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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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

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.)

zero forcing number vertex labeling graph coloring


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.

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  • L. Babai, “Automorphism groups, isomorphism, reconstruction”, Chapter 27, pp. 1447–1540 in Handbook of combinatorics, vol. 1, edited by R. L. Graham et al., Elsevier, Amsterdam, 1995.
  • F. Barioli, W. Barrett, S. Butler, S. M. Cioabă, D. Cvetković, S. M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanović, H. van der Holst, K. Vander Meulen, and A. Wangsness, “Zero forcing sets and the minimum rank of graphs”, Linear Algebra Appl. 428:7 (2008), 1628–1648.
  • F. Barioli, W. Barrett, S. M. Fallat, H. T. Hall, L. Hogben, B. Shader, P. van den Driessche, and H. van der Holst, “Zero forcing parameters and minimum rank problems”, Linear Algebra Appl. 433:2 (2010), 401–411.
  • L. DeLoss, J. Grout, T. McKay, J. Smith, and G. Tims, “Program for calculating bounds on the minimum rank of a graph using Sage”, preprint, 2008.
  • S. M. Fallat and L. Hogben, “The minimum rank of symmetric matrices described by a graph: a survey”, Linear Algebra Appl. 426:2-3 (2007), 558–582.
  • F. Harary and S. Hedetniemi, “The achromatic number of a graph”, J. Combinatorial Theory 8 (1970), 154–161.
  • D. D. Row, Zero forcing number: results for computation and comparison with other graph parameters, Ph.D. thesis, Iowa State University, Ames, IA, 2011, hook \posturlhook.