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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.

Complete, anticomplete and mixed vertex-set pairs in $P_{4}$

Statement

In the path $P_{4} = 1, 2, 3, 4$ :

$({2}, {1, 3})$ is complete;
$({1}, {3, 4})$ is anticomplete; and
$({2}, {3, 4})$ is mixed.

Facts & Assumptions

Given: $P_{4}$ with edge set ${12, 23, 34}$ .

[F1]

Complete means all cross pairs are edges, anticomplete means none are edges, and mixed means some but not all are edges (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

[F2]

The only edges of the path are consecutive pairs (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Verification

technique · direct

1.1F1F2

Both $21$ and $23$ are edges, so $({2}, {1, 3})$ is complete.

1.2F1F2

Neither $13$ nor $14$ is an edge, so $({1}, {3, 4})$ is anticomplete.

1.3F1F2

The pair $23$ is an edge but $24$ is not, so $({2}, {3, 4})$ is mixed.

2.1step 1.1step 1.2step 1.3∎

These verify the three claimed pair types.

Remarks

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

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

Complete, anticomplete and mixed vertex-set pairs in P4P4​