Alphabeta MathHow 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.
Cycles are 2-connected
Statement
The subgraph consisting of a simple cycle in a finite simple graph is 2-connected.
Statement references
Cutvertices, 2-connected pieces and nontrivial blocks
Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
Subgraphs, induced subgraphs and spanning subgraphs
Connected graphs and connected components defined by the existence of vertex paths
Facts & Assumptions
A simple cycle has at least three vertices, all distinct except for the repeated endpoint; its consecutive vertices are adjacent. Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
A graph is 2-connected when it has at least three vertices, is connected, and remains connected after deleting any vertex. Cutvertices, 2-connected pieces and nontrivial blocks
A graph is connected when its vertex set is nonempty and every two vertices are joined by a path. Connected graphs and connected components defined by the existence of vertex paths
Proof
Given: The graphs and hypotheses in the Statement.
Goal: The asserted conclusion.
Write the cycle in its cyclic vertex order. Following that order joins any two cycle vertices, so the cycle subgraph is connected and has at least three vertices.
After any vertex is deleted, the remaining vertices in cyclic order starting just after that vertex form a path containing all remaining vertices. Thus the remaining graph is connected, proving 2-connectivity.
Depends on
- Cutvertices, 2-connected pieces and nontrivial blocks
- Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
- A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
- Subgraphs, induced subgraphs and spanning subgraphs
- Connected graphs and connected components defined by the existence of vertex paths
Used by · 1
Dependency tree · 2 levels
Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.