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Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs

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Abstract

Let G be a simple graph with 2n vertices and a perfect matching. The forcing number f(G, M) of a perfect matching M of G is the smallest cardinality of a subset of M that is contained in no other perfect matching of G. Among all perfect matchings M of G, the minimum and maximum values of f(G, M) are called the minimum and maximum forcing numbers of G, denoted by f(G) and F (G), respectively. Then f(G) ≤ F (G) ≤ n − 1. Che and Chen (2011) proposed an open problem: how to characterize the graphs G with f(G) = n − 1. Later they showed that for a bipartite graph G, f(G)= n − 1 if and only if G is complete bipartite graph Kn,n. In this paper, we completely solve the problem of Che and Chen, and show that f(G)= n − 1 if and only if G is a complete multipartite graph or a graph obtained from complete bipartite graph Kn,n by adding arbitrary edges in one partite set. For all graphs G with F (G) = n − 1, we prove that the forcing spectrum of each such graph G forms an integer interval by matching 2-switches and the minimum forcing numbers of all such graphs G form an integer interval from \(\lfloor{n\over 2}\rfloor\) to n − 1.

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Acknowledgements

The authors are grateful to anonymous reviewers for giving valuable comments and suggestions in improving the manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P. R. China

    Qian Qian Liu & He Ping Zhang

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  1. Qian Qian Liu
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  2. He Ping Zhang
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Correspondence to He Ping Zhang.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 12271229) and Gansu Provincial Department of Education: Youth Doctoral fund project (Grant No. 2021QB-090)

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Liu, Q.Q., Zhang, H.P. Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs. Acta. Math. Sin.-English Ser. 39, 1289–1304 (2023). https://doi.org/10.1007/s10114-023-1020-6

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  • DOI: https://doi.org/10.1007/s10114-023-1020-6

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