Date on Honors Thesis

Spring 5-2025

Major

Mathematics/Data Science

Minor

Cognitive Science

Examining Committee Member

Dubravko Ivansic, PhD, Advisor

Examining Committee Member

Elizabeth Donovan, PhD, Committee Member

Examining Committee Member

Christopher Mecklin, PhD, Committee Member

Abstract/Description

Analysis of a childhood game has led us to the problem of maximum independent sets in planar graphs. We wrote a graph creation utility using R to generate a random planar map and its dual graph. This utility then finds a graph’s maximal independent set using a variety of six algorithms. We investigate statistical connections between graph structure, colorability, and the maximal independent sets found using these algorithms over an incredibly large and procedurally generated dataset. We find one can always win the coloring game if the resultant graph is two-colorable. The algorithms perform statistically and practically significantly better on two-colorable graphs, as well. We analyze pairwise statistical and practical significance between the algorithms to choose the best performing one. We additionally analyze performance of a purely random algorithm’s multiple iterations for possible strength in brute-force randomization. Lastly, we find hidden structure in the results indicative of the underlying structure inherent of two-colorable graphs and the performance improvements associated with them. This leads us to a newly found possible fractal related to the proportion of nodes contained in the maximal independent set found. Our research concludes that straightforward algorithmic design provides both implementation and execution time improvements in this limited case.

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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