TheoremStatement: AI-adaptedProof: AI-generatedverified 2026-09-24 (gpt-6-sol)

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.

Deleting a leaf and a complementary leaf preserves virality

Statement

Let $F$ be a finite family of finite graphs, and let $H_{1}, H_{2} \in F$ . Suppose $v_{1}$ is a leaf of $H_{1}$ and $v_{2}$ is a leaf of $\overline{H_{2}}$ . Put $H_{i}^{'} = H_{i} - v_{i}$ for $i = 1, 2$ . If

$F_{1} : = {H_{1}^{'}} \cup (F ∖ {H_{1}}), F_{2} : = {H_{2}^{'}} \cup (F ∖ {H_{2}})$

are viral, then $F$ is viral.

Facts & Assumptions

Given: The family and vertices in the statement.

[L1]

A leaf of $\overline{H_{2}}$ is a co-leaf of $H_{2}$ (Co-leaves of a finite graph).

[L2]

The finite-family leaf/co-leaf theorem proves virality from exactly these two modified-family hypotheses (Deleting a leaf and a co-leaf preserves virality of a finite forbidden family).

Proof

technique · direct

1.1givenL1L2

By [L1], $v_{2}$ is a co-leaf of $H_{2}$ . The modified families of [L2] for leaf $v_{1}$ and co-leaf $v_{2}$ are precisely $F_{1}$ and $F_{2}$ .

2.1givenstep 1.1L2∎

Both modified families are viral by hypothesis. Applying [L2] therefore gives virality of $F$ .

Source notes

The earlier version required two ordinary leaves. The cited Theorem 7.8 requires a leaf in $H_{1}$ and a leaf in $\overline{H_{2}}$ ; the bar on $H_{2}$ is essential. The old two-leaf claim is superseded, not used.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Nguyen, Scott and Seymour, Induced subgraph density IV, Theorem 7.8 (standard reference, not scraped)