Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001644
(load all 2 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
Description
The dimension of a graph. The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St001494
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001494: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => ([],2)
 => ([],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => ([],3)
 => ([],3)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
Description
The Alon-Tarsi number of a graph. Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
Matching statistic: St000786
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2 = 1 + 1
[1,2,3] => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3 = 2 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St001337
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2 = 1 + 1
[1,2,3] => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3 = 2 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,2,3,4,5,6,7] => ([],7)
 => ([],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[1,2,3,4,5,7,6] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,3,4,6,5,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,3,5,4,6,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,4,3,5,6,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,4,5,6,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,4,6,5,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,4,5,6,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,4,6,5,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,5,4,6,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,4,3,5,6,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,4,3,5,7,6] => ([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
Description
The upper domination number of a graph. This is the maximum cardinality of a minimal dominating set of $G$. The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2 = 1 + 1
[1,2,3] => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3 = 2 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4 = 3 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,2,3,4,5,6,7] => ([],7)
 => ([],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[1,2,3,4,5,7,6] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,3,4,6,5,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,3,5,4,6,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,4,3,5,6,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,4,5,6,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,4,6,5,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,4,5,6,7] => ([(5,6)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,4,6,5,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,5,4,6,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,3,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,4,3,5,6,7] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,4,3,5,7,6] => ([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
Description
The upper irredundance number of a graph. A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$. The upper irredundance number is the largest size of a maximal irredundant set. The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St000272
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,3,1,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,3,1,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,4,1,7,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,4,1,7,6,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,4,1,3,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,4,1,3,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,4,1,3,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,4,3,1,6,7,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,4,3,1,7,5,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,4,3,1,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,2,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,2,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,4,2,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,4,2,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,4,2,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,5,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,6,4,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,1,6,5,7,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,6,7,4,5] => ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,6,7,5,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,4,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,5,4,6] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,5,6,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,6,4,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
Description
The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000536
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,3,1,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,3,1,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,4,1,7,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,4,1,7,6,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,4,1,3,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,4,1,3,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,4,1,3,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,4,3,1,6,7,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,4,3,1,7,5,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,4,3,1,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,2,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,2,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,2,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,1,4,2,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,4,2,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,4,2,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,5,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,6,4,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,1,6,5,7,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,6,7,4,5] => ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,6,7,5,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,4,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,5,4,6] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,5,6,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,7,6,4,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
Description
The pathwidth of a graph.
Matching statistic: St000172
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => ([],2)
 => ([],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => ([],3)
 => ([],3)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,3,1,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,3,1,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,4,1,7,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,4,1,7,6,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,4,1,3,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,4,1,3,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,4,1,3,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,4,3,1,6,7,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,4,3,1,7,5,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,4,3,1,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,2,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,2,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,4,2,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,4,2,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,4,2,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,5,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,6,4,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,2,1,6,5,7,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,6,7,4,5] => ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,6,7,5,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,4,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,5,4,6] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,5,6,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,6,4,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
Description
The Grundy number of a graph. The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St001029
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => ([],2)
 => ([],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => ([],3)
 => ([],3)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,3,1,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,3,1,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,4,1,7,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,4,1,7,6,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,4,1,3,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,4,1,3,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,4,1,3,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,4,3,1,6,7,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,4,3,1,7,5,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,4,3,1,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,2,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,2,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,4,2,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,4,2,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,4,2,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,5,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,6,4,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,2,1,6,5,7,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,6,7,4,5] => ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,6,7,5,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,4,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,5,4,6] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,5,6,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,6,4,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
Description
The size of the core of a graph. The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Matching statistic: St001108
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001108: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => ([],2)
 => ([],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => ([],3)
 => ([],3)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,3,1,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,3,1,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,1,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,4,1,7,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,3,4,1,7,6,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,4,1,3,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,4,1,3,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,4,1,3,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[2,4,3,1,6,7,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,4,3,1,7,5,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[2,4,3,1,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,2,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,2,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,5,4,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,5,6,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,2,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,1,4,2,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,4,2,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,1,4,2,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,5,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,6,4,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[3,2,1,6,5,7,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,6,7,4,5] => ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,6,7,5,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,4,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,5,4,6] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,5,6,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[3,2,1,7,6,4,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
Description
The 2-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-dynamic chromatic number of a graph.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001116The game chromatic number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001963The tree-depth of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000822The Hadwiger number of the graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.