P, NP, NP-Complete, and NP-Hard

NP (verifiable in poly time) P (solvable in poly time) NP-Complete (hardest in NP) NP-Hard (includes NPC; may be outside NP)
NP P NPC NP-Hard NPC ⊆ NP-Hard NP-Hard may lie outside NP

Reading guide: P is wholly inside NP. The NP-Complete set sits inside NP and is included by the dashed NP-Hard boundary. The dashed NP-Hard ellipse intentionally extends outside NP to indicate NP-Hard problems that are not in NP (e.g., some undecidable or non‑verifiable problems).