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The Zero Forcing Number of Graphs with the Matching Number and the Cyclomatic Number

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Abstract

Let S be a subset of V(G). The vertices of S are colored black and the vertices of \(V(G)-S\) are colored white. The color-change-rule is defined as if there is a black vertex u having an unique white neighbor v, then change the color of v to black. The zero forcing number of a graph, denoted by Z(G), is the minimum cardinality of a set S such that all vertices of V(G) are black by repeating the color-change-rule, which was proposed at the AIM-minimum rank group in 2008 to bound the maximal nullity of G. Hence it has received much attention. Our interest is to study the zero forcing number of graphs with matching number \(\alpha '(G)\) and cyclomatic number c(G). In the paper, utilizing the alternating path and an operation associated with a vertex partition, we verify the sharp bounds that \(n(G)-2\alpha '(G)\le Z(G)\le n(G)-\alpha '(G)+c(G)-1\) for \(n(G)\ge 3\). The extremal graphs attain the upper bound are completely characterized.

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Acknowledgements

Shengjin Ji and Yu Jing were supported by Shandong Provincial Natural Science Foundation of China (Nos. ZR2019MA012, ZR2022MA077). Wenqian Zhang was supported by National Natural Science Foundation of Chnia (No. 12226304).

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  1. School of Mathematics and statistics, Shandong University of Technology, Zibo, 255000, Shandong, China

    Yu Jing, Wenqian Zhang & Shengjin Ji

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  1. Yu Jing
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Jing, Y., Zhang, W. & Ji, S. The Zero Forcing Number of Graphs with the Matching Number and the Cyclomatic Number. Graphs and Combinatorics 39, 72 (2023). https://doi.org/10.1007/s00373-023-02664-6

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