Rank-width rw(G): width of optimal rank-decomposition. At each cut (V₁,V₂), width = rank over GF(2) of the V₁×V₂ adjacency submatrix. Bounded rank-width ⟺ bounded clique-width. Oum-Seymour (2006): cubic FPT algorithm.
How to Use
Select graph:
- rw: Rank-width
- GF(2): Binary rank
- vs cw: Compare
GF(2) Rank
Adjacency matrix over GF(2) (binary field). Rank = dimension of column space mod 2. Captures structure of bipartite connections across cuts. Elegant algebraic graph parameter.
Algorithms
Oum-Seymour: O(n³) for fixed rw. Approximation: 3rw+1 in polynomial time. Much better than clique-width (which has no known FPT recognition). Rank-width is the 'right' parameter for dense graphs.
Step-by-Step Instructions
- 1Select graph.
- 2Compute rw.
- 3Binary rank.
- 4Compare cw.
- 5Apply FPT.