TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Deleting a leaf from each of two forbidden graphs preserves virality

Statement

Let $F$ be a finite family of finite graphs, and let $H_{1}, H_{2} \in F$ . For $i = 1, 2$ , let $v_{i}$ be a leaf of $H_{i}$ , and write $H_{i}^{'} : = H_{i} ∖ {v_{i}}$ . If

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

and

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

are both viral, then $F$ is viral.

Facts & Assumptions

Given: A finite family $F$ of finite graphs, members $H_{1}, H_{2} \in F$ , and leaves $v_{i} \in V (H_{i})$ for $i = 1, 2$ .

[F1]

The Nguyen-Scott-Seymour leaf-deletion theorem states exactly that, under these hypotheses, if the two modified families $F_{1}$ and $F_{2}$ are viral, then $F$ is viral.

Proof

technique · direct

1.1

Form the two modified families $F_{1}$ and $F_{2}$ by deleting the chosen leaves from $H_{1}$ and $H_{2}$ . The hypothesis says that both families are viral.

given

2.1

Applying [F1] to the data of step 1.1 gives that $F$ is viral.

step 1.1F1∎

Depends on

Used by

Dependency tree · two levels

10 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

Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. IV. New graphs with the Erdős-Hajnal property, Theorem 7.8 (standard reference, not scraped)