Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedprecheck 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:

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

Facts & Assumptions

Given: A graph G and disjoint sets A,B⊆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 G‾; likewise, no cross pair is an edge of G exactly when every cross pair is an edge of G‾.

F2
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

Dependency tree · two levels

5 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