Fully graphic degree sequences and P-stable degree sequences

TL;DR

This paper investigates the structure of P-stable degree sequences, showing that known infinite P-stable sets are fully graphic and establishing conditions under which P-stability and full graphicality are equivalent, thus advancing understanding of degree sequence families.

Contribution

It proves that all known infinite P-stable degree sequence sets are fully graphic and establishes equivalences between P-stability and full graphicality under certain conditions, strengthening existing theorems.

Findings

All known infinite P-stable degree sequence sets are fully graphic.

When the degree sum does not appear in the defining inequality, P-stability equals full graphicality.

Strengthens the theorem of Jerrum, McKay, and Sinclair regarding P-stability.

Abstract

The notion of $P$-stability of an infinite set of degree sequences plays influential role in approximating the permanents, rapidly sampling the realizations of graphic degree sequences, or even studying and improving network privacy. While there exist several known sufficient conditions for $P$-stability, we don't know any useful necessary condition for it. We also do not have good insight of possible structure of $P$-stable degree sequence families. At first we will show that every known infinite $P$-stable degree sequence set, described by inequalities of the parameters $n, c_1, c_2, \Sigma$ (the sequence length, the maximum and minimum degrees and the sum of the degrees) is ,,fully graphic" meaning that every degree sequence from the region with an even degree sum, is graphic. Furthermore, if $\Sigma$ does not occur in the determining inequality, then the notions of $P$-stability and full graphicality will be proved equivalent. In turns, this equality provides a strengthening of the well-known theorem of Jerrum, McKay and Sinclair about $P$-stability, describing the maximal $P$-stable sequence set by $n, c_1, c_2$. Furthermore we conjecture that similar equivalences occur in cases if $\Sigma$ also part of the defining inequality.