Date of Award

Summer 8-23-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Computational Analysis and Modeling

First Advisor

Galen Turner

Abstract

This dissertation introduces the class of equitably dissectible graphs — graphs whose vertex set can be recursively partitioned into two parts whose sizes differ by at most one where the two induced subgraphs are connected. In Chapter 2, we show that a graph is equitably dissectible if and only if it has a spanning tree that is equitably dissectible, and we use this result to establish connections between the class of equitably dissectible graphs and Hamiltonian graphs. We also fully characterize the equitably dissectible spiders of the form S3(a) and S3(a, b), identifying a family of unruly spiders that are not equitably dissectible. In Chapter 3, we investigate the subclass of equitably dissectible trees, establishing structural bounds on the maximal degree of such trees. Chapter 4 examines equitable dissections in complete multipartite graphs, including a complete characterization of equitably dissectible complete bipartite graphs. In Chapter 5, we address the computational complexity of determining whether a graph is equitably dissectible and present algorithms that detect equitable dissections in trees. Finally, Chapter 6 explores equitable dissections in broader contexts, including arbitrarily vertex-decomposable graphs and combinatorial observations.

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