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

Daligault, Rao and Thomassé 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 H1,H2 are forbidden. We confirm it for one of the two stubborn cases, namely for the (H1,H2) = (triangle, P2 + P4) case, by proving that the class of (triangle, P2 + P4)-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 (triangle, P1 +P5)-free graphs, which is known to have bounded cliquewidth, is well-quasi-ordered. Our results enable us to complete the classification of graphs H for which the class of (triangle,H)-free graphs is well-quasi-ordered.