Vertex cover τ(G): minimum vertices touching all edges. Complement of independence: τ=n-α. Gallai: τ+α=n. König: τ=ν for bipartite. NP-hard in general but admits a simple 2-approximation (take both endpoints of a maximal matching).
How to Use
Select graph:
- τ: Vertex cover
- Gallai: τ=n-α
- König: τ=ν bipartite
NP-Hardness
Finding minimum vertex cover is one of Karp's 21 NP-complete problems. However: (1) Bipartite: polynomial via König. (2) 2-approximation via maximal matching. (3) FPT in parameter τ: O(2^τ · n). (4) Kernelization reduces to 2τ vertices.
Gallai's Identity
τ(G)+α(G)=n always. A set S is a vertex cover iff V\S is an independent set. So min vertex cover = n - max independent set. This means vertex cover and independent set are equivalent problems.
Step-by-Step Instructions
- 1Select graph.
- 2Compute τ(G).
- 3Check Gallai.
- 4Try König.
- 5Compare to ν.