  Critical and in the highly frustrated regime also dynamical properties of the
$J_1-J_2$ Ising model with competing nearest-neighbor $J_1$ and
second-nearest-neighbor $J_2$ interactions on a honeycomb lattice are
investigated by standard Monte Carlo and parallel tempering simulations. The
phase boundary is determined as a function of the coupling ratio for the phase
transition between the paramagnetic and ferromagnetic states within $R \equiv
J_2/|J_1| \in [-1/4,0]$. It is confirmed that at least for $R \geq -0.2$ the
transition remains second-order and complies with the standard Ising
universality class. In the highly frustrated regime of $R < -0.2$ and low
temperatures the system tends to freeze to metastable domain states, separated
by large energy barriers, which show extremely sluggish dynamics. The resulting
huge equilibration and autocorrelation times hinder the analysis of critical
properties and thus the character of the transition in this region remains to
be determined.
