## Involve: A Journal of Mathematics

- Involve
- Volume 8, Number 1 (2015), 99-117.

### The failed zero forcing number of a graph

Katherine Fetcie, Bonnie Jacob, and Daniel Saavedra

#### 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.

#### Article information

**Source**

Involve, Volume 8, Number 1 (2015), 99-117.

**Dates**

Received: 21 June 2013

Revised: 30 July 2013

Accepted: 4 August 2013

First available in Project Euclid: 22 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1511370843

**Digital Object Identifier**

doi:10.2140/involve.2015.8.99

**Mathematical Reviews number (MathSciNet)**

MR3321714

**Zentralblatt MATH identifier**

1309.05071

**Subjects**

Primary: 05C15: Coloring of graphs and hypergraphs 05C78: Graph labelling (graceful graphs, bandwidth, etc.) 05C57: Games on graphs [See also 91A43, 91A46]

Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

**Keywords**

zero forcing number vertex labeling graph coloring

#### Citation

Fetcie, Katherine; Jacob, Bonnie; Saavedra, Daniel. The failed zero forcing number of a graph. Involve 8 (2015), no. 1, 99--117. doi:10.2140/involve.2015.8.99. https://projecteuclid.org/euclid.involve/1511370843