TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every nontrivial cograph is disconnected or has disconnected complement

Statement

Let $G$ be a cograph with at least two vertices. Then $G$ is disconnected or $\overline{G}$ is disconnected.

Facts & Assumptions

Given: A cograph $G$ with $∣ V (G) ∣ \geq 2$ .

[L1]

Every nontrivial cograph is obtained from two nonempty smaller cographs by one final disjoint-union step or one final complete-connection step (Cographs by the singleton, disjoint-union, and complete-connection recursion).

[L2]

The disjoint union of two nonempty graphs is disconnected (Connected graphs and connected components defined by the existence of vertex paths).

[L3]

If $G_{1}$ and $G_{2}$ are vertex-disjoint, then the complement of $G_{1} \nabla G_{2}$ is the disjoint union of $\overline{G_{1}}$ and $\overline{G_{2}}$ (The complete connection of two disjoint graphs, Graph isomorphisms, automorphisms and graph complements).

Proof

technique · direct

1.1

By [L1], there exist nonempty smaller cographs $G_{1}$ and $G_{2}$ such that $G$ is either the disjoint union of $G_{1}$ and $G_{2}$ , or the complete connection $G_{1} \nabla G_{2}$ .

L1given

2.1

In the disjoint-union case, [L2] immediately shows that $G$ is disconnected.

step 1.1L2

2.2

In the complete-connection case, [L3] shows that $\overline{G} = \overline{G_{1}} ⊔ \overline{G_{2}}$ . Both sides are nonempty because $G_{1}$ and $G_{2}$ are nonempty, so $\overline{G}$ is disconnected by [L2].

step 1.1L2L3

3.1

Therefore one of the two stated alternatives always holds: either $G$ is disconnected, or $\overline{G}$ is disconnected.

step 2.1step 2.2∎

Depends on

Used by

Dependency tree · two levels

11 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

Tero Harju, Lecture Notes on Combinatorial Structures in Graph Theory, Exercise 5.3 (standard reference, not scraped)