- FindStat

Identifier

Mp00198: Posets —incomparability graph⟶ Graphs
St000323: Graphs ⟶ ℤ

Values

([],1) => ([],1) => 0

([],2) => ([(0,1)],2) => 0

([(0,1)],2) => ([],2) => 0

([],3) => ([(0,1),(0,2),(1,2)],3) => 0

([(1,2)],3) => ([(0,2),(1,2)],3) => 0

([(0,1),(0,2)],3) => ([(1,2)],3) => 0

([(0,2),(2,1)],3) => ([],3) => 0

([(0,2),(1,2)],3) => ([(1,2)],3) => 0

([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0

([(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0

([(1,2),(1,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 0

([(0,1),(0,2),(0,3)],4) => ([(1,2),(1,3),(2,3)],4) => 0

([(0,2),(0,3),(3,1)],4) => ([(1,3),(2,3)],4) => 0

([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => 0

([(1,2),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 0

([(0,3),(3,1),(3,2)],4) => ([(2,3)],4) => 0

([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 0

([(0,3),(1,3),(3,2)],4) => ([(2,3)],4) => 0

([(0,3),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => 0

([(0,3),(1,2)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 0

([(0,3),(1,2),(1,3)],4) => ([(0,3),(1,2),(2,3)],4) => 0

([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,3),(1,2)],4) => 0

([(0,3),(2,1),(3,2)],4) => ([],4) => 0

([(0,3),(1,2),(2,3)],4) => ([(1,3),(2,3)],4) => 0

([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1

([(3,4)],5) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(2,3),(2,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(1,2),(1,3),(1,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,1),(0,2),(0,3),(0,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,2),(0,3),(0,4),(4,1)],5) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => ([(2,3),(2,4),(3,4)],5) => 0

([(1,3),(1,4),(4,2)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(4,1),(4,2)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => 0

([(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => 0

([(0,3),(0,4),(3,2),(4,1)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 0

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 0

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(1,4),(2,3)],5) => 0

([(2,3),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(1,4),(4,2),(4,3)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(4,1),(4,2),(4,3)],5) => ([(2,3),(2,4),(3,4)],5) => 0

([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(1,4),(2,4),(4,3)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(4,2),(4,3)],5) => ([(1,4),(2,3)],5) => 0

([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,4),(4,3)],5) => ([(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 0

([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,3),(4,2)],5) => ([(3,4)],5) => 0

([(0,4),(1,3),(2,3),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,3),(2,4)],5) => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,3),(3,4)],5) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(1,4),(2,3)],5) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 0

([(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,2),(1,4),(2,3)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 0

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 0

([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => ([(0,1),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => ([(1,4),(2,3)],5) => 0

([(0,4),(1,2),(1,4),(4,3)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 0

([(0,4),(1,2),(1,3)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 0

([(0,4),(1,2),(1,3),(1,4)],5) => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 0

([(0,2),(0,4),(3,1),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => 0

([(0,4),(1,2),(1,3),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 0

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(2,4),(3,4)],5) => 0

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,2),(1,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 0

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,1),(2,3),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 0

([(0,3),(1,2),(1,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => ([(0,1),(2,4),(3,4)],5) => 0

([(1,4),(3,2),(4,3)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0

([(0,3),(3,4),(4,1),(4,2)],5) => ([(3,4)],5) => 0

([(1,4),(2,3),(3,4)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,2),(2,4),(4,3)],5) => ([(2,4),(3,4)],5) => 0

([(0,3),(1,4),(4,2)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0

([(0,4),(3,2),(4,1),(4,3)],5) => ([(2,4),(3,4)],5) => 0

([(0,4),(1,2),(2,3),(2,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 0

([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 0

([(0,3),(1,2),(2,4),(3,4)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 0

([(0,4),(1,2),(2,3),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 0

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => 0

([],6) => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(4,5)],6) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(3,4),(3,5)],6) => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1

([(2,3),(2,4),(2,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1

([(1,2),(1,3),(1,4),(1,5)],6) => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1

([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => ([(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),(3,5),(4,5)],6) => ([(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),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(1,3),(1,4),(1,5),(5,2)],6) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(1,2),(1,3),(1,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 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

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.

The minimal crossing number of $G$ is then the minimal number of crossings of edges in a drawing of $G$.
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

incomparability graph

Description

The incomparability graph of a poset.