Chordal Graph Checker

perfect elimination order

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About Chordal Graph Checker

A chordal graph checker testing if every cycle of length ≥4 has a chord. Equivalent: has perfect elimination ordering (PEO). Chordal = intersection graph of subtrees. Perfect: χ=ω. Maximum cardinality search O(n+m). Client-side.

Chordal Graph Checker Features

  • Chordal test
  • PEO
  • χ = ω
  • MCS O(n+m)
  • Subtree intersection
Chordal graph: every cycle of length ≥ 4 has a chord (diagonal edge). Equivalently: has perfect elimination ordering — vertex ordering where each vertex's earlier neighbors form a clique. Recognizable in O(n+m) via Maximum Cardinality Search.

How to Use

Select graph:

  • Test: Chordal?
  • PEO: Elimination order
  • χ=ω: Perfect

Perfect Elimination Ordering

A vertex v is simplicial if N(v) is a clique. PEO: repeatedly remove simplicial vertices. Chordal iff PEO exists. MCS computes PEO in O(n+m): always pick the vertex with most already-numbered neighbors.

Applications

Sparse matrix factorization: Cholesky factor has fill-in pattern = chordal completion. Database theory: acyclic database schemas. Bayesian networks: moral graph is chordal. Extremely important in practice.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Run MCS.
  3. 3Check PEO.
  4. 4Find χ=ω.
  5. 5Apply algorithms.

Chordal Graph Checker — Frequently Asked Questions

What's Maximum Cardinality Search?+

MCS: number vertices n, n-1, ..., 1. At each step, pick unnumbered vertex with most numbered neighbors. The resulting ordering is a PEO if and only if G is chordal. O(n+m) using bucket sort.

Why are chordal graphs perfect?+

Chordal → PEO exists → greedy coloring on PEO uses exactly ω colors → χ=ω. The PEO ensures greedy never wastes colors because each vertex's earlier neighbors form a clique of size ≤ ω.

What's the subtree intersection model?+

Gavril (1974): G is chordal iff G = intersection graph of subtrees of a tree. Each vertex = subtree, two vertices adjacent iff subtrees intersect. Beautiful geometric characterization.

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