- FindStat

Identifier

Mp00247: Graphs —de-duplicate⟶ Graphs
St000270: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 38

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 28

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

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

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

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

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

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 28

([(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) => 49

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 31

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 52

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 82

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

([(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) => 38

([(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) => 76

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 128

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

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

([(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),(1,2),(1,3),(2,3)],4) => 38

([(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) => 291

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

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

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

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

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

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

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

([(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => 8

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

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

([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => 8

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

([(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => 14

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 8

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => 14

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 14

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

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

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => 8

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

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

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 28

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 8

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 8

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => 14

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 28

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 14

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

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => 8

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

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 28

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 14

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

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

([(0,5),(1,4),(2,3)],6) => ([(0,5),(1,4),(2,3)],6) => 8

([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 16

([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1),(2,5),(3,4),(4,5)],6) => 16

([(1,2),(3,4),(3,5),(4,5)],6) => ([(1,2),(3,4),(3,5),(4,5)],6) => 14

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 28

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 28

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 56

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 49

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 98

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 31

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 60

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 52

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 62

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Description

The number of forests contained in a graph.
That is, for a graph $G = (V,E)$ with vertices $V$ and edges $E$, the number of subsets $E' \subseteq E$ for which the subgraph $(V,E')$ is acyclic.
If $T_G(x,y)$ is the Tutte polynomial [2] of $G$, then the number of forests contained in $G$ is given by $T_G(2,1)$.

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.