Split graph: V = K ∪ I where K is a clique and I is independent set. Characterized by: no induced C₄, C₅, or 2K₂. Equivalently: both G and Ḡ are chordal. Hammer-Simeone (1977): split iff degree sequence satisfies Σᵢ₌₁ᵐ dᵢ = m(m-1) + Σᵢ₌ₘ₊₁ⁿ dᵢ.
How to Use
Select graph:
- Test: Split?
- Partition: K and I
- Degrees: Sequence check
Characterization
Split graph iff degree sequence d₁≥...≥dₙ satisfies: let m = max index with dₘ≥m-1. Then Σᵢ₌₁ᵐ dᵢ = m(m-1) + Σᵢ₌ₘ₊₁ⁿ dᵢ. Pure degree-sequence test! No adjacency matrix needed.
Properties
Split graphs are perfect (χ=ω). chordal ∩ co-chordal = split. Threshold graphs ⊂ split graphs. Many optimization problems polynomial: domination, covering, packing. Important in intersection graph theory.
Step-by-Step Instructions
- 1Select graph.
- 2Check split.
- 3Find K+I partition.
- 4Verify degrees.
- 5Compute χ=ω.