CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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CLIQUE is NP-complete

Statement

CLIQUE is NP-complete.

Facts & Assumptions

Given: An arbitrary CLIQUE instance $(G, k)$ .

[L1]

The previous theorem gives a polynomial-time many-one reduction from $3$ -SAT to CLIQUE, by 3SAT polynomial-time many-one reduces to CLIQUE.

[L2]

The language $3$ -SAT is NP-complete, by 3-SAT is NP-complete.

[L3]

A language is NP-complete exactly when it is NP-hard and belongs to $N P$ , by NP-hard and NP-complete languages.

[F1]

A yes-instance of CLIQUE consists of a graph with $k$ pairwise adjacent chosen vertices, by Clique, independent set, and vertex cover decision problems.

Proof

technique · direct

1.1

A certificate for $(G, k) \in C L I Q U E$ is a list of $k$ vertices. One checks in polynomial time that the listed vertices are distinct and that every pair is adjacent, so CLIQUE lies in $N P$ .

F1givenconstruct

1.2

Because [L2] says that $3$ -SAT is NP-complete, every language in $N P$ reduces to $3$ -SAT. Composing any such reduction with the fixed reduction [L1] shows that every language in $N P$ reduces to CLIQUE. Hence CLIQUE is NP-hard.

L1L2L3

2.1

Step 1.1 gives CLIQUE $\in N P$ , and step 1.2 gives NP-hardness. Therefore CLIQUE is NP-complete by [L3].

L3step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)
Fabrizio Iozzi and Luca Trevisan, Handout NP3 (standard reference, not scraped)