Geodesic
Complete forcing numbers of graphs
Ars Mathematica Contemporanea, Tome 23 (2023) no. 2, article no. 09, 20 p.

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The complete forcing number of a graph G with a perfect matching is the minimum cardinality of an edge set of G on which the restriction of each perfect matching M is a forcing set of M. This concept can be view as a strengthening of the concept of global forcing number of G. Došlić in 2007 has obtained that the global forcing number of a connected graph is at most its cyclomatic number. Motivated from this result, we obtain that the complete forcing number of a graph is no more than 2 times its cyclomatic number and characterize the matching covered graphs whose complete forcing numbers attain this upper bound and minus one, respectively. Besides, we present a method of constructing a complete forcing set of a graph. By using such method, we give closed formulas for the complete forcing numbers of wheels and cylinders.
DOI : 10.26493/1855-3974.2706.3c8
Keywords: Perfect matching, global forcing number, complete forcing number, cyclomatic number, wheel, cylinder 
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Xin He; Heping Zhang. Complete forcing numbers of graphs. Ars Mathematica Contemporanea, Tome 23 (2023) no. 2, article  no. 09, 20 p. doi : 10.26493/1855-3974.2706.3c8. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2706.3c8/

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