We define the strongly separating curve graph to be the full subgraph of the curve graph of a compact orientable surface, where the vertex set consists of all separating curves which do not bound a three-holed sphere. We show that, for all but finitely many surfaces, any automorphism of the strongly separating curve graph is induced by an element of the mapping class group.
Michigan Math. J. 65 (2016) 813-832.