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Complete forcing numbers of complete and almost-complete multipartite graphs

Author

Listed:
  • Xin He

    (Lanzhou University
    Lanzhou University of Technology)

  • Heping Zhang

    (Lanzhou University)

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 $$2n-3$$ 2 n - 3 and 2n-vertex balanced bipartite graphs with minimum degree at least $$n-2$$ n - 2 , by showing that each sufficiently short cycle is a nice cycle.

Suggested Citation

  • Xin He & Heping Zhang, 2023. "Complete forcing numbers of complete and almost-complete multipartite graphs," Journal of Combinatorial Optimization, Springer, vol. 46(2), pages 1-20, September.
    • Handle: RePEc:spr:jcomop:v:46:y:2023:i:2:d:10.1007_s10878-023-01078-7
    DOI: 10.1007/s10878-023-01078-7
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    References listed on IDEAS

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    1. Shou-Jun Xu & Heping Zhang & Jinzhuan Cai, 2015. "Complete forcing numbers of catacondensed hexagonal systems," Journal of Combinatorial Optimization, Springer, vol. 29(4), pages 803-814, May.
    2. Shou-Jun Xu & Xiu-Song Liu & Wai Hong Chan & Heping Zhang, 2016. "Complete forcing numbers of primitive coronoids," Journal of Combinatorial Optimization, Springer, vol. 32(1), pages 318-330, July.
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