Graphon convergence of random cographs

We study the behavior of random labeled and unlabeled cographs with n vertices as n tends to infinity. We show that both models admit a novel random graphon W 1/2 as distributional limit. Our main tool is an enhanced skeleton decomposition of the random Pólya tree with n leaves and no internal vertices having only one child. As a byproduct, we obtain limits describing the asymptotic shape of this model of random trees.