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.
The "split-substitute" graphs are the graphs starting with a split graph and substituting any split graph into any node. The forbidden induced subgraphs can be obtains from the forbidden induced subgraphs of split graphs (c4, C5, co-C4), but replacing the non-prime ones (c4 and co-c4) with their minimal prime extensions... (here primality is with respect with modular decomposition.)
So the 5 forbidden induced subgraphs are the C5, P5, co-P5, H6 and co-H6. This forms a self-complementary class which strictly contains the class of cographs.
The maxibrittle graphs are the perfectly orderable graphs for which a vertex sequence defined only by the degrees will produce a perfect order (more specifically, the degree sequence defines a brittle ordering). These graphs have exactly 5 minimal forbidden induced subgraphs.
Graphs where every connected component is a split graph are (C4, C5, P5, necktie, bowtie)-free (see A graph modification approach for finding core–periphery structures in protein interaction networks).