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We consider large uniform labeled random graphs in different classes with fewinduced $P_4$ ($P_4$ is the graph consisting of a single line of $4$ vertices)which generalize the case of cographs. Our main result is the convergence to aBrownian limit object in the space of graphons. As a by-product we obtain newasymptotic enumerative results for all these graph classes. We also obtaintypical density results for a wide variety of induced subgraphs. Theseasymptotics hold at a smaller scale than what is observable through the graphonconvergence. Our proofs rely on tree encoding of graphs. We then use mainlycombinatorial arguments, including the symbolic method and singularityanalysis.

Graph classes with few $P_4$'s: Universality and Brownian graphon limits