  The notion of P-stability played an influential role in approximating the
permanents, sampling rapidly the realizations of graphic degree sequences, or
even studying and improving network privacy. However, we did not have a good
insight of the structure of P-stable degree sequence families. In this paper we
develop a remedy to overstep this deficiency.
  We will show, that if an infinite set of graphic degree sequences,
characterized by some simple inequalities of their fundamental parameters, is
$P$-stable, then it is ``fully graphic'' -- meaning that every degree sequence
with an even sum, meeting the specified inequalities, is graphic. The reverse
statement also holds: an infinite, fully graphic set of degree sequences
characterized by some simple inequalities of their fundamental parameters is
P-stable.
  Along the way, we will significantly strengthen some well-known, older
results, and we construct new P-stable families of degree sequences.
