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The problem of checking blocking properties is studied for networks consisting of arbitrary numbers of finite-state discrete-event subsystems. The topology of the networks is that of a fully connected graph: any subsystem can potentially interact with any other. Two types of blocking are studied: component blocking, whereby a subsystem is potentially prevented from entering its set of marker states, and network blocking, whereby the subsystems are potentially unable to occupy marker states simultaneously. It is shown that if the subsystems are all identical and broadcast actions are permitted, both types of blocking properties are undecidable; but in the absence of broadcast actions, they become decidable. If the subsystems are not necessarily identical but only isomorphic, then blocking properties are in general undecidable; however, a template is proposed for ensuring adequate structure for decidability. It is claimed that this template is sufficiently general to admit many realistic examples.