Neighborhood diversity nd(G): number of equivalence classes under 'same type' (u≡v iff N[u]\{v}=N[v]\{u}). Each class is a clique or independent set. Lampis (2012): many problems FPT by nd. nd ≤ 2^tw. Computable in O(n+m).
How to Use
Select graph:
- nd: # types
- Classes: Twin groups
- FPT: Parameterized nd
Vertex Types
Two vertices have same type if they have identical neighborhoods (up to each other). Each type class is either a clique (all connected) or independent set (none connected). Graph compresses to nd-vertex 'type graph'.
FPT Results
Lampis 2012: independent set, domination, coloring, Hamiltonian — all FPT by nd. Uses ILP with nd variables. Very efficient for graphs with few vertex types (highly symmetric or regular structures).
Step-by-Step Instructions
- 1Select graph.
- 2Compute nd.
- 3Group by type.
- 4Identify cliques/ISets.
- 5Apply FPT.