Clique-width and Well-Quasi-Ordering of Triangle-Free Graph Classes
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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 H_1,H_2 are forbidden. We confirm it for one of the two stubborn cases, namely for the (H_1,H_2)=(triangle,P_2+P_4) case, by proving that the class of (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 (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 (triangle,H)-free graphs is well-quasi-ordered.
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