Identifier
Values
([(0,3),(1,2),(2,3)],4) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,4),(2,3),(3,4)],5) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(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) => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(2,5),(3,4),(4,5)],6) => ([(2,6),(3,5),(4,5),(4,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(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) => 2
([(0,1),(2,5),(3,4),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,3),(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) => 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,4),(1,2),(1,5),(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) => 2
([(0,5),(1,4),(2,3),(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) => 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,2),(1,4),(1,5),(2,3),(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) => 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,1),(1,5),(2,3),(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) => 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([(2,6),(3,5),(4,5),(4,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,2),(3,6),(4,5),(5,6)],7) => ([(2,6),(3,5),(4,5),(4,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => ([(2,6),(3,5),(4,5),(4,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,6),(2,3),(2,4),(3,5),(4,5),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,5),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
([(0,5),(1,2),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,4),(1,2),(1,6),(2,5),(3,5),(3,6),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,3),(1,5),(1,6),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
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Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Map
de-duplicate
Description
The de-duplicate of a graph.
Let $G = (V, E)$ be a graph. This map yields the graph whose vertex set is the set of (distinct) neighbourhoods $\{N_v | v \in V\}$ of $G$, and has an edge $(N_a, N_b)$ between two vertices if and only if $(a, b)$ is an edge of $G$. This is well-defined, because if $N_a = N_c$ and $N_b = N_d$, then $(a, b)\in E$ if and only if $(c, d)\in E$.
The image of this map is the set of so-called 'mating graphs' or 'point-determining graphs'.
This map preserves the chromatic number.
Let $G = (V, E)$ be a graph. This map yields the graph whose vertex set is the set of (distinct) neighbourhoods $\{N_v | v \in V\}$ of $G$, and has an edge $(N_a, N_b)$ between two vertices if and only if $(a, b)$ is an edge of $G$. This is well-defined, because if $N_a = N_c$ and $N_b = N_d$, then $(a, b)\in E$ if and only if $(c, d)\in E$.
The image of this map is the set of so-called 'mating graphs' or 'point-determining graphs'.
This map preserves the chromatic number.
Map
block-cut tree
Description
Sends a graph to its block-cut tree.
The block-cut tree has a vertex for each block and for each cut-vertex of the given graph, and there is an edge for each pair of block and cut-vertex that belongs to that block. A block is a maximal biconnected (or 2-vertex connected) subgraph. A cut-vertex is a vertex whose removal increases the number of connected components.
The block-cut tree has a vertex for each block and for each cut-vertex of the given graph, and there is an edge for each pair of block and cut-vertex that belongs to that block. A block is a maximal biconnected (or 2-vertex connected) subgraph. A cut-vertex is a vertex whose removal increases the number of connected components.
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