Graph without the annotations of one key or of one vertex or edge
AnnotationDelete[obj]
AnnotationDelete[{obj, itemspec}]
AnnotationDelete[spec, key]
g = CompleteGraph[4, VertexSize -> Small]; g; AnnotationDelete[{g, 4}, VertexStyle]; AnnotationDelete[{g, 1}, VertexLabels]; AnnotationDelete[{g, 1}, VertexSize]
→ Graph[<4>, <6>]