Scattering number sc(G) = max over cut sets S of ω(G-S) - |S|. If sc ≤ 0: graph might be Hamiltonian. Stars: sc = n-2 (many pieces, small separator). Complete graphs: sc = 2-n. Measures vulnerability from the attacker's perspective.
How to Use
Select graph:
- sc: Scattering #
- ω-|S|: Surplus
- Ham: sc ≤ 0?
Interpretation
sc > 0: attacker can create more fragments than vertices removed. Bad for network resilience! sc ≤ 0: defender always 'wins' — removing k vertices creates at most k pieces. Good for Hamiltonicity.
Extreme Cases
Star K_{1,n-1}: remove center → n-1 pieces, |S|=1, sc = n-2 (very vulnerable). K_n: remove k → 1 piece, sc = max(1-k) = 2-n (very robust). Path P_n: sc = 1 (moderately vulnerable).
Step-by-Step Instructions
- 1Select graph.
- 2Find max ω-|S|.
- 3Check sc ≤ 0?.
- 4Compare extremes.
- 5Assess vulnerability.