Vertex integrity I(G) = min_S (|S| + max component size of G-S). Balances separator size vs remaining component size. I=1 for edgeless. I=n for complete. Trees: I = O(log n). Bounded I implies bounded treedepth. NP-hard.
How to Use
Select graph:
- I: Integrity
- |S|: Separator
- max|C|: Component
Graph Vulnerability
Small I: graph is 'fragile'. Few vertices destroy connectivity into tiny pieces. Large I: graph is 'robust'. Need many vertices or large remnants persist. Military/infrastructure network resilience measure.
Key Results
I(G) ≤ td(G) (treedepth). I(K_n) = n. I(P_n) = ⌈log₂(n+1)⌉ + 1 (same structure as treedepth!). I(Star) = 2. Bounded vertex integrity is a strict subset of bounded treedepth.
Step-by-Step Instructions
- 1Select graph.
- 2Find optimal S.
- 3Compute |S|+max|C|.
- 4Compare with td.
- 5Assess vulnerability.