Modular-Width Calculator

modular decomposition fan-out

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About Modular-Width Calculator

A modular-width calculator computing mw(G): based on the modular decomposition. A module M is a set where all vertices outside see M uniformly. Modular-width = max fan-out of prime nodes. Bounded for cographs (0). FPT for many problems. Client-side.

Modular-Width Calculator Features

  • mw(G)
  • Module tree
  • Cograph=0
  • Prime nodes
  • Common graphs
Modular-width mw(G): max children of any prime node in the modular decomposition tree. Modules are vertex sets with uniform external neighborhoods. Cographs: mw=0 (no prime nodes). Series-parallel: small mw. Many NP-hard problems FPT parameterized by mw.

How to Use

Select graph:

  • mw: Modular-width
  • Tree: Decomposition
  • Prime: Max children

Modules

Module M ⊆ V: for every v ∉ M, v is adjacent to all of M or none. Trivial modules: ∅, V, singletons. Modular decomposition: unique tree of modules. Node types: series (join), parallel (union), prime (neither).

Algorithmic Use

Many problems (independent set, coloring, domination) solvable FPT by mw: solve on prime nodes (small), combine along tree. Modular decomposition computable in O(n+m). Very practical parameter.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Build mod. decomp.
  3. 3Find prime nodes.
  4. 4Compute max children.
  5. 5Apply FPT.

Modular-Width Calculator — Frequently Asked Questions

What's a module?+

A set M of vertices where every outside vertex sees all or none of M. Like a 'block' that behaves uniformly. Maximal non-trivial modules form a partition, recursively decomposed into the modular decomposition tree.

Why is modular-width useful?+

Captures graph structure at a higher level than treewidth. Many practical graphs (social networks, biological) have small modular-width. Problems solvable efficiently on each prime node, combined via the tree.

How does mw relate to clique-width?+

cw ≤ 2·mw (approximately). Bounded modular-width implies bounded clique-width. But mw is more structural: directly tied to the modular decomposition, computable in linear time.

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