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.
Nonbridge edges lie on 2-connected cycles
Statement
If an edge of a finite connected simple graph is not a bridge, it lies on a simple cycle, and that cycle subgraph is 2-connected.
Facts & Assumptions
Deleting an edge removes just that edge; a bridge is an edge whose deletion increases the number of connected components. Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors A bridge as an edge whose deletion increases the number of connected components
In a connected graph any two vertices can be joined by a path, whose vertices are distinct; a cycle closes such a path with one additional edge and has length at least three. Connected graphs and connected components defined by the existence of vertex paths Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
The subgraph consisting of a simple cycle is 2-connected, with 2-connectivity understood in the nontrivial block convention. Cycles are 2-connected Cutvertices, 2-connected pieces and nontrivial blocks
Every edge of a finite simple graph has two distinct endpoints, and each unordered pair of vertices occurs as at most one edge. A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
A subgraph is specified by a subset of the vertices and a subset of the edges whose endpoints lie in that vertex subset. Subgraphs, induced subgraphs and spanning subgraphs
Proof
Given: The graphs and hypotheses in the Statement.
Goal: The asserted conclusion.
Since is connected and is not a bridge, still has one connected component. Choose a to path in . Its length is at least two, because the graph is simple and the edge was deleted.
Closing this path with gives a simple cycle containing that edge. The cycle subgraph is 2-connected by the cycle lemma.
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
- Connected graphs and connected components defined by the existence of vertex paths
- Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors
- A bridge as an edge whose deletion increases the number of connected components
- Cycles are 2-connected
- A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
- Subgraphs, induced subgraphs and spanning subgraphs
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.