🤖 AI Summary
This study addresses the problem of characterizing graphic sequences and generating all their realizations. By introducing a “2-reduction” operation—subtracting one from each of two elements in the sequence—that preserves graphic equivalence, the authors develop a unified inductive reduction framework. This approach not only streamlines the reproducibility of classical characterization theorems such as Erdős–Gallai but also enables efficient construction of all graph realizations corresponding to a given sequence. Furthermore, it yields a novel equivalent characterization of graphic sequences, uncovering a deep connection between their intrinsic mathematical structure and underlying algorithmic mechanisms.
📝 Abstract
A sequence D=(d1, d2, ..., dn) of positive integers is graphic if it is the degree sequence of a simple graph, called in this case a {\em realization} of D. In this paper, we introduce the operation of 2-reduction, that subtracts 1 from two integers of D such that the resulting sequence D' is graphic if and only if D is graphic. We show that 2-reductions allow us to simply generate all the realizations of D, to prove existing characterizations of graphic sequences, as well as to propose new characterizations that highlight connections between mathematical and algorithmic aspects of graphic sequences.