New results on graph matching from degree preserving growth

TL;DR

This paper investigates the relationship between degree sequences and matchings in graph realizations, providing new bounds and equivalences relevant to the Degree Preserving Growth model for graph evolution.

Contribution

It establishes the equivalence between degree sequence extensions and matchings, and introduces bounds on maximum and minimal matchings in graph realizations.

Findings

Extension of degree sequences relates to existence of matchings in realizations.

Lower bounds for forcible matching numbers are provided.

Estimates for minimal and maximal matchings in realizations are derived.

Abstract

The recently introduced \emph{Degree Preserving Growth} model (Nature Physics, \DOI{10.1038/s41567-021-01417-7}) uses matchings to insert new vertices of prescribed degrees into the current graph of an ever-growing graph sequence. The process depends both on the size of the largest available matchings, which is our focus here, as well as on the actual choice of the matching. First we show that the question whether a graphic degree sequence, extended with a new degree $2\delta$ remains graphic is closely related to the available matchings in the realizations of the sequence. Namely we prove that the extension problem is equivalent to the existence of a realization of the original degree sequence with a matching of size $\delta$. Second we present lower bounds for the \emph{forcible matching number} of degree sequences. This number is the size of the maximum matchings in any realization of the degree sequence. We then study bounds on the size of maximal matchings in \emph{some} realizations of the sequence, known as the \emph{potential matching number}. We also estimate the minimum size of both the maximal and the maximum matchings, as determined by the degree sequence, independently of graphical realizations. Along this line we answer a question raised by Biedl, Demaine \emph{et al.} (\DOI{10.1016/j.disc.2004.05.003}).