Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes

TL;DR

This paper investigates the relationship between well-quasi-ordering and clique-width in graph classes defined by two forbidden induced subgraphs, providing new results that support the conjecture that finitely defined classes are well-quasi-ordered with bounded clique-width.

Contribution

It presents new results on the well-quasi-orderability of bigenic graph classes, advancing the understanding of their clique-width properties and supporting the conjecture for finitely forbidden subgraph classes.

Findings

New results on well-quasi-orderability of bigenic classes

Supports the conjecture that finitely forbidden classes are well-quasi-ordered

Advances understanding of clique-width in hereditary graph classes

Abstract

Daligault, Rao and Thomass\'e asked whether a hereditary class of graphs well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this is not true for classes defined by infinitely many forbidden induced subgraphs. However, in the case of finitely many forbidden induced subgraphs the question remains open and we conjecture that in this case the answer is positive. The conjecture is known to hold for classes of graphs defined by a single forbidden induced subgraph $H$, as such graphs are well-quasi-ordered and are of bounded clique-width if and only if $H$ is an induced subgraph of $P_4$. For bigenic classes of graphs, i.e. ones defined by two forbidden induced subgraphs, there are several open cases in both classifications. In the present paper we obtain a number of new results on well-quasi-orderability of bigenic classes, each of which supports the conjecture.