#### Campus Units

Mathematics

#### Document Type

Article

#### Publication Version

Accepted Manuscript

#### Publication Date

9-2012

#### Journal or Book Title

Discrete Applied Mathematics

#### Volume

160

#### First Page

1994

#### Last Page

2004

#### DOI

10.1016/j.dam.2012.04.003

#### Abstract

Zero forcing (also called graph infection) on a simple, undirected graph G is based on the color-change rule: if each vertex of G is colored either white or black, and vertex v is a black vertex with only one white neighbor w, then change the color of w to black. A minimum zero forcing set is a set of black vertices of minimum cardinality that can color the entire graph black using the color change rule. The propagation time of a zero forcing set B of graph G is the minimum number of steps that it takes to force all the vertices of G black, starting with the vertices in B black and performing independent forces simultaneously. The minimum and maximum propagation times of a graph are taken over all minimum zero forcing sets of the graph. It is shown that a connected graph of order at least two has more than one minimum zero forcing set realizing minimum propagation time. Graphs G having extreme minimum propagation times |G|−1, |G|−2, and 0 are characterized, and results regarding graphs having minimum propagation time 1 are established. It is shown that the diameter is an upper bound for maximum propagation time for a tree, but in general propagation time and diameter of a graph are not comparable.

#### Rights

This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

#### Copyright Owner

Elsevier B.V.

#### Copyright Date

2012

#### Language

en

#### File Format

application/pdf

#### Recommended Citation

Hogben, Leslie; Huynh, My; Kingsley, Nicole; Meyer, Sarah; Walker, Shanise; and Young, Michael, "Propagation time for zero forcing on a graph" (2012). *Mathematics Publications*. 70.

https://lib.dr.iastate.edu/math_pubs/70

## Comments

This is a manuscript of an article from

Discrete Applied Mathematics26 (2012): 1994, doi:10.1016/j.dam.2012.04.003. Posted with permission.