Turing reduction
As is common for complexity classes within the polynomial time hierarchy, a problem is called GI-hard if there is a polynomial-time Turing reduction from any problem in GI to that problem, i.e., a polynomial-time solution to a GI-hard problem would yield a polynomial-time solution to the graph isomorphism problem (and so all problems in GI). A problem formula_3 is called complete for GI, or GI-complete, if it is both GI-hard and a polynomial-time solution to the GI problem would yield a polynomial-time solution to formula_3.