Values
([],1) => ([],2) => 0
([],2) => ([],3) => 0
([(0,1)],2) => ([(1,2)],3) => 0
([],3) => ([],4) => 0
([(1,2)],3) => ([(2,3)],4) => 0
([(0,2),(1,2)],3) => ([(1,3),(2,3)],4) => 0
([(0,1),(0,2),(1,2)],3) => ([(1,2),(1,3),(2,3)],4) => 0
([],4) => ([],5) => 0
([(2,3)],4) => ([(3,4)],5) => 0
([(1,3),(2,3)],4) => ([(2,4),(3,4)],5) => 0
([(0,3),(1,3),(2,3)],4) => ([(1,4),(2,4),(3,4)],5) => 0
([(0,3),(1,2)],4) => ([(1,4),(2,3)],5) => 0
([(0,3),(1,2),(2,3)],4) => ([(1,4),(2,3),(3,4)],5) => 0
([(1,2),(1,3),(2,3)],4) => ([(2,3),(2,4),(3,4)],5) => 0
([(0,3),(1,2),(1,3),(2,3)],4) => ([(1,4),(2,3),(2,4),(3,4)],5) => 0
([(0,2),(0,3),(1,2),(1,3)],4) => ([(1,3),(1,4),(2,3),(2,4)],5) => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
([],5) => ([],6) => 0
([(3,4)],5) => ([(4,5)],6) => 0
([(2,4),(3,4)],5) => ([(3,5),(4,5)],6) => 0
([(1,4),(2,4),(3,4)],5) => ([(2,5),(3,5),(4,5)],6) => 0
([(0,4),(1,4),(2,4),(3,4)],5) => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
([(1,4),(2,3)],5) => ([(2,5),(3,4)],6) => 0
([(1,4),(2,3),(3,4)],5) => ([(2,5),(3,4),(4,5)],6) => 0
([(0,1),(2,4),(3,4)],5) => ([(1,2),(3,5),(4,5)],6) => 0
([(2,3),(2,4),(3,4)],5) => ([(3,4),(3,5),(4,5)],6) => 0
([(0,4),(1,4),(2,3),(3,4)],5) => ([(1,5),(2,5),(3,4),(4,5)],6) => 0
([(1,4),(2,3),(2,4),(3,4)],5) => ([(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(1,3),(1,4),(2,3),(2,4)],5) => ([(2,4),(2,5),(3,4),(3,5)],6) => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,4),(1,3),(2,3),(2,4)],5) => ([(1,5),(2,4),(3,4),(3,5)],6) => 0
([(0,1),(2,3),(2,4),(3,4)],5) => ([(1,2),(3,4),(3,5),(4,5)],6) => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
([],6) => ([],7) => 0
([(4,5)],6) => ([(5,6)],7) => 0
([(3,5),(4,5)],6) => ([(4,6),(5,6)],7) => 0
([(2,5),(3,5),(4,5)],6) => ([(3,6),(4,6),(5,6)],7) => 0
([(1,5),(2,5),(3,5),(4,5)],6) => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
([(2,5),(3,4)],6) => ([(3,6),(4,5)],7) => 0
([(2,5),(3,4),(4,5)],6) => ([(3,6),(4,5),(5,6)],7) => 0
([(1,2),(3,5),(4,5)],6) => ([(2,3),(4,6),(5,6)],7) => 0
([(3,4),(3,5),(4,5)],6) => ([(4,5),(4,6),(5,6)],7) => 0
([(1,5),(2,5),(3,4),(4,5)],6) => ([(2,6),(3,6),(4,5),(5,6)],7) => 0
([(0,1),(2,5),(3,5),(4,5)],6) => ([(1,2),(3,6),(4,6),(5,6)],7) => 0
([(2,5),(3,4),(3,5),(4,5)],6) => ([(3,6),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(2,4),(2,5),(3,4),(3,5)],6) => ([(3,5),(3,6),(4,5),(4,6)],7) => 0
([(0,5),(1,5),(2,4),(3,4)],6) => ([(1,6),(2,6),(3,5),(4,5)],7) => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,4),(2,3)],6) => ([(1,6),(2,5),(3,4)],7) => 0
([(1,5),(2,4),(3,4),(3,5)],6) => ([(2,6),(3,5),(4,5),(4,6)],7) => 0
([(0,1),(2,5),(3,4),(4,5)],6) => ([(1,2),(3,6),(4,5),(5,6)],7) => 0
([(1,2),(3,4),(3,5),(4,5)],6) => ([(2,3),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7) => 0
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Description
The minimal crossing number of a graph.
A drawing of a graph $G$ is a drawing in $\mathbb{R}^2$ such that
In particular, a graph is planar if and only if its minimal crossing number is $0$.
It is moreover conjectured that the crossing number of the complete graph $K_n$ [1] is
$$\frac{1}{4}\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{n-2}{2} \rfloor\lfloor \frac{n-3}{2} \rfloor,$$
and the crossing number of the complete bipartite graph $K_{n,m}$ [2] is
$$\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{m}{2} \rfloor\lfloor \frac{m-1}{2} \rfloor.$$
A general algorithm to compute the crossing number is e.g. given in [3].
This statistics data was provided by Markus Chimani [6].
A drawing of a graph $G$ is a drawing in $\mathbb{R}^2$ such that
- the vertices of $G$ are distinct points,
- the edges of $G$ are simple curves joining their endpoints,
- no edge passes through a vertex, and
- no three edges cross in a common point.
In particular, a graph is planar if and only if its minimal crossing number is $0$.
It is moreover conjectured that the crossing number of the complete graph $K_n$ [1] is
$$\frac{1}{4}\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{n-2}{2} \rfloor\lfloor \frac{n-3}{2} \rfloor,$$
and the crossing number of the complete bipartite graph $K_{n,m}$ [2] is
$$\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{m}{2} \rfloor\lfloor \frac{m-1}{2} \rfloor.$$
A general algorithm to compute the crossing number is e.g. given in [3].
This statistics data was provided by Markus Chimani [6].
Map
vertex addition
Description
Adds a disconnected vertex to a graph.
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