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.

翻译：P稳定性的概念在近似计算永续量、快速采样图度序列的实现，乃至研究与提升网络隐私方面发挥了重要影响。然而，我们对P稳定度序列族的结构缺乏深入理解。本文提出一种弥补这一不足的解决方案。我们将证明：若一个由基本参数的简单不等式刻画的无限图度序列集合是P稳定的，则它必然是“完全图化的”——即所有满足指定不等式的偶和度序列均为图序列。反之亦然：由基本参数简单不等式刻画的无限完全图度序列集合是P稳定的。在此过程中，我们还将显著加强若干经典旧有结论，并构造新的P稳定度序列族。