Skip to main content
SpringerLink
Log in

On the Semitotal Forcing Number of a Graph

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

Zero forcing is an iterative graph coloring process that starts with a subset S of “colored" vertices, all other vertices being “uncolored". At each step, a colored vertex with a unique uncolored neighbor forces that neighbor to be colored. If at the end of the forcing process all the vertices of the graph are colored, then the initial set S is called a zero forcing set. If in addition, every vertex in S is within distance 2 of another vertex of S, then S is a semitotal forcing set. The semitotal forcing number \(F_{t2}(G)\) of a graph G is the cardinality of the smallest semitotal forcing set of G. In this paper, we begin to study basic properties of \(F_{t2}(G)\), relate \(F_{t2}(G)\) to other domination parameters, and establish bounds on the effects of edge operations on the semitotal forcing number. We also investigate the semitotal forcing number for subfamilies of cubic graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. AIM Special Work Group: Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl. 428(7), 1628–1648 (2008)

  2. Brimkov, B., Mikesell, D., Smith. L.: Connected power domination in graphs. J. Combinatorial Optimization 38, 292–315 (2019)

  3. Brimkov, B., Davila, R.: Characterizations of the connected forcing number of a graph. arXiv preprint arXiv:1604.00740, (2016)

  4. Brimkov, B., Hicks, I.V.: Complexity and computation of connected zero forcing. Discrete Appl. Math. 229, 31–45 (2017)

    Article  MathSciNet  Google Scholar 

  5. Burgarth, D., Giovannetti, V.: Full control by locally induced relaxation. Phys. Rev. Lett. 99(10), 100501 (2007)

    Article  Google Scholar 

  6. Davila, R., Henning, M.A.: Total forcing and zero forcing in claw-free cubic graphs. Graphs and Combinatorics 34(6), 1371–1384 (2018)

    Article  MathSciNet  Google Scholar 

  7. Davila, R., Henning, M.A.: On the total forcing number of a graph. Discrete Appl. Math. 257, 115–127 (2019)

    Article  MathSciNet  Google Scholar 

  8. Davila, R., Henning, M.A.: Total forcing versus total domination in cubic graphs. Appl. Math. Comput. 354, 385–395 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Davila, R., Kalinowski, T., Stephen, S.: A lower bound on the zero forcing number. Discrete Appl. Math. 250, 363–367 (2018)

    Article  MathSciNet  Google Scholar 

  10. Davila, R., Henning, M.A.: Zero forcing versus domination in cubic graphs. J. Combinatorial Optim. 41, 553–577 (2021)

    Article  MathSciNet  Google Scholar 

  11. Edholm, C.J., Hogben, L., Huynh, M., et al.: Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph. Linear Algebra Appl. 436(12), 4352–4372 (2012)

    Article  MathSciNet  Google Scholar 

  12. Fuerst, M., Rautenbach, D.: A short proof for a lower bound on the zero forcing number. Discuss. Math. Graph Theory 40(1), 355–360 (2020)

    Article  MathSciNet  Google Scholar 

  13. Goddard, W., Henning, M.A., McPillan, C.A.: Semitotal domination in graphs. Util. Math. 94, 67–81 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Gravier, S.: Total domination number of grid graphs. Discret. Appl. Math. 121(1), 119–128 (2002)

    Article  MathSciNet  Google Scholar 

  15. Haynes, T. W., Hedetniemi, S., Slater, P.: Fundamentals of domination in graphs. CRC Press, 2013

  16. Henning, M.A., Löwenstein, C.: Locating-total domination in claw-free cubic graphs. Discrete Math. 312(21), 3107–3116 (2012)

    Article  MathSciNet  Google Scholar 

  17. Henning, M.A., Yeo, A.: Total domination in graphs. Springer, New York (2013)

    Book  Google Scholar 

  18. Rashidi, S., Poursalavati, N.S., Tavakkoli, M.: Computing the zero forcing number for generalized Petersen graphs. J Algebra Combinatorics Discrete Struct. Appl. 7(2), 183–193 (2020)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Nature Science Foundation of China (No. 11701542 ).

Author information

Authors and Affiliations

  1. College of Science, China Jiliang University, Hangzhou, 310018, People’s Republic of China

    Qin Chen

Authors
  1. Qin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qin Chen.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Xueliang Li.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Q. On the Semitotal Forcing Number of a Graph. Bull. Malays. Math. Sci. Soc. 45, 1409–1424 (2022). https://doi.org/10.1007/s40840-021-01236-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-021-01236-2

Keywords

Mathematics Subject Classification

Search

Navigation