Treedepth td(G): min height of elimination tree (or DFS tree of supergraph). td ≥ pw+1 ≥ tw+1. Paths: td = ⌈log₂(n+1)⌉. Stars: td = 2. Trees: td = O(log n). Captures how deeply nested the graph structure is.
How to Use
Select graph:
- td: Treedepth
- Elim: Elimination tree
- pw: Compare
Elimination Trees
Eliminate vertices one by one, adding edges between remaining neighbors. The order defines a rooted forest. Treedepth = min height over all elimination orderings. Captures recursion depth of divide-and-conquer.
Key Bounds
td ≥ pw+1 (treedepth dominates pathwidth). td ≤ n (trivially). For paths: td = ⌈log₂(n+1)⌉ (binary decomposition). For trees: td ≤ 1 + log₂ n. Small td ⟹ many algorithmic benefits (quantifier elimination).
Step-by-Step Instructions
- 1Select graph.
- 2Build elim. tree.
- 3Minimize height.
- 4Compare with pw.
- 5Check bounds.