CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Induced removal must permit adding edges as well as deleting them

Statement

For every $n \geq 3$ there is an $n$ -vertex graph with an induced empty three-vertex graph that cannot be destroyed by edge deletions, although adding one edge destroys that induced copy.

Facts & Assumptions

Given: An integer $n \geq 3$ .

[L1]

Induced removal permits changing adjacencies in both directions (Induced graph removal lemma for a fixed graph).

[L2]

The empty graph on three vertices has no edges (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[L3]

Labelled induced copies are injective maps preserving edges and nonedges (The induced-embedding count ${ind}_{H} (G)$ ).

Counterexample

technique · direct

1.1givenL2L3choose

Begin with $K_{n}$ , choose a triple $S$ , and delete exactly its three internal edges. By [L2] and [L3], $S$ induces the empty three-vertex graph.

2.1step 1.1

It is the unique unlabelled empty triple: every triple other than $S$ contains a vertex outside $S$ , and that vertex is adjacent to both other vertices.

2.2step 1.1

Deleting more edges never changes any of the three nonedges within $S$ into an edge, so the induced empty triple on $S$ survives every deletion-only operation.

3.1step 2.1step 2.2L1∎

Adding any one of the three missing edges within $S$ destroys this copy, and step 2.1 shows the resulting graph has no empty triple. Thus allowing additions, as [L1] does, is indispensable.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

D. Conlon and J. Fox, Graph removal lemmas, sec. 1 (standard reference, not scraped)