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Complete forcing numbers of catacondensed hexagonal systems

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Abstract

Let G be a graph with edge set E(G) that admits a perfect matching M. A forcing set of M is a subset of M contained in no other perfect matchings of G. A global forcing set of \(G\), introduced by Vukičević et al., is a subset of \(E(G)\) on which there are distinct restrictions of any two different perfect matchings of \(G\). Combining the above “forcing” and “global” ideas, we introduce and define a complete forcing set of G as a subset of \(E(G)\) on which the restriction of any perfect matching \(M\) of \(G\) is a forcing set of \(M\). The minimum cardinality of complete forcing sets is the complete forcing number of \(G\). First we establish some initial results about these two novel concepts, including a criterion for a complete forcing set, and comparisons between the complete forcing number and global forcing number. Then we give an explicit formula for the complete forcing number of a hexagonal chain. Finally a recurrence relation for the complete forcing number of a catacondensed hexagonal system is derived.

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Acknowledgments

This work is supported by NSFC (Grants No. 11001113, 61073046). The authors would like to sincerely thank the anonymous referees for the time they spent checking our proofs, as well as their many valuable suggestions.

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Authors and Affiliations

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

    Shou-Jun Xu & Heping Zhang

  2. Department of Applied Mathematics, Hainan University, Haikou, 570228, Hainan, People’s Republic of China

    Jinzhuan Cai

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  1. Shou-Jun Xu
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  2. Heping Zhang
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  3. Jinzhuan Cai
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Correspondence to Shou-Jun Xu.

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Xu, SJ., Zhang, H. & Cai, J. Complete forcing numbers of catacondensed hexagonal systems. J Comb Optim 29, 803–814 (2015). https://doi.org/10.1007/s10878-013-9624-x

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  • DOI: https://doi.org/10.1007/s10878-013-9624-x

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