Clique-width and Well-Quasi-Ordering of Triangle-Free Graph Classes

TL;DR

This paper proves that certain hereditary graph classes defined by forbidding a triangle and another specific subgraph are both well-quasi-ordered and have bounded clique-width, advancing the classification of such graph classes.

Contribution

It confirms for the first time that the class of (triangle, P2+P4)-free graphs has bounded clique-width and is well-quasi-ordered, completing part of the classification for these graph classes.

Findings

(triangle, P2+P4)-free graphs have bounded clique-width

(triangle, P2+P4)-free graphs are well-quasi-ordered

(triangle, P1+P5)-free graphs are well-quasi-ordered

Abstract

Daligault, Rao and Thomass\'e asked whether every hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev (JCTB 2017+) gave a negative answer to this question, but their counterexample is a class that can only be characterised by infinitely many forbidden induced subgraphs. This raises the issue of whether the question has a positive answer for finitely defined hereditary graph classes. Apart from two stubborn cases, this has been confirmed when at most two induced subgraphs $H_1,H_2$ are forbidden. We confirm it for one of the two stubborn cases, namely for the $(H_1,H_2)=(\mbox{triangle},P_2+P_4)$ case, by proving that the class of $(\mbox{triangle},P_2+P_4)$-free graphs has bounded clique-width and is well-quasi-ordered. Our technique is based on a special decomposition of $3$-partite graphs. We also use this technique to prove that the class of $(\mbox{triangle},P_1+P_5)$-free graphs, which is known to have bounded clique-width, is well-quasi-ordered. Our results enable us to complete the classification of graphs $H$ for which the class of $(\mbox{triangle},H)$-free graphs is well-quasi-ordered.