Conditional connectivity κ(G;P): remove minimum vertices so G-S disconnects AND all surviving components satisfy property P. P = 'no isolates' gives super-connectivity. P = 'min degree ≥ k' gives k-extra connectivity. Unified framework.
How to Use
Select graph and P:
- κ(G;P): Conn.
- P: Property
- vs κ: Compare
Common Properties
P₀: no property (standard κ). P₁: no isolated vertices (super κ). Pₖ: all components have min-degree ≥ k (extra connectivity). P_size: all components have ≥ k+1 vertices. Each gives stronger measure.
Hierarchy
κ(G;P₀) ≤ κ(G;P₁) ≤ κ(G;P₂) ≤ .... Stricter properties require more vertex removals. Each level gives finer network resilience information.
Step-by-Step Instructions
- 1Select graph.
- 2Choose property P.
- 3Compute κ(G;P).
- 4Compare hierarchy.
- 5Apply to design.