Complete forcing numbers of complete and almost-complete multipartite graphs

Xin He () and Heping Zhang ()
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Xin He: Lanzhou University
Heping Zhang: Lanzhou University

Journal of Combinatorial Optimization, 2023, vol. 46, issue 2, No 2, 20 pages

Abstract: 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.

Keywords: Perfect matching; Complete forcing number; Complete multipartite graph (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10878-023-01078-7

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