leanprover-community / leanprover-community/mathlib4
Graph theory def: Vertex connectivity
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Description
Vertex connectivity for simple graphs.
I suggest we're missing the following definitions:
def IsVertexReachable (G : SimpleGraph V) (k : ℕ∞) (u v : V) : Prop :=
-- `G'.Reachable u v` for every graph `G'` which is produced
-- by removing less than `k` other vertices from `G`
noncomputable def vertexReachability (G : SimpleGraph V) (u v : V) : ℕ∞ :=
⨆ (k : ℕ∞) (_ : G.IsVertexReachable k u v), k
def IsVertexPreconnected (G : SimpleGraph V) : Prop :=
∀ u v, G.IsVertexReachable k u v
def IsVertexConnected (G : SimpleGraph V) : Prop :=
k + 1 ≤ ENat.card V ∧ G.IsVertexPreconnected
noncomputable def vertexConnectivity (G : SimpleGraph V) : ℕ∞ :=
⨆ (k : ℕ∞) (_ : G.IsVertexConnected k), k
and basic API for them.
Contributor guide
First steps
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- Open a pull request that references the issue number.
Research direction
Start by reviewing the existing SimpleGraph reachability and connectivity API, along with the ℕ∞ and ENat.card concepts used in the proposal. Define the scope of the requested vertex-reachability and vertex-connectivity definitions and basic API, with the proposed predicates and values supported consistently.
Written by the indexing model from the issue text.
Assessment
- Domain
- backend
- Issue type
- Feature
- Difficulty
- 5/5
- Estimated time
- Over a week
- Activity status
- Stale
- Clarity
- Mostly clear
- Newbie friendliness
- 35/100