Arboricity a(G): minimum edge-disjoint forests covering all edges. Nash-Williams (1964): a(G) = max_{H⊆G} ⌈|E(H)|/(|V(H)|-1)⌉. a(tree)=1. a(K_n)=⌈n/2⌉. Planar: a≤3. Measures 'density' from a forest decomposition perspective.
How to Use
Select graph:
- a(G): Arboricity
- Formula: Nash-Williams
- Planar: a ≤ 3
Nash-Williams Formula
a(G) = max over all subgraphs H with |V(H)|≥2 of ⌈|E(H)|/(|V(H)|-1)⌉. This is a min-max formula computable in polynomial time via matroid intersection. Elegant connection to matroid theory.
Applications
Data structure design: arboricity bounds query time. Graph orientation: can orient edges so max in-degree = a(G). Sparse graph algorithms: many run in O(a·n) time. Network reliability: forest decompositions give redundancy.
Step-by-Step Instructions
- 1Select graph.
- 2Compute a(G).
- 3Check Nash-Williams.
- 4Find forests.
- 5Compare to density.