Adjacency relationships forced by a degree sequence

TL;DR

This paper establishes necessary and sufficient conditions for adjacency relationships in all realizations of a degree sequence, generalizing threshold graph characterizations and exploring the structure of degree sequences with forced adjacencies.

Contribution

It provides a comprehensive characterization of when vertex adjacencies are fixed across all graphs with a given degree sequence, extending previous threshold graph results.

Findings

Conditions for fixed adjacency in all realizations

Generalization of threshold graph characterizations

Degree sequences with forced adjacencies form an upward-closed set

Abstract

There are typically several nonisomorphic graphs having a given degree sequence, and for any two degree sequence terms it is often possible to find a realization in which the corresponding vertices are adjacent and one in which they are not. We provide necessary and sufficient conditions for two vertices to be adjacent (or nonadjacent) in every realization of the degree sequence. These conditions generalize degree sequence and structural characterizations of the threshold graphs, in which every adjacency relationship is forcibly determined by the degree sequence. We further show that degree sequences for which adjacency relationships are forced form an upward-closed set in the dominance order on graphic partitions of an even integer.