Graph Coarseness Calculator

edge-disjoint non-planar

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About Graph Coarseness Calculator

A graph coarseness calculator computing ξ(G): maximum number of edge-disjoint non-planar subgraphs of G. Dual of skewness. ξ=0 iff planar. For K_n: ξ = ⌊(n+1)(n-3)/12⌋. Measures how 'densely non-planar' G is. Client-side.

Graph Coarseness Calculator Features

  • ξ(G)
  • Planar=0
  • Disjoint
  • vs skewness
  • Common graphs
Coarseness ξ(G): maximum edge-disjoint non-planar subgraphs. ξ=0 ⟺ planar. Dual viewpoint to skewness (which minimizes edge deletions). K_n: ξ = ⌊(n+1)(n-3)/12⌋. Measures density of non-planarity.

How to Use

Select graph:

  • ξ: Coarseness
  • =0?: Planar?
  • Disjoint: Edge-disjoint

Complete Graphs

K_n: ξ = ⌊(n+1)(n-3)/12⌋. Beautiful closed form! Each non-planar subgraph needs ≥9 edges (K_{3,3} or ≥10 for K_5). Elegant number theory connections.

Dual of Skewness

Skewness: minimum edges to remove for planarity. Coarseness: maximum non-planar 'packing'. Related: sk(G) ≥ 9·ξ(G) - 3 approximately. Both measure non-planarity from opposite perspectives.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute ξ.
  3. 3Pack non-planar.
  4. 4Compare with sk.
  5. 5Apply formula.

Graph Coarseness Calculator — Frequently Asked Questions

What's the relationship to skewness?+

Dual measures! Skewness = minimum edges to remove (make planar). Coarseness = maximum non-planar subgraphs to pack (edge-disjoint). Skewness is a covering problem; coarseness is a packing problem.

Why edge-disjoint?+

Edge-disjoint non-planar subgraphs: no edge appears in two subgraphs. Measures how many 'independent' non-planar structures exist. More subgraphs = more fundamentally non-planar.

What about K_5 and K_{3,3}?+

K_5: has 10 edges, ξ(K_5) = 1 (one non-planar subgraph). K_{3,3}: 9 edges, ξ = 1. K_6: 15 edges, ξ = 1. K_7: 21 edges, ξ = 2 (two edge-disjoint K_{3,3} copies!).

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