LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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The anticonnected components of $G$ are exactly the connected components of $\overline G$

Statement

For every graph $G$ , its anticomponents are exactly the vertex sets of the connected components of $\overline G$ . In particular, they partition $V(G)$ .

Facts & Assumptions

Given: A finite graph $G$ .

[F1]

Anticomponents are defined to be the component vertex sets of $\overline G$ (Anticonnected graphs and anticonnected components).

[L1]

Connected components partition a graph's vertex set (The connected components of a graph partition its vertex set and are its maximal connected subgraphs).

[L2]

$\overline{G[A]}=\overline G[A]$ ( $\overline{G[W]}=\overline G[W]$ for every vertex set $W$ ).

Proof

technique · direct

1.1

By F1, a set $A$ is an anticomponent of $G$ exactly when it is the vertex set of a connected component of $\overline G$ .

2.1

Equivalently, $\overline G[A]=\overline{G[A]}$ is connected and $A$ is maximal with this property.

step 1.1L2

3.1

The component partition theorem applied to $\overline G$ shows that these sets partition $V(\overline G)=V(G)$ .

step 1.1L1∎

Depends on

Used by

Distinct connected components are anticomplete, and distinct anticonnected components are complete Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Valerio Boncompagni, On hereditary graph classes defined by forbidding Truemper configurations (PhD thesis, 2018) (standard reference, not scraped)

The anticonnected components of GGG are exactly the connected components of G‾\overline GG