We show that, in the aforementioned models of testing graph properties, characterization by such invaraint local conditions is closely related to proximity oblivious testing (as defined by Goldreich and Ron, STOC'09). In contrast to this relation, we show that, in general, characterization by invaraint local conditions is neither necessary nor sufficient for proximity oblivious testing. Furthermore, we show that easy testability is {\em not}\/ guaranteed even when the property is characterized by local conditions that are invariant under a 1-transitive group of permutations.