- FindStat

Identifier

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000300: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 16

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],5) => 32

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(2,3),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(3,4)],5) => 16

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

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

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

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

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

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

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

([(0,4),(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) => 17

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

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

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

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

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

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

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

([(0,4),(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) => 13

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

([(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 16

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],6) => 64

([(4,5)],6) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(4,5)],6) => 48

([(3,4),(3,5)],6) => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => 40

([(2,3),(2,4),(2,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,5),(3,5),(4,5)],6) => 36

([(1,2),(1,3),(1,4),(1,5)],6) => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 34

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

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

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

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

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

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

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

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

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

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$F_{1} = q^{2}$

$F_{2} = q^{3} + q^{4}$

$F_{3} = q^{4} + 2\ q^{5} + q^{6} + q^{8}$

$F_{4} = q^{5} + 3\ q^{6} + 3\ q^{7} + 2\ q^{8} + 3\ q^{9} + 2\ q^{10} + q^{12} + q^{16}$

$F_{5} = q^{6} + 4\ q^{7} + 6\ q^{8} + 6\ q^{9} + 9\ q^{10} + 8\ q^{11} + 7\ q^{12} + 4\ q^{13} + 5\ q^{14} + 2\ q^{15} + 2\ q^{16} + 2\ q^{17} + 3\ q^{18} + 2\ q^{20} + q^{24} + q^{32}$

$F_{6} = q^{7} + 5\ q^{8} + 10\ q^{9} + 13\ q^{10} + 20\ q^{11} + 26\ q^{12} + 26\ q^{13} + 28\ q^{14} + 23\ q^{15} + 26\ q^{16} + 17\ q^{17} + 23\ q^{18} + 14\ q^{19} + 17\ q^{20} + 10\ q^{21} + 14\ q^{22} + 2\ q^{23} + 9\ q^{24} + 5\ q^{25} + 6\ q^{26} + 3\ q^{27} + 5\ q^{28} + 2\ q^{30} + 2\ q^{32} + 2\ q^{33} + 2\ q^{34} + 3\ q^{36} + 2\ q^{40} + q^{48} + q^{64}$

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.

Map

complement

Description

The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.

Map

incomparability graph

Description

The incomparability graph of a poset.