We introduce a new family of graphical models that consists of graphs with possibly directed, undirected and bidirected edges but without directed cycles. We show that these models are suitable for representing causal models with additive error terms. We provide a set of sufficient graphical criteria for the identification of arbitrary causal effects when the new models contain directed and undirected edges but no bidirected edge. We also provide a necessary and sufficient graphical criterion for the identification of the causal effect of a single variable on the rest of the variables. Moreover, we develop an exact algorithm for learning the new models from observational and interventional data via answer set programming. Finally, we introduce gated models for causal effect identification, a new family of graphical models that exploits context specific independences to identify additional causal effects. (C) 2017 Elsevier Inc. All rights reserved.