Recursive max-linear vectors provide models for causal dependence between large values of random variables that are supported on directed acyclic graphs, but the standard assumption that all nodes of such a graph are observed can be unrealistic. We give necessary and sufficient conditions for a partially observed recursive max-linear vector to be representable as a recursive max-linear (sub-)model and provide a graphical algorithm to construct the latter. Our conditions concern the max-weighted paths of a directed acyclic graph and its minimal representation, which play a key role for such models. In the framework of regular variation we translate these conditions into checkable criteria and establish a connection between max-weighted paths and the extremal dependence measure of transformed variables for pairs of nodes. We propose a statistical algorithm to detect bivariate regularly varying recursive max-linear models among the node variables of a directed acyclic graph and show consistency and asymptotic normality of the estimators of the extremal dependence measure under a thresholding procedure. Simulations show that our algorithm performs satisfactorily. We apply it to nutrition intake data.