The Clar number of a (hydro)carbon molecule, introduced by Clar [E. Clar, \emph{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 planar 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 [H. G. Abeledo and G. W. Atkinson, \emph{Unimodularity of the clar number problem}, Linear algebra and its applications \textbf{420} (2007), no. 2, 441--448] 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 NP-hard. We also prove 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 FPT algorithm that determines the Clar number of a given 2-connected plane graph. The parameter of the algorithm is the length of the shortest odd join in the planar dual graph. For fullerenes this is not yet a polynomial algorithm, but for certain carbon nanotubes it gives an efficient algorithm.
[1]
Heping Zhang,et al.
An Upper Bound for the Clar Number of Fullerene Graphs
,
2007
.
[2]
Bojan Mohar,et al.
Face Covers and the Genus Problem for Apex Graphs
,
2001,
J. Comb. Theory, Ser. B.
[3]
Gary W. Atkinson,et al.
A min-max theorem for plane bipartite graphs
,
2010,
Discret. Appl. Math..
[4]
Heping Zhang,et al.
Extremal fullerene graphs with the maximum Clar number
,
2009,
Discret. Appl. Math..
[5]
E. Clar.
The aromatic sextet
,
1972
.
[6]
S. C. O'brien,et al.
C60: Buckminsterfullerene
,
1985,
Nature.
[7]
A. Frank.
Connections in Combinatorial Optimization
,
2011
.