  Self-testing identifies quantum states and correlations that exhibit
nonlocality, distinguishing them, up to local transformations, from other
quantum states. Due to their strong nonlocality, it is known that all graph
states can be self-tested in the standard setting - where parties are not
allowed to communicate. Recently it has been shown that graph states display
nonlocal correlations even when bounded classical communication on the
underlying graph is permitted, a feature that has found applications in proving
a circuit-depth separation between classical and quantum computing. In this
work, we develop self testing in the framework of bounded classical
communication, and we show that certain graph states can be robustly
self-tested even allowing for communication. In particular, we provide an
explicit self-test for the circular graph state and the honeycomb cluster state
- the latter known to be a universal resource for measurement based quantum
computation. Since communication generally obstructs self-testing of graph
states, we further provide a procedure to robustly self-test any graph state
from larger ones that exhibit nonlocal correlations in the communication
scenario.
