PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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A graph is bull-free if and only if its complement is bull-free

Statement

A finite simple graph $G$ is bull-free if and only if its complement $\overline{G}$ is bull-free.

Facts & Assumptions

Given: A finite simple graph $G$ .

[F1]

The bull has vertices ${x_{1}, x_{2}, x_{3}, y, z}$ and edges $x_{1} x_{2}$ , $x_{2} x_{3}$ , $x_{1} x_{3}$ , $x_{1} y$ , and $x_{2} z$ (The bull graph).

[F2]

In the complement graph, two distinct vertices are adjacent exactly when they are nonadjacent in the original graph (Graph isomorphisms, automorphisms and graph complements).

[F3]

A graph is bull-free exactly when it has no induced bull (A bull-free graph).

Proof

technique · direct

1.1

By [F1] and [F2], the complement of the bull is again a bull: the bijection $x_{1} \mapsto y$ , $x_{2} \mapsto z$ , $x_{3} \mapsto x_{3}$ , $y \mapsto x_{2}$ , $z \mapsto x_{1}$ sends nonedges of the bull to edges of the bull.

F1F2algebra

2.1

If $G$ contains an induced bull on a vertex set $S$ , then $\overline{G} [S]$ is the complement of that bull, hence another bull by step 1.1. The same argument with $G$ and $\overline{G}$ interchanged proves the converse implication.

step 1.1F2

3.1

Therefore $G$ has an induced bull exactly when $\overline{G}$ does, so [F3] gives the equivalence of bull-freeness.

step 2.1F3∎

Depends on

Used by

Dependency tree · two levels

6 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

Maria Chudnovsky and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Section 1 (standard reference, not scraped)
Maria Chudnovsky, The structure of bull-free graphs III: global structure, Section 2.1 (standard reference, not scraped)