On realization graphs of degree sequences

TL;DR

This paper explores the structure of realization graphs of degree sequences, linking Cartesian products to canonical decompositions, and characterizes sequences with realization graphs that are triangle-free or hypercubes.

Contribution

It establishes a connection between realization graphs and canonical decompositions, and characterizes degree sequences with special realization graph structures.

Findings

Connected realization graphs can be characterized via Cartesian products.

Identifies degree sequences with realization graphs that are triangle-free.

Identifies degree sequences with realization graphs that are hypercubes.

Abstract

Given the degree sequence $d$ of a graph, the realization graph of $d$ is the graph having as its vertices the labeled realizations of $d$, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.