  We show that by restricting the degrees of the vertices of a graph to an
arbitrary set \( \Delta \), the threshold point $ \alpha(\Delta) $ of the phase
transition for a random graph with $ n $ vertices and $ m = \alpha(\Delta) n $
edges can be either accelerated (e.g., $ \alpha(\Delta) \approx 0.381 $ for $
\Delta = \{0,1,4,5\} $) or postponed (e.g., $ \alpha(\{ 2^0, 2^1, \cdots, 2^k,
\cdots \}) \approx 0.795 $) compared to a classical Erd\H{o}s--R\'{e}nyi random
graph with $ \alpha(\mathbb Z_{\geq 0}) = \tfrac12 $. In particular, we prove
that the probability of graph being nonplanar and the probability of having a
complex component, goes from $ 0 $ to $ 1 $ as $ m $ passes $ \alpha(\Delta) n
$. We investigate these probabilities and also different graph statistics
inside the critical window of transition (diameter, longest path and
circumference of a complex component).
