Carving width: binary recursive partitioning of edges. Width = max number of edges crossing any partition boundary. Dual to branchwidth for 3-connected graphs. Used in fixed-parameter algorithms. Computable in O(n) for fixed parameter.
How to Use
Select graph:
- cw: Carving width
- Tree: Decomposition
- vs bw: Branchwidth
Carving Decomposition
Binary tree T: leaves = edges of G. Each internal node partitions edge set into two. Width at node = number of vertices in both sides. Carving width = minimum over all such trees of the maximum width.
Algorithms
Seymour-Thomas: for 3-connected graphs, carvingwidth = branchwidth. FPT: O(n) for fixed carving width. Dynamic programming on carving decompositions solves many hard problems efficiently.
Step-by-Step Instructions
- 1Select graph.
- 2Build carving tree.
- 3Compute partition widths.
- 4Minimize maximum.
- 5Apply DP.