Tree decomposition: organize vertices into bags forming a tree. Three rules: (1) every vertex in ≥1 bag, (2) every edge has both endpoints in a bag, (3) bags containing each vertex form a connected subtree. Width = max bag size - 1. Treewidth = min width.
How to Use
Select graph:
- Bags: Vertex groups
- Tree: Bag connections
- Width: Max bag - 1
Properties
Running intersection property: bags containing vertex v form subtree. This ensures 'locality'. Width 1: graph is a forest. Width 2: series-parallel. Width 3: includes planar graphs. Many problems FPT in treewidth.
Algorithms on Decompositions
Dynamic programming on tree decomposition: 3-coloring in O(3^tw·n), independent set in O(2^tw·n), Hamiltonian cycle in O(2^tw·n). Courcelle's theorem: any MSO property in O(f(tw)·n). Extremely powerful framework.
Step-by-Step Instructions
- 1Select graph.
- 2View decomposition.
- 3Examine bags.
- 4Check properties.
- 5Compute width.