Cubicity cub(G): minimum d for G = intersection graph of unit hypercubes in ℝ^d. Stricter than boxicity (unit constraint). cub(G) ≥ box(G). Unit interval: cub=1. cub ≤ ⌈2n/3⌉. Related to L∞ dimension of a graph.
How to Use
Select graph:
- cub: Cubicity
- vs box: Compare
- Unit: Equal sides
L∞ Connection
G has cubicity d iff vertices can be embedded in ℝ^d with L∞ metric such that u~v iff ||u-v||_∞ ≤ 1. Cubicity = L∞ dimension. Beautiful metric geometry interpretation.
Bounds
cub ≤ ⌈2n/3⌉ (tight). cub ≤ box · ⌈log n⌉ approximately. For trees: cub = ⌈log₂ n⌉ (when Δ large). Computing cubicity is NP-hard.
Step-by-Step Instructions
- 1Select graph.
- 2Compute cubicity.
- 3Compare with boxicity.
- 4Check L∞ embedding.
- 5Apply bounds.