Boxicity box(G): minimum d for G to be intersection graph of d-dimensional axis-aligned boxes. box=1 iff G is interval graph. Roberts (1969): box(G) ≤ ⌊n/2⌋. Planar: box≤3. Related to treewidth: box ≤ tw+1.
How to Use
Select graph:
- box: Boxicity
- Dim: Embedding
- Interval: box=1?
Roberts' Theorem
box(G) = minimum number of interval graphs whose edge-intersection is G. Each interval graph captures one dimension. Beautiful connection between geometric and combinatorial graph theory.
Bounds
box ≤ ⌊n/2⌋ (Roberts). box ≤ tw+1 (Chandran-Sivadasan). Planar: box ≤ 3 (Thomassen). Outerplanar: box ≤ 2. Computing box is NP-hard for general graphs.
Step-by-Step Instructions
- 1Select graph.
- 2Compute boxicity.
- 3Check interval.
- 4Apply bounds.
- 5Visualize boxes.