Tenacity T(G) = min_S (|S| + τ(G-S)) / ω(G-S). Balances separator size, largest remnant, and number of pieces. T ≥ 1 implies Hamiltonian (conjecture, proved for many classes). Stronger than toughness as vulnerability measure.
How to Use
Select graph:
- T: Tenacity
- |S|+τ: Numerator
- ω: Components
The Formula
T(G) = min over cut sets S of (|S| + τ(G-S)) / ω(G-S). Numerator: cost of separator + largest surviving piece. Denominator: fragmentation count. Higher T = more resilient network.
Hamiltonian Connection
Conjecture: T ≥ 1 implies Hamiltonian. Proved for special classes. Stronger than Chvátal's toughness condition. Active research area in Hamiltonian graph theory.
Step-by-Step Instructions
- 1Select graph.
- 2Find optimal S.
- 3Compute (|S|+τ)/ω.
- 4Compare with toughness.
- 5Check Hamiltonian.