ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-01

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$P_{4}$ is both connected and anticonnected

Statement

The path $P_{4}$ is connected and anticonnected. Indeed, $\overline{P_{4}} ≅ P_{4}$ .

Facts & Assumptions

Given: $P_{4}$ with vertices $1, 2, 3, 4$ and edges $12, 23, 34$ .

[F1]

A path is connected (Connected graphs and connected components defined by the existence of vertex paths, Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

A graph is anticonnected when its complement is connected (Anticonnected graphs and anticonnected components).

[F3]

Complement edges are precisely the nonedges between distinct vertices (Graph isomorphisms, automorphisms and graph complements).

Verification

technique · direct

1.1

The graph $P_{4}$ is connected.

1.2

Its complement has edges $13, 14, 24$ , which form the path $3, 1, 4, 2$ . Thus $\overline{P_{4}} ≅ P_{4}$ and is connected.

2.1

Therefore $P_{4}$ is anticonnected as well as connected.

step 1.1step 1.2F2∎

Remarks


\draw[<->,line width=.9pt] (4.2,0)--node[above,font=\scriptsize] {complement} (5.7,0);

\node[vertex] (b3) at (6.15,0) {$3$}; \node[vertex] (b1) at (7.4,0) {$1$}; \node[vertex] (b4) at (8.65,0) {$4$}; \node[vertex] (b2) at (9.9,0) {$2$}; \draw[edge] (b3)--(b1)--(b4)--(b2); \node[caption] at (8.025,-.7) {$\overline{P_4}$: edges $13,14,24$}; \end{tikzpicture} ```

Depends on

Used by

Nothing in the library uses this result yet.

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)

P4P4​ is both connected and anticonnected