DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs

Definition

Let $G$ be a finite simple graph and let $A,B\subseteq V(G)$ be disjoint. An edge between $A$ and $B$ is an edge $ab$ with $a\in A$ and $b\in B$ .

The pair $(A,B)$ is:

complete when every $a\in A$ is adjacent to every $b\in B$ ;
anticomplete when no $a\in A$ is adjacent to any $b\in B$ ;
pure when it is complete or anticomplete; and
mixed when it is neither complete nor anticomplete.

Adjacency is the symmetric edge relation of $G$ (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets, Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree). If $A=\varnothing$ or $B=\varnothing$ , the pair is both complete and anticomplete, hence pure and not mixed.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 8 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)