  We propose an efficient method for the numerical approximation of a general
class of two dimensional semilinear parabolic problems on polygonal meshes. The
proposed approach takes advantage of the properties of the serendipity version
of the Virtual Element Method, which not only reduces the number of degrees of
freedom compared to the original Virtual Element Method, but also allows the
introduction of an approximation of the nonlinear term that is computable from
the degrees of freedom of the discrete solution with a low computational cost,
thus significantly improving the efficiency of the method. An error analysis
for the semi-discrete formulation is carried out, and an optimal estimate for
the error in the $L_2$-norm is obtained. The accuracy and efficiency of the
proposed method when combined with a second order Strang operator splitting
time discretization is illustrated in our numerical experiments, with
approximations up to order $6$.
