On 2-switches and isomorphism classes

TL;DR

This paper investigates how 2-switch operations affect graph isomorphism classes, identifying specific configurations that change isomorphism and providing characterizations of certain graph families.

Contribution

It characterizes when 2-switches alter graph isomorphism classes and introduces new criteria for such changes, advancing understanding of graph realizations.

Findings

4 configurations where 2-switches change isomorphism

A sufficient condition for 2-switches to alter isomorphism

Characterization of matrogenic graphs and unique degree sequence realizations

Abstract

A 2-switch is an edge addition/deletion operation that changes adjacencies in the graph while preserving the degree of each vertex. A well known result states that graphs with the same degree sequence may be changed into each other via sequences of 2-switches. We show that if a 2-switch changes the isomorphism class of a graph, then it must take place in one of four configurations. We also present a sufficient condition for a 2-switch to change the isomorphism class of a graph. As consequences, we give a new characterization of matrogenic graphs and determine the largest hereditary graph family whose members are all the unique realizations (up to isomorphism) of their respective degree sequences.