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

  • Yan ZhaoEmail author
  • Lily Chen
  • Hengzhe Li
Article
  • 67 Downloads

Abstract

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 \(k\le n-2\).

Keywords

k-forcing k-forcing number Zero forcing set 

Mathematics Subject Classification

05C15 05C69 

Notes

Acknowledgements

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).

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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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