On Tight Bounds for the k-Forcing Number of a Graph

  • Yan ZhaoEmail author
  • Lily Chen
  • Hengzhe Li


The k-forcing number of a graph G, denoted by \(F_k(G)\), was introduced by Amos et al. It is a generalization of the zero forcing number of a graph G, denoted by Z(G). Amos et al. proved that for a connected graph G of order n with maximum degree \(\varDelta \ge 2\), \(Z(G)=F_1(G)\le \frac{(\varDelta -2)n+2}{\varDelta -1}\), and this inequality is sharp. Moreover, they posed a conjecture that \(Z(G)=F_1(G)=\frac{(\varDelta -2)n+2}{\varDelta -1}\) if and only if \(G=C_n\), \(G=K_{\varDelta +1}\) or \(G=K_{\varDelta ,\varDelta }\). In this paper, we prove that this conjecture is true. Moreover, we point out a mistake in their paper and get a stronger result which shows that \(F_{n-1}(G)=1\) if and only if G is connected and \(F_k(G)=n-k\) if and only if \(G=K_n\) for \(k\le n-2\).


k-forcing k-forcing number Zero forcing set 

Mathematics Subject Classification

05C15 05C69 



Yan Zhao was partially supported by the Natural Science Foundation of Jiangsu Province (No. BK20160573), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 16KJD110005).


  1. 1.
    Barioli, F., Barrett, W., Butler, S., Cioaba, S., Cvetkovic, D., Fallat, S., Godsil, C., Haemers, W., Hogben, L., Mikkelson, R., Narayan, S., Pryporova, O., Sciriha, I., So, W., Stevanovic, D., van der Holst, H., Meulen, K.V., Wehe, A.W., AIM Minimum Rank-Special Graphs Work Group: Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl. 428, 1628–1648 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amos, D., Caro, Y., Davila, R., Pepper, R.: Upper bounds on the \(k\)-forcing number of a graph. Discrete Appl. Math. 181, 1–10 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burgarth, D., Giovannetti, V.: Full control by locally induced relaxation. Phys. Rev. Lett. 99, 100501 (2007)CrossRefGoogle Scholar
  4. 4.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan Press Ltd, London (1976)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bondy, J.A., Murty, U.S.R.: Graph Theory, GTM 244. Springer, New York (2008)CrossRefGoogle Scholar
  6. 6.
    Caro, Y., Pepper, R.: Dynamic approach to \(k\)-forcing, arXiv:1405.7573 (2014)
  7. 7.
    Chilakammari, K.B., Dean, N., Kang, C.X., Yi, Eunjeong: Iteration index of a zero forcing set in a graph. Bull. Inst. Combin. Appl. 64, 57–72 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Edholm, C.J., Hogben, L., Huynh, M., LaGrange, J., Row, D.D.: Vertex and edge spread of the zero forcing number, maximum nullity, and minimum rank of a graph. Linear Algebra Appl. 436, 4352–4372 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hogben, L., Huynh, M., Kingsley, N., Meyer, S., Walker, S., Young, M.: Propagation time for zero forcing on a graph. Discrete Appl. Math. 160, 1994–2005 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Meyer, S.A.: Zero forcing sets and bipartite circulants. Linear Algebra Appl. 436, 888–900 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Row, D.: A technique for computing the zero forcing number of a graph with a cut-vertex. Linear Algebra Appl. 436, 4423–4432 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of MathematicsTaizhou UniversityTaizhouChina
  2. 2.School of Mathematics ScienceHuaqiao UniversityQuanzhouChina
  3. 3.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangChina

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