LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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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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A set is $c$ -sparse in $G$ exactly when it is $c$ -dense in $\overline{G}$ , so $c$ -restrictedness is complement-invariant

Statement

Let $G$ be a finite simple graph, let $c \geq 0$ , and let $X \subseteq V (G)$ be nonempty. Then $X$ is $c$ -sparse in $G$ if and only if $X$ is $c$ -dense in $\overline{G}$ . Consequently $X$ is $c$ -restricted in $G$ if and only if it is $c$ -restricted in $\overline{G}$ .

Facts & Assumptions

Given: A finite simple graph $G$ , a real $c \geq 0$ , and a nonempty set $X \subseteq V (G)$ .

[L1]

For distinct vertices $x, y$ , they are adjacent in $\overline{G}$ exactly when they are nonadjacent in $G$ (Graph isomorphisms, automorphisms and graph complements, Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

[L2]

The definitions of $c$ -sparse, $c$ -dense, and $c$ -restricted are those of $c$ -sparse, $c$ -dense and $c$ -restricted vertex sets.

Proof

technique · direct

1.1

For $x \in X$ , the set $N_{\overline{G}} (x) \cap X$ is exactly $(X ∖ {x}) ∖ N_{G} (x)$ by [L1].

2.1

Therefore the inequality defining $c$ -sparsity in $\overline{G}$ is exactly the inequality defining $c$ -density in $G$ , and vice versa, by [L2].

step 1.1L2

3.1

Since $c$ -restricted means the disjunction of the sparse and dense conditions, step 2.1 shows that restrictedness is unchanged by complementation.

step 2.1L2∎

Depends on

Used by

Dependency tree · two levels

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Sources

M. Chudnovsky, A. Scott, P. Seymour, and S. Spirkl, Strengthening Rödl's theorem, sec. 1 (standard reference, not scraped)

A set is cc-sparse in GG exactly when it is cc-dense in G‾G, so cc-restrictedness is complement-invariant