Forcing polynomials of benzenoid parallelogram and its related benzenoids

Klein and Randić introduced the innate degree of freedom (forcing number) of a Kekulé structure (perfect matching) M of a graph G as the smallest cardinality of subsets of M that are contained in no other Kekulé structures of G, and the innate degree of freedom of the entire G as the sum over the forcing numbers of all perfect matchings of G. We proposed the forcing polynomial of G as a counting polynomial for perfect matchings with the same forcing number. In this paper, we obtain recurrence relations of the forcing polynomial for benzenoid parallelogram and its related benzenoids. In particular, for benzenoid parallelogram, we derive explicit expressions of its forcing polynomial and innate degree of freedom by generating functions.