Abstract
Domination theory is a subfield within graph theory that aims to describe subsets of the vertices of a graph which satisfy certain distance properties. The original domination problem asked one to find subsets of the vertices of a graph (with minimal cardinality) so that every vertex in the graph was either in the set or adjacent to a vertex in the set. Since its development, thousands of papers on domination theory and its many variants have appeared in the literature. We focus our study on (t, r) broadcast domination, a variant with a connection to the placement of cellphone towers, where some vertices send out a signal to nearby vertices (with the signal decaying linearly along edges according to distance), and where all vertices must receive a minimum predetermined amount of this signal. The overall goal is to minimize the number of tower vertices needed to have all vertices receive the appropriate amount of signal reception. We summarize our past work with students and present many remaining open problems in this field. We end the chapter by providing some advice on how we continue to develop new research projects with and for students; although the mathematical content of the chapter is in domination theory, the suggestions can be implemented in any area.
