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Identifier

St000300: Graphs ⟶ ℤ

Values

([],0) => 1

([],1) => 2

([],2) => 4

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

([],3) => 8

([(1,2)],3) => 6

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

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

([],4) => 16

([(2,3)],4) => 12

([(1,3),(2,3)],4) => 10

([(0,3),(1,3),(2,3)],4) => 9

([(0,3),(1,2)],4) => 9

([(0,3),(1,2),(2,3)],4) => 8

([(1,2),(1,3),(2,3)],4) => 8

([(0,3),(1,2),(1,3),(2,3)],4) => 7

([(0,2),(0,3),(1,2),(1,3)],4) => 7

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

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

([],5) => 32

([(3,4)],5) => 24

([(2,4),(3,4)],5) => 20

([(1,4),(2,4),(3,4)],5) => 18

([(0,4),(1,4),(2,4),(3,4)],5) => 17

([(1,4),(2,3)],5) => 18

([(1,4),(2,3),(3,4)],5) => 16

([(0,1),(2,4),(3,4)],5) => 15

([(2,3),(2,4),(3,4)],5) => 16

([(0,4),(1,4),(2,3),(3,4)],5) => 14

([(1,4),(2,3),(2,4),(3,4)],5) => 14

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 13

([(1,3),(1,4),(2,3),(2,4)],5) => 14

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 12

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 12

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 11

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 11

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 10

([(0,4),(1,3),(2,3),(2,4)],5) => 13

([(0,1),(2,3),(2,4),(3,4)],5) => 12

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 11

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 10

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 11

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 10

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 9

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 10

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 10

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 9

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 9

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 8

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7

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

([],6) => 64

([(4,5)],6) => 48

([(3,5),(4,5)],6) => 40

([(2,5),(3,5),(4,5)],6) => 36

([(1,5),(2,5),(3,5),(4,5)],6) => 34

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 33

([(2,5),(3,4)],6) => 36

([(2,5),(3,4),(4,5)],6) => 32

([(1,2),(3,5),(4,5)],6) => 30

([(3,4),(3,5),(4,5)],6) => 32

([(1,5),(2,5),(3,4),(4,5)],6) => 28

([(0,1),(2,5),(3,5),(4,5)],6) => 27

([(2,5),(3,4),(3,5),(4,5)],6) => 28

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 26

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 26

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 25

([(2,4),(2,5),(3,4),(3,5)],6) => 28

([(0,5),(1,5),(2,4),(3,4)],6) => 25

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 24

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 23

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 24

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 24

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 22

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 22

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 22

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 21

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 22

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 21

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 20

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 20

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 20

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 19

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 19

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18

([(0,5),(1,4),(2,3)],6) => 27

([(1,5),(2,4),(3,4),(3,5)],6) => 26

([(0,1),(2,5),(3,4),(4,5)],6) => 24

([(1,2),(3,4),(3,5),(4,5)],6) => 24

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 22

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 22

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 21

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 20

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 20

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 19

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 22

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 20

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 20

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Description

The number of independent sets of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is St000093The cardinality of a maximal independent set of vertices of a graph.

References

[1] wikipedia:Vertex_cover

Code

from sage.graphs.independent_sets import IndependentSets
def statistic(G):
    return IndependentSets(G).cardinality()

Created

Nov 26, 2015 at 12:23 by Christian Stump

Updated

Mar 31, 2021 at 11:11 by Martin Rubey