Graph immersion: H immersed in G if vertices of H map to G and edges of H map to edge-disjoint paths. Intermediate between topological minor (subdivision) and minor. Robertson-Seymour: immersion well-quasi-ordering. Deep structural theory.
How to Use
Select graphs:
- H ≤ G: Immersion?
- Paths: Edge-disjoint
- Compare: vs minor
Immersion vs Minor vs Subdivision
Subdivision (topological minor): paths must be internally vertex-disjoint. Minor: allows edge contractions. Immersion: paths must be edge-disjoint (may share vertices). subdivision → immersion → (sometimes) minor relationships.
Well-Quasi-Ordering
Robertson-Seymour: graphs are well-quasi-ordered under immersion (and under minors). No infinite antichain! For any graph property closed under immersion: finitely many minimal obstructions. Profound structure theorem.
Step-by-Step Instructions
- 1Select H and G.
- 2Check immersion.
- 3Route paths.
- 4Compare with minor.
- 5Apply WQO.