  The critical phenomena associated to the liquid to solid transition of
quasi-two-dimensional vibrated granular systems is studied using molecular
dynamics simulations of the inelastic hard sphere model. The critical
properties are associated to the fourfold bond-orientational order parameter
$\chi_4$, which measures the level of square crystallization of the system.
Previous experimental results have shown that the transition of $\chi_4$, when
varying the vibration amplitude, can be either discontinuous or continuous, for
two different values of the height of the box. Exploring the amplitude-height
phase space, a transition line is found, which can be either discontinuous or
continuous, merging at a tricritical point and the continuous branch ends in an
upper critical point. In the continuous transition branch, the critical
properties are studied. The exponent associated to the amplitude of the order
parameter is $\beta=1/2$, for various system sizes, in complete agreement with
the experimental results. However, the fluctuations of $\chi_4$ do not show any
critical behavior, probably due to crossover effects by the close presence of
the tricritical point. Finally, in quasi-one-dimensional systems, the
transition is only discontinuous, limited by one critical point, indicating
that two is the lower dimension for having a tricritical point.
