Relative complements and a `switch'-classification of simple graphs

TL;DR

This paper introduces a new classification of finite simple graphs based on switch-equivalence, providing an inductive method to count switch-types and identifying exactly 16 types on 6 vertices.

Contribution

It presents a novel switch-classification framework for graphs and an inductive approach to enumerate switch-types, including a complete classification for graphs on 6 vertices.

Findings

Exactly 16 switch-types on 6 vertices

An inductive method to compute switch-types

Introduction of a switch-classification for simple graphs

Abstract

In the paper we introduce and study a classification of finite (simple, undirected, loopless) graphs with respect to a switch-equivalence (`local-complement' equivalence of \cite{pascvebl}, an analogue of the complement-equivalence of \cite{conell}). In the paper we propose a simple inductive method to compute the number of switch-types of graphs on $n$ vertices and we show that there are exactly 16 such types of graphs on 6 vertices.