Clar Sets and Maximum Forcing Numbers of Hexagonal Systems

Let H be a hexagonal system with a perfect matching. Xu et al. discovered that the maximum forcing number of H equals its Clar number. In this article we obtain a result: for any resonant set K of a peri-condensed hexagonal system H consisting of disjoint hexagons not meeting the boundary of H, if the subgraph obtained from H by deleting K and the boundary of H has a perfect matching or is empty, then the Clar number of H is at least |K| + 2. This fact improves the previous corresponding result due to Zheng and Chen. Based on the result, we prove that for each perfect matching M of H with the maximum forcing number, there exists a Clar set consisting of disjoint M-alternating hexagons of H.