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Complete forcing numbers of complete and almost-complete multipartite graphs

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Abstract

A complete forcing set of a graph G with a perfect matching is a subset of E(G) on which the restriction of each perfect matching M is a forcing set of M. The complete forcing number of G is the minimum cardinality of complete forcing sets of G. It was shown that a complete forcing set of G also antiforces each perfect matching. Previously, some closed formulas for the complete forcing numbers of some types of hexagonal systems including cata-condensed hexagonal systems and parallelograms have been derived. In this paper, we show that the subset of E(G) obtained from E(G) by deleting all edges that are incident with some vertices of a 2-independent set of G is a complete forcing set. As applications, we give some expressions for the complete forcing numbers of complete multipartite graphs, 2n-vertex graphs with minimum degree at least \(2n-3\) and 2n-vertex balanced bipartite graphs with minimum degree at least \(n-2\), by showing that each sufficiently short cycle is a nice cycle.

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Acknowledgements

The authors are grateful to two reviewers for their careful reading and valuable suggestions to improve this manuscript.

Funding

This work is supported by National Natural Science Foundation of China (Grant No. 12271229).

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Authors and Affiliations

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

    Xin He & Heping Zhang

  2. School of Science, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People’s Republic of China

    Xin He

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  1. Xin He
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Correspondence to Heping Zhang.

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He, X., Zhang, H. Complete forcing numbers of complete and almost-complete multipartite graphs. J Comb Optim 46, 11 (2023). https://doi.org/10.1007/s10878-023-01078-7

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