On the maximum forcing and anti-forcing numbers of (4,6)-fullerenes

Let G be a (4,6)-fullerene graph. We show that the maximum forcing number of G is equal to its Clar number, and the maximum anti-forcing number of G is equal to its Fries number, which extend the known results for hexagonal systems with a perfect matching (Xu et al., 2013; Lei et al., 2016). Moreover, we obtain two formulas dependent only on the order of G to count the Clar number and Fries number of G respectively. Hence we can compute the maximum forcing number of a (4,6)-fullerene graph in linear time. This answers an open problem proposed by Afshani et al. (2004) in the case of (4,6)-fullerene graphs.