Relations between Clar structures, Clar covers, and the sextet-rotation tree of a hexagonal system

Sextet rotations of the perfect matchings of a hexagonal system H are represented by the sextet-rotation-tree R(H), a directed tree with one root. In this article we find a one-to-one correspondence between the non-leaves of R(H) and the Clar covers of H, without alternating hexagons. Accordingly, the number (nl) of non-leaves of R(H) is not less than the number (cs) of Clar structures of H. We obtain some simple necessary and sufficient conditions, and a criterion for cs=nl, that are useful for the calculation of Clar polynomials. A procedure for constructing hexagonal systems with cs<nl is provided in terms of normal additions of hexagons.