Anti-forcing polynomials for benzenoid systems with forcing edges

Abstract

The anti-forcing number of a perfect matching M of a graph G is the minimal number of edges of G not in M whose deletion results in a subgraph with a unique perfect matching M. The maximum anti-forcing number of a benzenoid system H equals the Fries number of H. Hwang et al. defined the anti-forcing polynomial of a graph as a counting polynomial for perfect matchings with the same anti-forcing number, and obtained explicit expressions of the anti-forcing polynomial of hexagonal chains and crowns. In this paper, we focus on benzenoid systems with forcing edges, and obtain a recurrence relation of their anti-forcing polynomial. Furthermore, we get an explicit expression of the anti-forcing polynomial of benzenoid parallelograms.