As a strengthening of the concept of global forcing number of a graph G, the complete forcing number of G is the cardinality of a minimum edge subset of G to which the restriction of every perfect matching M is a forcing set of M. Xu et al. (J Comb Opt 29: 803–814, 2015) revealed that a complete forcing set of G also antifores each perfect matching, and obtained that for a catacondensed hexagonal system, the complete forcing number is equal to the Clar number plus the number of hexagons (Chan et al. MATCH Commun Math Comput Chem 74: 201–216, 2015). In this paper, we consider general hexagonal systems H, and present sharp upper bound on the complete forcing number of H in terms of elementary edge-cut cover and lower bound via graph decomposition as well. Through such approaches, we obtain some closed formulas for the complete forcing numbers of some types of hexagonal systems including parallelogram, regular hexagon- and rectangle-shaped hexagonal systems.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
H. Abeledo, G.W. Atkinson, A min-max theorem for plane bipartite graphs. Discrete Appl. Math. 158, 375–378 (2010)
J.A. Bondy, U.S.R. Murty, Graph Theory with Applications (American Elsevier, New York, Macmillan, London, 1976)
A.T. Balaban, M. Randić, Coding canonical Clar structures of polycyclic benzenoid hydrocarbons. MATCH Commun. Math. Comput. Chem. 82, 139–162 (2019)
Z. Che, Z. Chen, Forcing on perfect matchings-A survey. MATCH Commun. Math. Comput. Chem. 66, 93–136 (2011)
E. Clar, The Aromatic Sextet (Wiley, London, 1972)
S.J. Cyvin, I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons (Springer, Berlin, 1988)
J. Cai, H. Zhang, Global forcing number of some chemical graphs. MATCH Commun. Math. Comput. Chem. 67, 289–312 (2012)
W. Chan, S. Xu, G. Nong, A linear-time algorithm for computing the complete forcing number and the Clar number of catacondensed hexagonal systems. MATCH Commun. Math. Comput. Chem. 74, 201–216 (2015)
T. Došlić, Global forcing number of benzenoid graphs. J. Math. Chem. 41, 217–229 (2007)
I. Gutman, S.J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons (Springer, Berlin, 1989)
W.C. Herndon, Resonance theory and the enumeration of Kekulé structures. J. Chem. Educ. 51, 10–15 (1974)
F. Harary, D.J. Klein, T.P. Živković, Graphical properties of polyhexes: perfect matching vector and forcing. J. Math. Chem. 6, 295–306 (1991)
P. Hansen, M. Zheng, Upper bounds for the Clar number of benzenoid hydrocarbons. J. Chem. Soc. Faraday Trans. 88, 1621–1625 (1992)
P. Hansen, M. Zheng, Normal components of benzenoid systems. Theor. Chim. Acta 85, 335–344 (1993)
P. Hansen, M. Zheng, The Clar number of a benzenoid hydrocarbon and linear programming. J. Math. Chem. 15, 93–107 (1994)
D.J. Klein, M. Randić, Innate degree of freedom of a graph. J. Comput. Chem. 8, 516–521 (1987)
B. Liu, H. Bian, H. Yu, Complete forcing numbers of polyphenyl systems. Iran. J. Math. Chem. 7, 39–46 (2016)
B. Liu, H. Bian, H. Yu, J. Li, Complete forcing number of spiro hexagonal systems. Polyc. Arom. Comp. (2019). https://doi.org/10.1080/10406638.2019.1600560
L. Lovász, M.D. Plummer, Matching Theory, Annals of Discrete Mathematics, vol. 29 (North-Holland, Amsterdam, 1986)
J. Langner, H.A. Witek, Interface theory of benzenoids. MATCH Commun. Math. Comput. Chem. 84, 143–176 (2020)
J. Langner, H.A. Witek, Interface theory of benzenoids: Basic applications. MATCH Commun. Math. Comput. Chem. 84, 177–215 (2020)
E.S. Mahmoodian, R. Naserasr, M. Zaker, Defining sets in vertex colorings of graphs and Latin rectangles. Discrete Math. 167, 451–460 (1997)
H. Sachs, Perfect matchings in hexagonal system. Combinatorica 4, 89–99 (1984)
J. Sedlar, The global forcing number of the parallelogram polyhex. Discrete Appl. Math. 160, 2306–2313 (2012)
D. Vukičević, T. Došlić, Global forcing number of grid graphs. Aust. J. Combin. 38, 47–62 (2007)
D. Vukičević, J. Sedlar, Total forcing number of the triangular grid. Math. Commun. 9, 169–179 (2004)
S. Xu, X. Liu, W. Chan, H, Zhang, Complete forcing numbers of primitive coronoids. J. Comb. Opt. 32, 318–330 (2016)
S. Xu, H. Zhang, J. Cai, Complete forcing numbers of catacondensed hexagonal systems. J. Comb. Opt. 29, 803–814 (2015)
F. Zhang, R. Chen, When each hexagon of a hexagonal system covers it. Discrete Appl. Math. 30, 63–75 (1991)
F. Zhang, R. Chen, X. Guo, Perfect matchings in hexagonal systems. Graphs Combin. 1, 383–386 (1985)
H. Zhang, J. Cai, On the global forcing number of hexagonal systems. Discrete Appl. Math. 162, 334–347 (2014)
H. Zhang, F. Zhang, The Clar covering polynomial of hexagonal systems I. Discrete Appl. Math. 69, 147–167 (1996)
H. Zhang, H. Yao, D. Yang, A min-max result on outerplane bipartite graphs. Appl. Math. Lett. 20, 199–205 (2007)
H. Zhang, F. Zhang, Plane elementary bipartite graphs. Discrete Appl. Math. 105, 473–490 (2000)
Conflict of interest
The authors declare that they have no conflict of interest.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by NSFC (Grant No. 11871256).
About this article
Cite this article
He, X., Zhang, H. Complete forcing numbers of hexagonal systems. J Math Chem 59, 1767–1784 (2021). https://doi.org/10.1007/s10910-021-01261-3
- Hexagonal system
- Perfect matching
- Complete forcing set
- Complete forcing number