- FindStat

Identifier

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

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],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) => 7

([(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) => 8

([(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) => 9

([(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) => 10

([(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) => 11

([(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) => 12

([(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) => 14

([(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) => 16

([(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) => 18

([(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) => 20

([(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) => 13

([(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) => 16

([(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) => 15

([(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) => 17

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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.

Map

incomparability graph

Description

The incomparability graph of a poset.