CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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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

1.1F1givenconstruct

First reject malformed instances and any $k > ∣ V (G) ∣$ ; such an instance cannot contain a $k$ -clique. Otherwise $k \leq ∣ V (G) ∣$ , so a certificate is a list of at most $∣ V (G) ∣$ vertices. Check that exactly $k$ listed vertices are distinct and every pair is adjacent. The certificate length and verification time are polynomial in the graph encoding length, even when $k$ is written in binary. Thus CLIQUE lies in $N P$ .

1.2L1L2L3

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.

2.1L3step 1.1step 1.2∎

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

Depends on

Used by

Dependency tree · two levels

12 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)