Abstract
Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The cardinality of a forcing set of M with the smallest size is called the forcing number of M, denoted by f(G, M). The forcing spectrum of G is defined as: Spec(G) = {f(G, M)∣ M is a perfect matching of G}. In this paper, by applying the Z-transformation graph (resonance graph) we show that for any polyomino with perfect matchings and any even polygonal chain, their forcing spectra are integral intervals. Further we obtain some sharp bounds on maximum and minimum forcing numbers of hexagonal chains with given number of kinks. Forcing spectra of two extremal chains are determined.
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References
Adams, P., Mahdian, M., Mahmoodian, E. On the forced matching numbers of bipartite graphs. Discrete Math., 281: 1–12 (2004)
Afshani, P., Hatami, H., Mahmoodian, E. On the spectrum of the forcing matching number of graphs. Austral. J. Combin., 30: 147–160 (2004)
Cai, J., Zhang, H. Global forcing number of some chemical graphs. MATCH Commun. Math. Comput. Chem., 67: 289–312 (2012)
Che, Z., Chen, Z. Forcing hexagons in hexagonal systems. MATCH Commun. Math. Comput. Chem., 56: 649–668 (2006)
Che, Z., Chen, Z. Forcing on perfect matchings—a survey. MATCH Commun. Math. Comput. Chem., 66: 93–136 (2011)
Došlić, T. Global forcing number of benzenoid graphs. J. Math. Chem., 41: 217–229 (2007)
Gründler, W. Signifikante Elektronenstrukturen fur benzenoide Kohlenwasserstoffe. Wiss. Z. Univ. Halle, 31: 97–116 (1982)
Harary, F., Klein, D., Živković, T. Graphical properties of polyhexes: perfect matching vector and forcing. J. Math. Chem., 6: 295–306 (1991)
Hansen, P., Zheng, M. Bonds fixed by fixing bonds. J. Chem. Inform. Comput. Sci., 34: 297–304 (1994)
Klein, D., Randić, M. Innate degree of freedom of a graph. J. Comput. Chem., 8: 516–521(1987)
Kleinerman, S. Bounds on the forcing numbers of bipartite graphs. Discrete Math., 306: 66–73 (2006)
Lam, F., Pachter, L. Forcing numbers of stop signs. Theor. Comput. Sci., 303: 409–416 (2003)
Li, C. Spectrum of matching forcing numbers of graph C3 × P2n. preprint, 2008.
Pachter, L., Kim, P. Forcing matchings on square grids. Discrete Math., 190: 287–294 (1998)
Randić, M. Aromaticity of polycyclic conjugated hydrocarbons. Chem. Reviews, 103(9): 3449–3605 (2003)
Randić, M. Resonance in catacondensed benzenoid hydrocarbons. Int. J. Quantum Chem., 63: 585–600 (1997)
Riddle, M. The minimum forcing number for the torus and hypercube. Discrete Math., 245: 283–292 (2002)
Shi, L., Zhang, H., Lin, R. Characterizing the fullerene graphs with the minimum forcing number 3. Discrete Appl. Math., 294: 181–204 (2021)
Vukičević, D., Trinajstić, N. On the anti-Kekulé and anti-forcing number of cata-condensed benzenoids. J. Math. Chem., 43: 719–726 (2008)
Wang, H., Ye, D., Zhang, H. The forcing number of toroidal polyhexes. J. Math. Chem., 43: 457–475 (2008)
Yang, Q., Ye, D., Zhang, H., Lin, Y. The anti-Kekulé number of fullerene graphs. MATCH Commun. Math. Comput. Chem., 67: 281–288 (2012)
Zhang, F., Guo, X., Chen, R. Z-transformation graph of perfect matchings of hexagonal systems. Discrete Math., 72: 405–415 (1988)
Zhang, F., Guo, X., Chen, R. The connectivity of Z-transformation graphs of perfect matchings of hexagonal systems. Acta Math. Appl. Sinica (English Ser.), 4: 131–135 (1988)
Zhang, F., Li, X. Forcing bonds of a benzenoid system. Acta Math. Appl. Sinica (English Ser.), 12: 209–215 (1996)
Zhang, F., Li, X. Hexagonal systems with forcing edges. Discrete Math., 140: 253–263 (1995)
Zhang, H. Z-transformation graphs of perfect matchings of plane bipartite graphs: a survey, MATCH Commun. Math. Comput. Chem., 56: 457–476 (2006)
Zhang, H. The connectivity of Z-transformation graphs of perfect matchings of polyominoes. Discrete Math., 158: 257–272 (1996)
Zhang, H., Ye, D., Shiu, W. Forcing matching numbers of fullerene graphs. Discrete Appl. Math., 158: 573–582 (2010)
Zhang, H., Zhang, F. Perfect matchings of polyomimo graphs. Graphs Combin., 13: 295–304 (1997)
Zhang, H., Zhang, F. Plane elementary bipartite graphs. Discrete Appl. Math., 105: 291–311 (2000)
Zhang, H., Zhang, F., Yao, H. Z-transformation graphs of perfect matchings of plane bipartite graphs. Discrete Math., 276: 393–404 (2004)
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The authors are grateful to the referees for their careful reading and many valuable suggestions.
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This work is supported by the National Natural Science Foundation of China (Nos. 11871256, 11371180, 11226286).
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Zhang, Hp., Jiang, Xy. Continuous Forcing Spectra of Even Polygonal Chains. Acta Math. Appl. Sin. Engl. Ser. 37, 337–347 (2021). https://doi.org/10.1007/s10255-021-1010-3
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DOI: https://doi.org/10.1007/s10255-021-1010-3