This article deals with dynamical networks in which the relations between node signals are described by proper transfer functions and external signals can influence each of the node signals. We are interested in graph-theoretic conditions for identifiability of such dynamical networks, where we assume that only a subset of nodes is measured but the underlying graph structure of the network is known. This problem has recently been investigated from a generic viewpoint. Roughly speaking, generic identifiability means that the transfer functions in the network can be identified for “almost all” network matrices associated with the graph. In this article, we investigate the stronger notion of identifiability for all network matrices. To this end, we introduce a new graph-theoretic concept called the graph simplification process. Based on this process, we provide necessary and sufficient topological conditions for identifiability. Notably, we also show that these conditions can be verified by polynomial time algorithms. Finally, we explain how our results generalize existing sufficient conditions for identifiability.