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

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Abstract

Given a graph G, the zero forcing number of G, Z(G), 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(G), 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


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