Abstract
The Clar number of a (hydro)carbon molecule, introduced by Clar (The aromatic sextet, 1972), is the maximum number of mutually disjoint resonant hexagons in the molecule. Calculating the Clar number can be formulated as an optimization problem on 2-connected plane graphs. Namely, it is the maximum number of mutually disjoint even faces a perfect matching can simultaneously alternate on. It was proved by Abeledo and Atkinson (Linear Algebra Appl 420(2):441–448, 2007) that the Clar number can be computed in polynomial time if the plane graph has even faces only. We prove that calculating the Clar number in general 2-connected plane graphs is \(\mathsf {NP}\)-hard. We also prove \(\mathsf {NP}\)-hardness of the maximum independent set problem for 2-connected plane graphs with odd faces only, which may be of independent interest. Finally, we give an exact algorithm that determines the Clar number of a given 2-connected plane graph. The algorithm has a polynomial running time if the length of the shortest odd join in the planar dual graph is fixed, which gives an efficient algorithm for some fullerene classes, such as carbon nanotubes.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10910-017-0799-8/MediaObjects/10910_2017_799_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10910-017-0799-8/MediaObjects/10910_2017_799_Fig2_HTML.gif)
Similar content being viewed by others
References
H.G. Abeledo, G.W. Atkinson, Unimodularity of the Clar number problem. Linear Algebra Appl. 420(2), 441–448 (2007)
H.G. Abeledo, G.W. Atkinson, A min–max theorem for plane bipartite graphs. Discrete Appl. Math. 158(5), 375–378 (2010)
E. Clar, The Aromatic Sextet (Wiley, 1972). ISBN 9780471158400
D. Erdős, A. Frank, K. Kun, Sink-stable sets of digraphs. SIAM J. Discrete Math. 28(4), 1651–1674 (2014)
M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman & Co., New York, 1979)
M. Ghorbani, E. Naserpour, The Clar number of fullerene \({C}_{24n}\) and carbon nanocone \({CNC}_4 [n]\). Iran. J. Math. Chem. 2, 53–59 (2011)
P. Hansen, M. Zheng, The Clar number of a benzenoid hydrocarbon and linear programming. J. Math. Chem. 15(1), 93–107 (1994)
H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, R.E. Smalley et al., \({C}_{60}\): Buckminsterfullerene. Nature 318(6042), 162–163 (1985)
B. Mohar, Face covers and the genus problem for apex graphs. J. Comb. Theory Ser. B 82(1), 102–117 (2001)
A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York, 1986)
A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24 (Springer, Berlin, 2003)
K. Shalem, H. Abeledo, Alternative integer-linear-programming formulations of the Clar problem in hexagonal systems. J. Math. Chem. 39(3), 605–610 (2006)
D. Ye, H. Zhang, Extremal fullerene graphs with the maximum Clar number. Discrete Appl. Math. 157(14), 3152–3173 (2009)
H. Zhang, D. Ye, An upper bound for the Clar number of fullerene graphs. J. Math. Chem. 41(2), 123–133 (2007)
H. Zhang, D. Ye, L. Yunrui, A combination of Clar number and Kekulé count as an indicator of relative stability of fullerene isomers of \({C}_{60}\). J. Math. Chem. 48(3), 733–740 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the Hungarian Scientific Research Fund (OTKA, Grant Number K109240).
Rights and permissions
About this article
Cite this article
Bérczi-Kovács, E.R., Bernáth, A. The complexity of the Clar number problem and an exact algorithm. J Math Chem 56, 597–605 (2018). https://doi.org/10.1007/s10910-017-0799-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-017-0799-8