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Anti-forcing spectra of perfect matchings of graphs

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Abstract

Let M be a perfect matching of a graph G. The smallest number of edges whose removal to make M as the unique perfect matching in the resulting subgraph is called the anti-forcing number of M. The anti-forcing spectrum of G is the set of anti-forcing numbers of all perfect matchings in G, denoted by \(\hbox {Spec}_{af}(G)\). In this paper, we show that any finite set of positive integers can be the anti-forcing spectrum of a graph. We present two classes of hexagonal systems whose anti-forcing spectra are integer intervals. Finally, we show that determining the anti-forcing number of a perfect matching of a bipartite graph with maximum degree four is a NP-complete problem.

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Acknowledgments

Supported by NSFC (Grant No. 11371180).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Kai Deng & Heping Zhang

  2. School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, 750027, Ningxia, People’s Republic of China

    Kai Deng

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  1. Kai Deng
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  2. Heping Zhang
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Correspondence to Heping Zhang.

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Deng, K., Zhang, H. Anti-forcing spectra of perfect matchings of graphs. J Comb Optim 33, 660–680 (2017). https://doi.org/10.1007/s10878-015-9986-3

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  • DOI: https://doi.org/10.1007/s10878-015-9986-3

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