Perfect Graph Checker

χ = ω everywhere

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About Perfect Graph Checker

A perfect graph checker testing if χ(H)=ω(H) for all induced subgraphs H. SPGT (Chudnovsky et al., 2006): perfect iff no odd hole or odd antihole. Polynomial recognition. Includes bipartite, chordal, interval, comparability. Client-side.

Perfect Graph Checker Features

  • Perfect test
  • SPGT
  • χ = ω
  • No odd hole
  • Graph classes
Perfect graph: χ(H) = ω(H) for every induced subgraph H. Strong Perfect Graph Theorem (Chudnovsky-Robertson-Seymour-Thomas, 2006): G perfect iff no odd hole (C_{2k+1}, k≥2) or odd antihole. One of the greatest achievements in combinatorics.

How to Use

Select graph:

  • Test: Perfect?
  • Holes: Odd hole check
  • Antiholes: Odd antihole

Strong Perfect Graph Theorem

Conjectured by Berge (1961). Proved by Chudnovsky et al. (2006): 179-page proof! G is perfect iff G has no induced C_{2k+1} (k≥2) and no induced complement of C_{2k+1} (k≥2). Monumental result.

Perfect Graph Classes

Bipartite ⊂ comparability ⊂ perfect. Chordal ⊂ perfect. Interval ⊂ chordal ⊂ perfect. Split ⊂ chordal ⊂ perfect. Cographs ⊂ perfect. Each class has specialized algorithms.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Check perfect.
  3. 3Find odd holes.
  4. 4Check complement.
  5. 5Apply SPGT.

Perfect Graph Checker — Frequently Asked Questions

What's an odd hole?+

An induced cycle C_{2k+1} with k≥2 (5,7,9,...). C_3 (triangle) is allowed! Only C_5, C_7, C_9,... are forbidden. An odd antihole is the complement of an odd hole.

Why is SPGT important?+

Berge's conjecture was open for 45 years. The proof is one of the longest in combinatorics. It gives exact characterization: just check for odd holes and antiholes. Polynomial-time recognition follows.

What algorithms work better on perfect graphs?+

Maximum clique: O(n^3) via semidefinite programming (Grötschel-Lovász-Schrijver). Coloring: χ=ω, computable in polynomial time. Maximum independent set: equivalent to max clique in complement. Weight versions too.

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