimal connectivity phenomena.

Chapter 2 introduces cycle-contraction minors of graphs and investigates their structural properties. We establish a connection between cc-minors and induced subgraphs via graph duality. The main result gives an unavoidable-families characterization for cc-minors of sufficiently large loopless $2$-connected graphs.

Chapter 3 studies super-minimally $3$-connected graphs, namely $3$-connected graphs that have no proper $3$-connected subgraphs. We establish extremal bounds on structural parameters of these graphs, including the minimum number of degree-$3$ vertices and the maximum number of edges.

Chapter 4 develops the matroid analogue of super-minimal $3$-connectivity. We prove extremal results for super-minimally $3$-connected matroids, including bounds on the maximum number of elements and density of elements contained in triads. These results extend methods from graph connectivity theory to matroids and illustrate parallels between the two settings.

Date

4-9-2026

Committee Chair

Oxley, James

LSU Acknowledgement

1

LSU Accessibility Acknowledgment

1

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