Interval Graph Checker

interval intersection model

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About Interval Graph Checker

An interval graph checker testing if G is the intersection graph of intervals on ℝ. Equivalent: chordal + AT-free (no asteroidal triple). O(n+m) recognition via PQ-trees or LBFS. Perfect: χ=ω. Applications in scheduling, bioinformatics. Client-side.

Interval Graph Checker Features

  • Interval test
  • PQ-tree
  • χ = ω
  • O(n+m)
  • Scheduling
Interval graph: intersection graph of intervals on ℝ. Vertex = interval, edge iff intervals overlap. Equivalent: chordal + no asteroidal triple. Recognizable in O(n+m). Perfect: χ=ω. Cornerstone of algorithmic graph theory.

How to Use

Select graph:

  • Test: Interval?
  • Model: Interval assign
  • Perfect: χ=ω

Recognition

Fulkerson-Gross (1965): G is interval iff clique matrix has consecutive 1s property. PQ-tree algorithm: O(n+m). LBFS-based: 6 sweeps suffice. Also: chordal + AT-free (Lekkerkerker-Boland).

Applications

Job scheduling: intervals = time slots, conflicts = overlaps. DNA fragment assembly: intervals on genome. Register allocation in compilers. Temporal reasoning. Resource allocation with time windows.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Check interval.
  3. 3Find interval model.
  4. 4Compute χ=ω.
  5. 5Apply scheduling.

Interval Graph Checker — Frequently Asked Questions

What's an asteroidal triple?+

Three vertices where between any two, there's a path avoiding the neighborhood of the third. AT-free graphs have no asteroidal triple. Interval = chordal + AT-free. AT-freeness ensures 'linearity'.

How are interval graphs used in scheduling?+

Jobs = intervals, conflicts = overlaps. Minimum colors = minimum machines. Since interval graphs are perfect: χ=ω, so greedy on PEO gives optimal schedule. Classic O(n log n) algorithm.

What's a unit interval graph?+

All intervals have the same length. Equivalent to proper interval graphs (no interval contains another). Also equivalent to K_{1,3}-free interval graphs. Even more structured: indifference graphs.

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