Skip to main content
SpringerLink
Log in

A New Lower Bound for Positive Zero Forcing

  • Conference paper
  • First Online:
Frontiers in Algorithmics (FAW 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10336))

Included in the following conference series:

  • 533 Accesses

Abstract

The positive zero forcing number is a variant of the zero forcing number, which is an important parameter in the study of minimum rank/maximum nullity problems. In this paper, we first introduce the propagation decomposition of graphs; then we use this decomposition to prove a lower bound for the positive zero forcing number of a graph. We apply this lower bound to find the positive zero forcing number of matching-chain graphs. We prove that the positive zero forcing number of a matching-chain graph is equal to its zero forcing number. As a consequence, we prove the conjecture about the positive zero forcing number of the Cartesian product of two paths, and partially prove the conjecture about the positive zero forcing number of the Cartesian product of a cycle and a path. We also show that the positive zero forcing number and the zero forcing number agree for claw-free graphs. We prove that it is NP-complete to find the positive zero forcing number of line graphs.

Research supported in part by an NSERC Discovery Research Grant, Application No.: RGPIN-2013-261290.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Article  MathSciNet  Google Scholar 

  2. Barioli, F., Barrett, W., Fallat, S., Hall, H.T., Hogben, L., Shader, B., van den Driessche, P., van der Holst, H.: Parameters related to tree-width, zero forcing, and maximum nullity of a graph. J. Graph Theory 72, 146–177 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barioli, F., Barrett, W., Fallat, S., Hall, H.T., Hogben, L., Shader, B., van den Driessche, P., van der Holst, H.: Zero forcing parameters and minimum rank problems. Linear Algebra Appl. 433(2), 401–411 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barioli, F., Fallat, S., Mitchell, L., Narayan, S.: Minimum semidefinite rank of outerplanar graphs and the tree cover number. Electron. J. Linear Algebra 22, 10–21 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Booth, M., Hackney, P., Harris, B., Johnson, C.R., Lay, M., Mitchell, L.H., Narayan, S.K., Pascoe, A., Steinmetz, K., Sutton, B.D., Wang, W.: On the minimum rank among positive semidefinite matrices with a given graph. SIAM J. Matrix Anal. Appl. 30, 731–740 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  7. Ekstrand, J., Erickson, C., Hall, H.T., Hay, D., Hogben, L., Johnson, R., Kingsley, N., Osborne, S., Peters, T., Roat, J., Ross, A., Row, D., Warnberg, N., Young, M.: Positive semidefinite zero forcing. Linear Algebra Appl. 439, 1862–1874 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ekstrand, J., Erickson, C., Hay, D., Hogben, L., Roat, J.: Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial 2-trees. Electron. J. Linear Algebra 23, 79–87 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fallat, S., Hogben, L.: Minimum rank, maximum nullity, and zero forcing number of graphs. In: Handbook of Linear Algebra, Discrete Mathematics and its Applications, chap. 46, pp. 775–810. CRC Press (2013)

    Google Scholar 

  10. Fallat, S., Meagher, K., Yang, B.: On the complexity of the positive semidefinite zero forcing number. Linear Algebra Appl. 491, 101–122 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fallat, S., Meagher, K., Soltani, A., Yang, B.: Positive zero forcing and edge clique coverings. In: Zhu, D., Bereg, S. (eds.) FAW 2016. LNCS, vol. 9711, pp. 53–64. Springer, Cham (2016). doi:10.1007/978-3-319-39817-4_6

    Google Scholar 

  12. Fallat, S., Soltani, A.: Line graphs: their maximum nullities and zero forcing numbers. Czechoslov. Math. J. 66, 743–755 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, L.-H., Chang, G.J., Yeh, H.-G.: On minimum rank and zero forcing sets of a graph. Linear Algebra Appl. 432, 2961–2973 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamiltonian paths in grid graphs. SIAM J. Comput. 11(4), 676–686 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lovász, L., Plummer, M.D.: Matching Theory. North Holland Publishing, Amsterdam (1986)

    MATH  Google Scholar 

  16. Peters, T.: Positive semidefinite maximum nullity and zero forcing number. Electron. J. Linear Algebra 23, 815–830 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Warnberg, N.: Positive semidefinite propagation time. Discret. Appl. Math. 198, 274–290 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)

    Google Scholar 

  19. Yang, B.: Fast-mixed searching and related problems on graphs. Theoret. Comput. Sci. 507(7), 100–113 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yang, B.: Fast edge-searching and fast searching on graphs. Theoret. Comput. Sci. 412, 1208–1219 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang, B.: Lower bounds for positive semidefinite zero forcing and their applications. J. Comb. Optim. 33, 81–105 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Regina, Regina, SK, Canada

    Boting Yang

Authors
  1. Boting Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boting Yang .

Editor information

Editors and Affiliations

  1. University of Electronic Science and Technology of China, Chengdu, China

    Mingyu Xiao

  2. University of Bergen, Bergen, Norway

    Frances Rosamond

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Yang, B. (2017). A New Lower Bound for Positive Zero Forcing. In: Xiao, M., Rosamond, F. (eds) Frontiers in Algorithmics. FAW 2017. Lecture Notes in Computer Science(), vol 10336. Springer, Cham. https://doi.org/10.1007/978-3-319-59605-1_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-59605-1_23

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59604-4

  • Online ISBN: 978-3-319-59605-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation