Restricted edge connectivity λ'(G): minimum edges to remove such that no resulting component is a single vertex. λ' ≥ λ always. Graph is super-edge-connected if λ' > λ. Esfahanian-Hakimi (1988): λ' exists for connected graphs with n ≥ 4.
How to Use
Select graph:
- λ': Restricted λ
- vs λ: Compare
- Super: λ'>λ?
Super-Edge-Connected
Graph is λ-optimal if λ' = ξ(G) where ξ = min{d(u)+d(v)-2 : uv ∈ E}. Many regular and vertex-transitive graphs are λ-optimal. Optimal networks have predictable failure behavior.
Bounds
λ ≤ λ' ≤ ξ(G). For k-regular: λ' ≤ 2k-2. For vertex-transitive: often λ' = 2k-2 (optimal!). Hypercubes, Cayley graphs: typically λ-optimal.
Step-by-Step Instructions
- 1Select graph.
- 2Compute λ'.
- 3Compare with λ.
- 4Check λ-optimal.
- 5Apply to network.