Alphabeta Math
CounterexampleConstruction: 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.

The class of connected graphs is not hereditary

Statement refuted

The class of connected finite graphs is hereditary.

Facts & Assumptions

Given: The path P3=v0v1v2.

[F1]

P3 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 Pn and Cn have n vertices).

[F2]

The induced subgraph on {v0,v2} has no edge (Subgraphs, induced subgraphs and spanning subgraphs).

[F3]

A hereditary class must contain every induced subgraph of each member (Hereditary graph classes).

Counterexample

technique · direct
1.1

The graph P3 belongs to the class of connected graphs.

F1
1.2

Its induced subgraph on the endpoints is K‾2, which is disconnected.

F2
2.1

Hence this class is not closed under induced subgraphs and is not hereditary.

step 1.1step 1.2F3∎

Remarks


\draw[->,line width=.9pt] (3.55,0)--node[above,font=\scriptsize] {induce on $\{v_0,v_2\}$} (5.25,0);

\node[vertex] (w0) at (5.8,0) {$v_0$}; \node[vertex] (w2) at (7.3,0) {$v_2$}; \node[caption] at (6.55,-.75) {$\overline K_2$ is disconnected}; \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