Variations of zero forcing and power domination

Abstract

Zero forcing is a propagation process on a graph that turns white vertices into blue vertices. In this process, an initial set of vertices in a graph $G$ are chosen to be blue and all others are colored white, then a color-change rule is iteratively applied until all of $G$ becomes blue. A large amount of research has gone into finding the size of a minimum zero forcing set in graphs as the process has connections in linear algebra, computer science, and even physics. In power networks, the zero forcing process is used to measure the phase throughout the system. This process is known as the power domination process, and it is a well researched area of graph theory. Power domination has since been generalized using different variations of zero forcing. The generalization that will be discussed in this dissertation is known as $k$-power domination which uses the generalization of zero forcing known as $k$-forcing. Recently, a new variation of zero forcing known as leaky forcing was introduced to research concerns involving faulty vertices in a system. An $\ell$-leaky forcing set is a zero forcing set that is resistant to any $\ell$ ``broken'' vertices (leaks) in a graph. In this dissertation, results proving $\ell$-leaky forcing sets and $\ell$-edge leaky forcing sets are equivalent are provided. Furthermore, bounds for minimum sized $k$-power dominating sets in hypergraphs are proven.