Graph Skewness Calculator

distance from planarity

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About Graph Skewness Calculator

A graph skewness calculator computing sk(G): minimum |F| such that G\F is planar. sk=0 iff planar. sk = m - (3n-6) for dense graphs. Related to crossing number: cr(G) ≤ sk(G)·(Δ-2). Measures 'distance from planarity'. NP-hard. Client-side.

Graph Skewness Calculator Features

  • sk(G)
  • Planar=0
  • Edge removal
  • vs cr(G)
  • Common graphs
Skewness sk(G): minimum edges to delete for planarity. sk=0 ⟺ planar. Lower bound: sk ≥ m - (3n-6). Related to crossing number: cr ≤ sk·(Δ-2). Measures how 'non-planar' a graph is. NP-hard to compute.

How to Use

Select graph:

  • sk: Skewness
  • =0?: Planar?
  • vs cr: Compare

Bounds

sk ≥ m - (3n-6) (planarity bound). sk ≤ m - n + 1 (spanning tree). K_5: sk=1with only 1 edge removal. K_{3,3}: sk=1. K_n: sk = n(n-1)/2 - (3n-6) for n≥3.

Applications

VLSI layout: minimize wire crossings. Network design: make networks planar for routing. Graph drawing: simplify by removing few edges. Algorithm design: near-planar graph algorithms.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute skewness.
  3. 3Check planarity.
  4. 4Apply lower bound.
  5. 5Compare with cr.

Graph Skewness Calculator — Frequently Asked Questions

How is skewness different from crossing number?+

Skewness: delete edges to make planar. Crossing number: minimum crossings in any drawing. cr ≤ sk·(Δ-2). Both measure non-planarity but in different ways. Skewness is edge deletion; crossing number is drawing-based.

What's the lower bound?+

Planar graphs have ≤ 3n-6 edges. So sk ≥ m - (3n-6). Tight when deleted edges don't affect the constraint. For K_n: sk = n(n-1)/2 - 3n+6 = (n²-7n+12)/2.

Is skewness NP-hard?+

Yes! Even for specific k: 'is sk(G) ≤ k?' is NP-complete. But good approximations exist, and the lower bound m-(3n-6) is often tight or close.

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