Component connectivity cκₖ(G): minimum vertices to produce ≥ k components. cκ₂ = κ(G). cκ₃ = minimum to create ≥ 3 pieces. Hierarchy: cκ₂ ≤ cκ₃ ≤ .... For K_n: cκₖ = n-k+1 (trivial). For hypercubes: elegant formulas.
How to Use
Select graph and k:
- cκₖ: Component conn.
- k: # components
- Hierarchy: cκ₂≤cκ₃≤...
Multi-Way Cuts
Standard connectivity: 2-way cut (disconnect into ≥ 2 pieces). Component connectivity: k-way cut. How hard to shatter into k pieces? Naturally generalizes connectivity to multi-fragmentation scenarios.
Applications
Distributed computing: ensuring k fault-tolerant regions. Network partitioning: minimum cost to create k zones. Military: minimum attacks to fragment into k isolated groups. Supply chains: k-way disruption resistance.
Step-by-Step Instructions
- 1Select graph.
- 2Choose k.
- 3Compute cκₖ.
- 4Build hierarchy.
- 5Apply to partition.