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

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Purity is symmetric; complementation swaps complete and anticomplete pairs and preserves mixed pairs

Statement

For disjoint vertex sets $A,B$ in a graph $G$ :

$(A,B)$ is complete, anticomplete, pure or mixed exactly when $(B,A)$ has the same property;
complementation swaps complete pairs with anticomplete pairs; and
complementation preserves pure pairs and mixed pairs.

Facts & Assumptions

Given: A graph $G$ and disjoint sets $A,B\subseteq V(G)$ .

[F1]

Complete, anticomplete, pure and mixed pairs are defined by the cross-pair adjacency pattern (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

[F2]

Graph adjacency is symmetric, and complementation exchanges adjacency with nonadjacency between distinct vertices (Graph isomorphisms, automorphisms and graph complements).

Proof

technique · direct

1.1

Since $ab$ and $ba$ describe the same edge, reversing the ordered pair of sets changes none of the four properties.

F1F2

1.2

Every cross pair is an edge of $G$ exactly when no cross pair is an edge of $\overline G$ ; likewise, no cross pair is an edge of $G$ exactly when every cross pair is an edge of $\overline G$ .

2.1

Hence complementation swaps complete and anticomplete pairs. It therefore preserves their union, the pure pairs, and its complement, the mixed pairs.

step 1.2F1

3.1

Together with symmetry from step 1.1, this proves all assertions.

step 1.1step 2.1∎

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: 8 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)
ISGCI, Self-complementary classes (standard reference, not scraped)