Graph Thickness Calculator

planar decomposition layers

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About Graph Thickness Calculator

A graph thickness calculator computing θ(G): minimum planar subgraphs covering all edges. θ=1 iff planar. θ(K_n) = ⌈(n+7)/6⌉ for n≥3 (except n=9,10). θ(K_{m,n}) = ⌈mn/(2(m+n-2))⌉. NP-hard in general. Client-side.

Graph Thickness Calculator Features

  • θ(G) value
  • Planar layers
  • K_n formula
  • Earth-moon
  • Common graphs
Graph thickness θ(G): minimum planar subgraphs whose union is G. θ=1 iff planar. Tutte (1963): θ(K_9)=3. Beineke-Harary: θ(K_n) = ⌈(n+7)/6⌉ (n≥3, except n=9,10). Earth-moon problem: χ of thickness-2 graphs ≤ 12.

How to Use

Select graph:

  • θ: Thickness
  • Layers: Planar decomposition
  • Bounds: Known formulas

Earth-Moon Problem

Ringel (1959): what's the chromatic number of thickness-2 graphs? Known: 9 ≤ answer ≤ 12. The 'earth-moon' problem because thickness-2 maps model two-sided planet coloring. Open for 60+ years!

Known Formulas

θ(K_n) = ⌈(n+7)/6⌉ for most n. θ(K_{m,n}) = ⌈mn/(2(m+n-2))⌉. For random graphs: θ ≈ |E|/(3n-6) (close to lower bound). Computing θ is NP-hard.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute thickness.
  3. 3Find decomposition.
  4. 4Apply formulas.
  5. 5Check bounds.

Graph Thickness Calculator — Frequently Asked Questions

How does thickness relate to arboricity?+

Arboricity a(G): minimum forest decomposition. Thickness θ(G): minimum planar decomposition. a(G) ≤ 3θ(G) (each planar graph has arboricity ≤ 3). θ gives coarser decomposition.

What's the outerthickness?+

Minimum outerplanar subgraphs. Always ≥ θ. For K_n: outerthickness = ⌈n/4⌉. Each layer is outerplanar (all vertices on outer face). More restrictive than thickness.

Can thickness be computed efficiently?+

NP-hard in general (Mansfield, 1983). For specific graphs (complete, complete bipartite): exact formulas. Lower bound: θ ≥ ⌈|E|/(3n-6)⌉. Heuristics decompose into near-planar layers.

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