On the threshold of intractability

The computational complexity of the graph modification problems Threshold Editing and Chain has been an important open question in theory for more than 15 years. These consist adding deleting fewest possible edges to transform a into threshold or chain graph. We show that both are NP-hard, resolving conjecture by Natanzon, Shamir, Sharan from 2001. On positive side, we admit quadratic vertex kernels give subexponential time parameterized algorithms solving 2O(klog⁡k)+nO(1) time. Few natural known be this class. results also extend completion deletion variants problems, threshold/chain graphs only classes which all three versions—completion, deletion, editing—are NP-complete solvable