Siblings of countable cographs

TL;DR

This paper proves that every countable cograph has either exactly one or infinitely many siblings, addressing a conjecture by Thomassé using well quasi ordering and labelled ordered trees.

Contribution

It establishes a partial proof of Thomassé's conjecture by characterizing the number of siblings of countable cographs.

Findings

Countable cographs have either one or infinitely many siblings.

The proof uses well quasi ordering and labelled ordered trees.

Addresses a partial case of Thomassé's conjecture.

Abstract

We show that every countable cograph has either one or infinitely many siblings. This answers, very partially, a conjecture of Thomass\'e. The main tools are the notion of well quasi ordering and the correspondence between cographs and some labelled ordered trees.