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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000383
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => 1
([],2)
=> [2] => [2] => 2
([(0,1)],2)
=> [1,1] => [1,1] => 1
([],3)
=> [3] => [3] => 3
([(1,2)],3)
=> [1,2] => [2,1] => 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
([],4)
=> [4] => [4] => 4
([(2,3)],4)
=> [1,3] => [3,1] => 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,2] => 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => 3
([],5)
=> [5] => [5] => 5
([(3,4)],5)
=> [1,4] => [4,1] => 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,1,2] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,3] => 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,1,1,2] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,1,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,1,1,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,2,1,1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
Description
The last part of an integer composition.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => [1] => 1
([],2)
=> [2] => [2] => [2] => 2
([(0,1)],2)
=> [1,1] => [1,1] => [1,1] => 1
([],3)
=> [3] => [3] => [3] => 3
([(1,2)],3)
=> [1,2] => [2,1] => [1,2] => 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => [1,1,1] => 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => [2,1] => 2
([],4)
=> [4] => [4] => [4] => 4
([(2,3)],4)
=> [1,3] => [3,1] => [1,3] => 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => [1,1,2] => 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,2,1] => [2,1,1] => 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => [1,1,1,1] => 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => [2,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,2,1] => [2,1,1] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => [1,2,1] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => [3,1] => 3
([],5)
=> [5] => [5] => [5] => 5
([(3,4)],5)
=> [1,4] => [4,1] => [1,4] => 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => [1,1,3] => 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => [2,1,2] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,3,1] => [3,1,1] => 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => [2,3] => 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => [1,1,1,2] => 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => [1,1,1,2] => 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => [2,3] => 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => [1,1,1,2] => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,2,1,1] => [2,1,1,1] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => [2,1,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => [1,2,2] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,2,1,1] => [2,1,1,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => [1,2,2] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => [1,2,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => [3,2] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => [1,2,1,1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => [1,1,2,1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [2,2,1] => 2
Description
The first part of an integer composition.
Matching statistic: St001322
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1
([],2)
=> [2] => [2] => ([],2)
=> 2
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 3
([(1,2)],3)
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2
([],4)
=> [4] => [4] => ([],4)
=> 4
([(2,3)],4)
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 3
([],5)
=> [5] => [5] => ([],5)
=> 5
([(3,4)],5)
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St001339
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001339: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001339: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1
([],2)
=> [2] => [2] => ([],2)
=> 2
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 3
([(1,2)],3)
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2
([],4)
=> [4] => [4] => ([],4)
=> 4
([(2,3)],4)
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 3
([],5)
=> [5] => [5] => ([],5)
=> 5
([(3,4)],5)
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(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)
=> ? = 2
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(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)
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(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)
=> ? = 2
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> [1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(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)
=> ? = 2
Description
The 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 irredundance number is the smallest size of a maximal irredundant set.
Matching statistic: St001363
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001363: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001363: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1
([],2)
=> [2] => [2] => ([],2)
=> 2
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 3
([(1,2)],3)
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2
([],4)
=> [4] => [4] => ([],4)
=> 4
([(2,3)],4)
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 3
([],5)
=> [5] => [5] => ([],5)
=> 5
([(3,4)],5)
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(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)
=> ? = 2
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(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)
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(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)
=> ? = 2
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> [1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(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)
=> ? = 2
Description
The Euler characteristic of a graph according to Knill.
This is ∑k≥1(−1)k−1ck,
where ck is the number of cliques with k vertices.
Matching statistic: St000273
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1
([],2)
=> [2] => [2] => ([],2)
=> 2
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 3
([(1,2)],3)
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2
([],4)
=> [4] => [4] => ([],4)
=> 4
([(2,3)],4)
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 3
([],5)
=> [5] => [5] => ([],5)
=> 5
([(3,4)],5)
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,6),(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)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(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)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
Description
The domination number of a graph.
The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
Matching statistic: St000544
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1
([],2)
=> [2] => [2] => ([],2)
=> 2
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 3
([(1,2)],3)
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2
([],4)
=> [4] => [4] => ([],4)
=> 4
([(2,3)],4)
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 3
([],5)
=> [5] => [5] => ([],5)
=> 5
([(3,4)],5)
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,6),(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)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(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)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
Description
The cop number of a graph.
This is the minimal number of cops needed to catch the robber. The algorithm is from [2].
Matching statistic: St000916
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000916: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000916: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1
([],2)
=> [2] => [2] => ([],2)
=> 2
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 3
([(1,2)],3)
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2
([],4)
=> [4] => [4] => ([],4)
=> 4
([(2,3)],4)
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 3
([],5)
=> [5] => [5] => ([],5)
=> 5
([(3,4)],5)
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,6),(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)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(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)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
Description
The packing number of a graph.
This is the size of a largest subset of vertices of a graph, such that any two distinct vertices in the subset have disjoint closed neighbourhoods, or, equivalently, have distance greater than two.
Matching statistic: St001829
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1
([],2)
=> [2] => [2] => ([],2)
=> 2
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 3
([(1,2)],3)
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2
([],4)
=> [4] => [4] => ([],4)
=> 4
([(2,3)],4)
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 3
([],5)
=> [5] => [5] => ([],5)
=> 5
([(3,4)],5)
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,6),(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)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(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)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(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
Description
The common independence number of a graph.
The common independence number of a graph G is the greatest integer r such that every vertex of G belongs to some independent set X of vertices of cardinality at least r.
Matching statistic: St000772
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> ? = 2
([(0,1)],2)
=> 1
([],3)
=> ? = 3
([(1,2)],3)
=> ? = 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> ? = 4
([(2,3)],4)
=> ? = 1
([(1,3),(2,3)],4)
=> ? = 1
([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> ? = 2
([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ? = 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([],5)
=> ? = 5
([(3,4)],5)
=> ? = 1
([(2,4),(3,4)],5)
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> ? = 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> ? = 2
([(1,4),(2,3),(3,4)],5)
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> ? = 1
([(2,3),(2,4),(3,4)],5)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([],6)
=> ? = 6
([(4,5)],6)
=> ? = 1
([(3,5),(4,5)],6)
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(2,5),(3,4)],6)
=> ? = 2
([(2,5),(3,4),(4,5)],6)
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> ? = 1
([(3,4),(3,5),(4,5)],6)
=> ? = 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ? = 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,5),(1,4),(2,3)],6)
=> ? = 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ? = 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 1.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore also statistic 1.
The graphs with statistic n−1, n−2 and n−3 have been characterised, see [1].
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000287The number of connected components of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000264The girth of a graph, which is not a tree.
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