TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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 $|V(G)|\ge2$ .

[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 $xy\in E(\overline G)$ .

L1F2

1.3

If $x\ne y$ and they lie in the same component, choose a vertex $z$ in a different component. Then $xz,zy\in E(\overline G)$ , 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

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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