Strong NP-completeness

An NP decision problem is strongly NP-complete when its NP-hardness persists with numerical parameters written in the unary numeral system. An important proof method bounds every relevant numerical value by a polynomial in the remaining explicit input size. For the clique decision problem, values $k>|V(G)|$ can be mapped to one fixed no-instance; otherwise writing $1^k$ takes only polynomial time. The distinction is representation-sensitive, so an explicit graph encoding must be specified.