#### Vol. 8, No. 1, 2015

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The failed zero forcing number of a graph

### Katherine Fetcie, Bonnie Jacob and Daniel Saavedra

Vol. 8 (2015), No. 1, 99–117
##### Abstract

Given a graph $G$, the zero forcing number of $G$, $Z\left(G\right)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the color change rule results in all vertices joining $S$. The color change rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ joins $S$ in the next iteration.

In this paper, we introduce a new graph parameter, the failed zero forcing number of a graph. The failed zero forcing number of $G$, $F\left(G\right)$, is the maximum cardinality of any set of vertices on which repeated applications of the color change rule will never result in all vertices joining the set.

We establish bounds on the failed zero forcing number of a graph, both in general and for connected graphs. We also classify connected graphs that achieve the upper bound, graphs whose failed zero forcing numbers are zero or one, and unusual graphs with smaller failed zero forcing number than zero forcing number. We determine formulas for the failed zero forcing numbers of several families of graphs and provide a lower bound on the failed zero forcing number of the Cartesian product of two graphs.

We conclude by presenting open questions about the failed zero forcing number and zero forcing in general.

##### Keywords
zero forcing number, vertex labeling, graph coloring
##### Mathematical Subject Classification 2010
Primary: 05C15, 05C78, 05C57
Secondary: 05C50
##### Milestones
Received: 21 June 2013
Revised: 30 July 2013
Accepted: 4 August 2013
Published: 10 December 2014

Communicated by Joseph A. Gallian
##### Authors
 Katherine Fetcie Department of Civil Engineering Technology, Environmental Management and Safety Rochester Institute of Technology 1 Lomb Memorial Drive Rochester, NY 14623 United States Bonnie Jacob Science and Mathematics Department, National Technical Institute for the Deaf Rochester Institute of Technology 52 Lomb Memorial Drive Rochester, NY 14623 United States Daniel Saavedra Department of Packaging Science College of Applied Science and Technology Rochester Institute of Technology 1 Lomb Memorial Drive Rochester, NY 14623 United States