TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-01

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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.

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Every graph with at least two vertices is connected or anticonnected

Statement

Every finite graph with at least two vertices is connected or anticonnected. Equivalently, if $G$ is disconnected and nontrivial, then $\overline{G}$ is connected.

Facts & Assumptions

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

[F1]

$G$ is anticonnected exactly when $\overline{G}$ is connected (Anticonnected graphs and anticonnected components).

[L1]

If $G$ is disconnected, its connected components partition $V (G)$ into at least two nonempty parts (The connected components of a graph partition its vertex set and are its maximal connected subgraphs, Connected graphs and connected components defined by the existence of vertex paths).

[F2]

Vertices in different components of $G$ are nonadjacent in $G$ and hence adjacent in $\overline{G}$ (Graph isomorphisms, automorphisms and graph complements).

Proof

technique · direct

1.1

If $G$ is connected, the first alternative holds. Suppose instead that $G$ is disconnected.

Given

1.2

Let $x, y \in V (G)$ . If $x = y$ , the length-zero path joins them. If they are distinct and lie in different components, then $x y \in E (\overline{G})$ .

L1F2

1.3

If $x \neq y$ and they lie in the same component, choose a vertex $z$ in a different component. Then $x z, z y \in E (\overline{G})$ , so $x, z, y$ is an $x$ - $y$ path in $\overline{G}$ .

L1F2

2.1

Every two vertices are therefore joined in $\overline{G}$ , so $\overline{G}$ is connected and $G$ is anticonnected.

step 1.2step 1.3F1∎

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

Cographs survey (Princeton PDF) (standard reference, not scraped)