- FindStat

Identifier

Mp00250: Graphs —clique graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001325: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],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) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => ([(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) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(0,4),(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),(5,6)],7) => ([],7) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph.
A graph is a comparability graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges and $(a,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

Map

Ore closure

Description

The Ore closure of a graph.
The Ore closure of a connected graph $G$ has the same vertices as $G$, and the smallest set of edges containing the edges of $G$ such that for any two vertices $u$ and $v$ whose sum of degrees is at least the number of vertices, then $(u,v)$ is also an edge.
For disconnected graphs, we compute the closure separately for each component.

Map

clique graph

Description

The clique graph of a graph.
The clique graph of a graph $G$ has as vertex set the set of maximal cliques $G$ and an edge between vertices corresponding to cliques that intersect.
In other words, it is the intersection graph of the maximal cliques of $G$.

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.