Boolean-width boolw(G): at each cut (A, V\A), count distinct neighborhoods N(v)∩A for v in V\A. Width = log₂ of max such count. boolw ≤ rw. Many NP-hard problems solvable in time n^O(1) · 2^O(boolw). Elegant counting-based parameter.
How to Use
Select graph:
- boolw: Boolean-width
- Types: Distinct nbrs
- vs rw: Compare
Distinct Neighborhoods
For a cut (A, B): two vertices in B have the same 'type' if they see the same subset of A. Boolean-width counts (log₂ of) how many distinct types exist. Fewer types → simpler structure.
Algorithmic Applications
Maximum independent set, dominating set, chromatic number: all solvable in O*(2^boolw) time. Boolean-width can be much smaller than treewidth for dense graphs. Practical for many real-world networks.
Step-by-Step Instructions
- 1Select graph.
- 2Compute boolw.
- 3Count types.
- 4Compare with rw.
- 5Apply algorithms.