- FindStat

Identifier

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001746: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],5) => ([],1) => ([],1) => ([],1) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

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

$F_{6} = q + 3\ q^{2} + 9\ q^{3} + 22\ q^{4} + 43\ q^{5} + 78\ q^{6}$

$F_{7} = q + 3\ q^{2} + 13\ q^{3} + 37\ q^{4} + 117\ q^{5} + 285\ q^{6} + 588\ q^{7}$

Description

The coalition number of a graph.
This is the maximal cardinality of a set partition such that each block is either a dominating set of cardinality one, or is not a dominating set but can be joined with a second block to form a dominating set.

Map

weak duplicate order

Description

The weak duplicate order of the de-duplicate of a graph.
Let

$G=(V, E)$ be a graph and let

$N=\{ N_v | v\in V\}$ be the set of (distinct) neighbourhoods of

$G$ .
This map yields the poset obtained by ordering

$N$ by reverse inclusion.

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

to graph

Description

Returns the Hasse diagram of the poset as an undirected graph.