The forcing number, denoted by f(G), of a graph G with a perfect matching is the minimum number of independent edges that completely determine the perfect matching of G. In this paper, we consider the forcing number of a toroidal polyhex H(p,q,t) with a torsion t, a cubic graph embedded on torus with every face being a hexagon. We obtain that f(H(p,q,t)) ≥ min{p,q}, and equality holds for p ≤ q or p > q and t∈{ 0,p−q,p−q + 1,..., p−1}. In general, we show that f(H(p,q,t)) is equal to the side length of a maximum triangle on H(p,q,t). Based on this result, we design a linear algorithm to compute the forcing number of H(p,q,t).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams P, Mahdian M, Mahmoodian E.S, (2004) . Discrete Math. 281: 1–12
Afshani P, Hatami H, Mahmoodian E.S, (2004) . Aust. J. Comb. 30: 147–160
T. Došlić, J. Math. Chem. (2006). Online First, DOI: 10.1007/s10910-006-9056-2.
Harary F, Klein D.J, Živković T.P., (1991) . J. Math. Chem. 6, 295–306
Klein D.J, Randić M., (1987) . J. Comput. Chem. 8, 516–521
Klein D.J, Zhu H, (1996) . Discrete Appl. Math. 67, 157–173
Kleinerman S, (2006) . Discrete Math. 306, 66–73
Kirby E.C, Klein D.J, Mallion R.B, Pollak P, Sachs H, (2004) . Croat. Chem. Acta 77(1–2): 263–278
Kirby E.C, Mallion R.B, Pollak P, (1993) . J. Chem. Soc. Faraday Trans. 89(12): 1945–1953
Kirby E.C, Pollak P, (1998) . J. Chem. Inf. Comput. Sci. 38: 66–70
Kutnar K, Malnič A., Marušič D., (2005) . J. Chem. Inf. Comput. Sci. 45: 1527–1535
Lam F, Pachter L, (2003) . Theor. Comput. Sci. 303, 409–416
Liu J, Dai H, Hafner J.H, Colbert D.T, Smalley R.E, Tans S.J, Dekker C, (1997) . Nature 385, 780–781
Pechter L, Kim P, (1998) . Discrete Math. 190, 287–294
M. Randić and Klein D.J, in: Mathematical and Computational Concepts in Chemsitry, ed. N. Trinajstić (Wiley, New York, 1985) pp. 274–282
Riddle M.E, (2002) . Discrete Math. 245, 283–292
Shiu W.C, Lam P.C.B., Zhang H, (2005) . J. Math. Chem. 38(4): 451–466
Vukičević D., Kroto H.W, Randić M., (2005) . Croat. Chem. Acta 78: 223–234
Vukičević D., Sedlar J, (2004) . Math. Commun. 9, 169–179
D. Vukičević and N. Trinajstić, J. Math. Chem. (2006) Online-first, DOI: 10.1007/s10910-006-9133-6.
Zhang F, Li X, (1995) . Discrete Math. 140, 253–263
Zhang F, Li X, (1996) . Acta Math. Appl. Sinica (English Series) 12(2): 209–215
Zhang F, Zhang H, (1995) . J. Mol. Struct. (Theochem) 331, 255–260
Zhang H, Zhang F, (2000) . Discrete Appl. Math. 105, 291–311
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, H., Ye, D., Zhang, H. et al. The forcing number of toroidal polyhexes. J Math Chem 43, 457–475 (2008). https://doi.org/10.1007/s10910-006-9208-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-006-9208-4