Labelled Well Quasi Ordered Classes of Bounded Linear Clique-Width

TL;DR

This paper develops an algorithm to determine if classes of finite graphs with labelled vertices are well-quasi-ordered by induced subgraphs, confirming a conjecture for classes with bounded linear clique-width and linking graph orderings to embedding relations.

Contribution

It introduces an algorithm for deciding well-quasi-ordering in labelled graph classes with bounded linear clique-width and provides a new proof connecting graph orderings to embedding relations.

Findings

Algorithm successfully decides well-quasi-ordering for labelled graph classes.

Confirmed Pouzet's conjecture under bounded linear clique-width.

Established a connection between well-quasi-orderings and the gap embedding relation.

Abstract

We are interested in characterizing which classes of finite graphs are well-quasi-ordered by the induced subgraph relation. To that end, we devise an algorithm to decide whether a class of finite graphs well-quasi-ordered by the induced subgraph relation when the vertices are labelled using a finite set. In this process, we answer positively to a conjecture of Pouzet, under the extra assumption that the class is of bounded linear clique-width. As a byproduct of our approach, we obtain a new proof of an earlier result from Daliagault, Rao, and Thomass\'e, by uncovering a connection between well-quasi-orderings on graphs and the gap embedding relation of Dershowitz and Tzameret.