Splitting number sp(G): minimum vertex splits to achieve planarity. Split vertex v: replace with v₁,v₂, partition neighbors between them. sp=0 ⟺ planar. Related to skewness: sp ≤ sk. More refined measure of non-planarity.
How to Use
Select graph:
- sp: Splitting #
- Split: Vertex operation
- Planar: sp=0?
Vertex Splitting
Take vertex v with neighbors N(v). Split into v₁, v₂. Partition N(v) into N₁∪N₂. Edges from v₁ to N₁, v₂ to N₂. Reduces vertex degree. Enough splits → planar graph.
sp vs sk
sp ≤ sk: each edge deletion can be simulated by a split. But sp can be much smaller: splitting one vertex can save many edge crossings. More efficient non-planarity repair.
Step-by-Step Instructions
- 1Select graph.
- 2Compute sp.
- 3Find split vertices.
- 4Verify planarity.
- 5Compare with sk.