Comparability graph: edges can be transitively oriented (if a→b and b→c, then a→c). Equivalently: graph of a partial order. Co-comparability = permutation graph. Perfect: χ=ω. Recognizable in O(n+m) via modular decomposition.
How to Use
Select graph:
- Test: Comparability?
- Orient: Transitive
- Poset: Partial order
Transitive Orientation
Can edges be directed so transitivity holds? Gallai (1967): G is comparability iff every odd cycle has two consecutive edges forming a 'triangle'. Modular decomposition gives O(n+m) algorithm.
Dilworth's Theorem
For posets (comparability graphs): max antichain = min chain cover. Graph-theoretically: α(G) = χ̄(G) (independence number = clique cover of complement). Beautiful duality theorem connecting order theory and graph theory.
Step-by-Step Instructions
- 1Select graph.
- 2Check comparability.
- 3Find orientation.
- 4Build poset.
- 5Apply Dilworth.