The complexity of the Clar number problem and an FPT algorithm

TL;DR

This paper proves that computing the Clar number in general 2-connected planar graphs is NP-hard, introduces an FPT algorithm based on the shortest odd join, and discusses its applicability to fullerenes and nanotubes.

Contribution

It establishes NP-hardness of the Clar number problem in general graphs and presents a fixed-parameter tractable algorithm based on the shortest odd join.

Findings

NP-hardness of Clar number computation in general 2-connected planar graphs

NP-hardness of maximum independent set in 2-connected plane graphs with only odd faces

FPT algorithm for Clar number based on shortest odd join in the dual graph

Abstract

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.