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

We consider large uniform labeled random graphs in different classes with few induced $P_4$ ($P_4$ is the graph consisting of a single line of $4$ vertices) which generalize the case of cographs. Our main result is the convergence to a Brownian limit object in the space of graphons. As a by-product we obtain new asymptotic enumerative results for all these graph classes. We also obtain typical density results for a wide variety of induced subgraphs. These asymptotics hold at a smaller scale than what is observable through the graphon convergence. Our proofs rely on tree encoding of graphs. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.