Speaker:
Leslie Hogben
Iowa State University and American Institute of Mathematics

Title:
Power domination and zero forcing: Using graphs to model real-world problems

Abstract:
A graph $G = (V, E)$ is a set of vertices $V = {1, dots , n}$ and set of edges $E$ of
two element sets of vertices. A graph can be used to model connections between
vertices, such as airline routes between cities, internet connections, a quantum
system, or an electric power network.
Power domination and zero forcing are related coloring processes on graphs.
We start with a set of vertices colored blue and the rest colored white. We apply
a color change rule to color the white vertices blue. A set of blue vertices that
can color all vertices blue by using the power domination color change rule (or
zero forcing color change rule) is called a power dominating set (or a zero forcing
set). Finding a such set allows us to solve various problems, and a minimum
such set can provide an optimal solution.
In an electric power network, a power dominating set (blue vertices) gives
a set of locations from which monitoring units can observe the entire network.
In a quantum system, a zero forcing set (blue vertices) gives a set of locations
from which the entire system can be controlled.
This talk will describe power domination and zero forcing processes on
graphs and some of their applications.