I’m interested in hereditary graph classes characterized by a small number of excluded subgraphs. There are some well-known graph classes that are characterized by three or four obstructions — examples are threshold graphs, chain grapns and trivially perfect graphs. My question is: are there natural graph classes characterized by five obstructions? (No relation to the eponymous movie). It may be possible to obtain some by considering the P4-structure of the graph.
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