IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v39y2020i2d10.1007_s10878-019-00475-1.html
   My bibliography  Save this article

On the zero forcing number of a graph involving some classical parameters

Author

Listed:
  • Shuchao Li

    (Central China Normal University)

  • Wanting Sun

    (Central China Normal University)

Abstract

Given a simple graph G, let $$Z(G),\, p(G),\, \Phi (G), ex(G)$$Z(G),p(G),Φ(G),ex(G) and M(G), respectively, be the zero forcing number, the number of pendant vertices, the cyclomatic number, the number of exterior major vertices and the maximum nullity of G. Wang et al. (Linear Multilinear Algebra, 2018. https://doi.org/10.1080/03081087.2018.1545829) established upper and lower bounds on Z(G) with respect to $$p(G),\, ex(G)$$p(G),ex(G) and $$\Phi (G)$$Φ(G): $$p(G)-ex(G)-1\leqslant Z(G)\leqslant p(G)+2\Phi (G)+1$$p(G)-ex(G)-1⩽Z(G)⩽p(G)+2Φ(G)+1. Hence, it is interesting to study the distribution of the zero forcing number Z(G) in the interval $$[p(G)-ex(G)-1,\, p(G)+2\Phi (G)+1]$$[p(G)-ex(G)-1,p(G)+2Φ(G)+1]. Wang et al. (2018) determined all the connected graphs G with $$Z(G)=p(G)-ex(G)$$Z(G)=p(G)-ex(G) and $$Z(G)=p(G)+2\Phi (G)-1.$$Z(G)=p(G)+2Φ(G)-1. In this paper we identify all the connected graphs G with $$Z(G)=p(G)-ex(G)+1$$Z(G)=p(G)-ex(G)+1 and $$Z(G)=p(G)+2\Phi (G)-2.$$Z(G)=p(G)+2Φ(G)-2. On the other hand, ‘AIM Minimum Rank-Special Graphs Work Group’ (Linear Algebra Appl 428(7):1628–1648, 2008) established the inequality $$Z(G)\geqslant M(G)$$Z(G)⩾M(G). The authors posted an attractive question: What is the class of graphs G for which $$Z(G)=M(G)$$Z(G)=M(G)? In this paper, we show that the equality holds for threshold graphs.

Suggested Citation

  • Shuchao Li & Wanting Sun, 2020. "On the zero forcing number of a graph involving some classical parameters," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 365-384, February.
    • Handle: RePEc:spr:jcomop:v:39:y:2020:i:2:d:10.1007_s10878-019-00475-1
    DOI: 10.1007/s10878-019-00475-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-019-00475-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10878-019-00475-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Chassidy Bozeman & Boris Brimkov & Craig Erickson & Daniela Ferrero & Mary Flagg & Leslie Hogben, 2019. "Restricted power domination and zero forcing problems," Journal of Combinatorial Optimization, Springer, vol. 37(3), pages 935-956, April.
    2. Maguy TREFOIS & Jean-Charles DELVENNE, 2015. "Zero forcing number, constrained matchings and strong structural controllability," LIDAM Reprints CORE 2785, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Boris Brimkov & Derek Mikesell & Illya V. Hicks, 2021. "Improved Computational Approaches and Heuristics for Zero Forcing," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1384-1399, October.
    2. Prosenjit Bose & Valentin Gledel & Claire Pennarun & Sander Verdonschot, 2020. "Power domination on triangular grids with triangular and hexagonal shape," Journal of Combinatorial Optimization, Springer, vol. 40(2), pages 482-500, August.
    3. Brimkov, Boris & Fast, Caleb C. & Hicks, Illya V., 2019. "Computational approaches for zero forcing and related problems," European Journal of Operational Research, Elsevier, vol. 273(3), pages 889-903.
    4. Prosenjit Bose & Valentin Gledel & Claire Pennarun & Sander Verdonschot, 0. "Power domination on triangular grids with triangular and hexagonal shape," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-19.
    5. Gu-Fang Mou & Tian-Fei Wang & Zhong-Shan Li, 2020. "The Bipartite Zero Forcing Set for a Full Sign Pattern Matrix," Mathematics, MDPI, vol. 8(3), pages 1-14, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:39:y:2020:i:2:d:10.1007_s10878-019-00475-1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.