We generalize the order-theoretic variant of the Myhill-Nerode theorem to graph languages, and characterize the recognizable graph languages as the class of languages for which the Myhill-Nerode quasi order is a well quasi order. In the second part of the paper we restrict our attention to graphs of bounded interface size, and use Myhill-Nerode quasi orders to verify that, for such bounded graphs, a recognizable graph property is an invariant of a graph transformation system. A recognizable graph property is a recognizable graph language, given as an automaton functor. Finally, we present an algorithm to approximate the Myhill-Nerode ordering.
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