  Graph states are a large class of multipartite entangled quantum states that
form the basis of schemes for quantum computation, communication, error
correction, metrology, and more. In this work, we consider verification of
graph states generated by an untrusted source and shared between a network of
possibly dishonest parties. This has implications in certifying the application
of graph states for various distributed tasks. We present a protocol which is
globally efficient for a large family of useful graph states, including cluster
states, GHZ states, cycle graph states and more. For general graph states,
efficiency with respect to the security parameter is maintained, though there
is a cost increase with the size of the graph state. The protocols are
practical, requiring only multiple copies of the graph state, local
measurements and classical communication.
