benchmarks from region **(1)**, and the green ones to benchmarks from region **(2).***[]{data-label="Mhn"}](f2.eps "fig:"){width="8.5cm" height="6.5cm"}\
![*The dependence of the quantity $\protect\upsilon _{c}/T_{c}$ on the lightest Higgs mass, on the lightest neutralino mass, and the corresponding critical temperature, taking into account different values of the parameters in (\[pra\]) is shown. The red points refer to benchmarks from region **(1)**, and the green ones to benchmarks from region **(2).***[]{data-label="Mhn"}](f3.eps "fig:"){width="8.5cm" height="6.5cm"}

As it is clear from Fig. \[Mhn\], the EWPT could be strongly first-order in the two regions **(1)** and **(2)**, for a large range of the of the values of the lightest Higgs mass, and lightest neutralino mass. These masses are estimated at tree-level, which are expected to be boosted to larger values when considering one-loop corrections. One remarks that the phase transition strength does depend on the lightest Higgs and neutralino masses. One also remarks that the majority of the benchmarks that give a weak phase transition have the value of $\upsilon \left( T_{c}\right)
/T_{c}\sim 0.14-0.18$, and their corresponding critical temperature is between $T_{c}\sim 300-600$ GeV. While all the corresponding values of $%
T_{c} $ to the strong phase transition are $T_{c}\leq 180$ GeV.

The mixing with the singlet will 