# Zero forcing and its applications

January 30 to February 3, 2017

at the

American Institute of Mathematics, San Jose, California

organized by

Shaun Fallat, Simone Severini, and Michael Young

## Original Announcement

This workshop will be devoted to the theory of zero forcing and its applications. Zero forcing is a propagation on graphs described by the following process. Consider a graph $G$ and color each of its vertices blue or white. A blue vertex $v$ can force a white vertex $w$ to be blue if $w$ is the only white vertex in the neighborhood of $v$. A zero forcing set of $G$ is a set of vertices $S \subset V(G)$ such that if the vertices of $S$ are colored blue and the remaining vertices are colored white, then every vertex can eventually become blue, after a repeated application of the forcing process. The zero forcing number of $G, Z(G),$ is the minimum cardinality of a zero forcing set.

The concept of zero forcing has been used in multiple branches of science and mathematics for many years. This workshop will discuss and study the zero forcing number of graphs, and its applications to linear algebra, computer science, power networks, and mathematical physics. We will also look at the contemporary problems in computing zero forcing numbers and the propagation time of zero forcing. Other types of zero forcing (e.g. positive semidefinite zero forcing) have been defined and each type has been defined on graphs, directed graphs, and graphs with loops. These related parameters may be investigated also.

The main topics for the workshop are:

• Applications of zero forcing to inverse eigenvalue problems, PMU placement problems, and quantum control problems.
• Connections to graph searching and certain minor-monotone parameters.
• Computational methods of zero forcing.
• Propagation times of zero forcing.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Rigid linkages and partial zero forcing
by  Daniela Ferrero, Mary Flagg, H. Tracy Hall, Leslie Hogben, Jephian C.-H. Lin, Seth Meyer, Shahla Nasserasr, and Bryan Shader
Properties of a q-analogue of zero forcing
by  Steve Butler, Craig Erickson, Shaun Fallat, H. Tracy Hall, Brenda Kroschel, Jephian C.-H. Lin, Bryan Shader, Nathan Warnberg, and Boting Yang
Restricted power domination and zero forcing problems
by  Chassidy Bozeman, Boris Brimkov, Craig Erickson, Daniela Ferrero, Mary Flagg, and Leslie Hogben
Throttling for the game of Cops and Robbers on graphs
by  Jane Breen, Boris Brimkov, Joshua Carlson, Leslie Hogben, K. E. Perry, and Carolyn Reinhart
Throttling positive semidefinite zero forcing propagation time on graphs
by  Joshua Carlson, Leslie Hogben, Jurgen Kritschgau, Kate Lorenzen, Michael S. Ross, Seth Selken, and Vicente Valle Martinez
Families of graphs with maximum nullity equal to zero-forcing number
by  J.S. Alameda, E. Curl, A. Grez, L. Hogben, O-N. Kingston, A. Schulte, D. Young, and M. Young,  Spec. Matrices 6 (2018), 56–67  MR3760976