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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
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    References listed on IDEAS

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    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).
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