Graph Minor Checker

H ≤_m G minor relation

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About Graph Minor Checker

A graph minor checker testing if H is a minor of G (obtainable by edge deletion, contraction, vertex deletion). Robertson-Seymour (2004): every minor-closed family has finite forbidden minors. Planar: no K_5 or K_{3,3} minor. Client-side.

Graph Minor Checker Features

  • Minor check
  • Wagner theorem
  • Robertson-Seymour
  • Forbidden minors
  • Common graphs
Graph minor H ≤_m G: H obtainable from G by deleting edges/vertices and contracting edges. Robertson-Seymour Graph Minor Theorem (2004): the minor order is a well-quasi-order. Every minor-closed property has finitely many forbidden minors!

How to Use

Select graph pair:

  • Minor? H ≤_m G
  • Forbidden: Known obstructions
  • Property: Minor-closed?

Graph Minor Theorem

Robertson-Seymour (1983-2004, 20 papers!): graphs are well-quasi-ordered by the minor relation. Every minor-closed family has a finite list of forbidden minors. Arguably the deepest theorem in graph theory.

Forbidden Minors

Planar: {K_5, K_{3,3}} (Wagner/Kuratowski). Outerplanar: {K_4, K_{2,3}}. Series-parallel: {K_4}. Linklessly embeddable: Petersen family (7 graphs). The forbidden minors for treewidth ≤ k are known for k ≤ 3.

Step-by-Step Instructions

  1. 1Select G, H.
  2. 2Check H ≤_m G.
  3. 3Find contractions.
  4. 4Identify forbidden.
  5. 5Apply theorem.

Graph Minor Checker — Frequently Asked Questions

What's the difference from subdivision?+

Subdivision: only vertex insertions on edges. Minor allows edge contraction (merging endpoints). Every subdivision is a minor but not vice versa. Kuratowski uses subdivisions; Wagner uses minors. For K_5: both equivalent.

Why is Robertson-Seymour important?+

Any property preserved under taking minors (planarity, bounded treewidth, linkless embedding) has a finite characterization! This is non-constructive: we know the finite list exists but may not know what it is. Revolutionary for structural graph theory.

Can minors be tested efficiently?+

Testing H ≤_m G is NP-complete for general H. But for FIXED H: O(n³) by Robertson-Seymour! So 'is G planar?' (fixed H=K_5,K_{3,3}) is polynomial. The FPT algorithm has enormous constants.

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