Master's Research

For the past year I have been studying the domination number of grid graphs and similar structures with Dr. Erik Insko, Katie Johnson, and Christie Mauretour. This has been an exposure to the trials of mathematic research, and a chance to try my hand at computer aided proofs via SAGE.

A preprint of the paper is available on arxiv.org.

I recently gave a MAA Contributed Paper Session talk at the Joint Mathematics Meeting 2014 in Baltimore, slides available.

Domination is a way of measuring the complexity of a graph, through selecting a subset of vertices so that every vertex is either in the dominating set or adjacent to it. If we take the infimum over all possible domination sets, we come up with an invariant of the graph, the domination number. The topic of domination of grids has been a topic of consideration for over 20 years.

For a proof of Chang's conjecture (1992) of the domination number of large grids see The Domination Number of Grids in order to get a flavor for what this area of mathematics is all about.

We concern ourselves with a generalization of domination number, (t,r)-broadcast domination, which is classical domination when t = 2 and r = 1. As an example, consider the (3,2)-broadcast domination of a 5 by 5 grid from a paper in the final stages of writing.

We are currently in the process of proving closed form formulas for the domination number of several small cases, ie, (3 x n) for all n in the (2,2)-broadcast domination. This style of domination has the potential for applications in network area coverage, security of large areas, ect.

Below you will find my slides from the USTARS 2013 presentation at Purdue University where I gave a presentation on the domination number of half grids.