Involve: A Journal of Mathematics
- Volume 8, Number 1 (2015), 99-117.
The failed zero forcing number of a graph
Given a graph , the zero forcing number of , , is the smallest cardinality of any set of vertices on which repeated applications of the color change rule results in all vertices joining . The color change rule is: if a vertex is in , and exactly one neighbor of is not in , then joins 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 , , 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.
Involve, Volume 8, Number 1 (2015), 99-117.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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.)
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