Abstract
The forcing number of a perfect matching M in a graph G is the smallest number of edges inside M that can not be contained in other perfect matchings of G. The anti-forcing number of M is the smallest number of edges outside M whose removal results in a spanning subgraph with a unique perfect matching, that is M. Recently, in order to investigate the distributions of forcing and anti-forcing numbers of all perfect matchings of a graph, the forcing and anti-forcing polynomials were proposed, respectively. In this paper, we compute the forcing and anti-forcing polynomials of a type of polyomino graphs, and the recurrence relations of the forcing polynomial and anti-forcing polynomial are derived, respectively. As consequences, the forcing and anti-forcing spectra are determined, and the asymptotic behaviors of sums over the forcing numbers and anti-forcing numbers of all perfect matchings are revealed, respectively.
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Communicated by Leonardo de Lima.
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This work is supported by Fundamental Research Funds for the Central Universities (Grant no. 2021JCYJ05), Natural Science Foundation of Ningxia (Grant no. 2022AAC03285) and National Natural Science Foundation of China (Grant no. 12161002).
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Deng, K., Lü, H. & Wu, T. Forcing and anti-forcing polynomials of a type of polyomino graphs. Comp. Appl. Math. 42, 91 (2023). https://doi.org/10.1007/s40314-023-02228-7
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DOI: https://doi.org/10.1007/s40314-023-02228-7