Identifier
Values
([],1) => ([],1) => ([],1) => 1
([],2) => ([],1) => ([],1) => 1
([(0,1)],2) => ([(0,1)],2) => ([(0,1)],2) => 2
([],3) => ([],1) => ([],1) => 1
([(1,2)],3) => ([(1,2)],3) => ([(0,1)],2) => 2
([(0,2),(1,2)],3) => ([(0,1)],2) => ([(0,1)],2) => 2
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([],4) => ([],1) => ([],1) => 1
([(2,3)],4) => ([(1,2)],3) => ([(0,1)],2) => 2
([(1,3),(2,3)],4) => ([(1,2)],3) => ([(0,1)],2) => 2
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => ([(0,1)],2) => 2
([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => ([(0,1)],2) => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([],5) => ([],1) => ([],1) => 1
([(3,4)],5) => ([(1,2)],3) => ([(0,1)],2) => 2
([(2,4),(3,4)],5) => ([(1,2)],3) => ([(0,1)],2) => 2
([(1,4),(2,4),(3,4)],5) => ([(1,2)],3) => ([(0,1)],2) => 2
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 2
([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,1),(2,4),(3,4)],5) => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,3),(1,4),(2,3),(2,4)],5) => ([(1,2)],3) => ([(0,1)],2) => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([(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) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([],6) => ([],1) => ([],1) => 1
([(4,5)],6) => ([(1,2)],3) => ([(0,1)],2) => 2
([(3,5),(4,5)],6) => ([(1,2)],3) => ([(0,1)],2) => 2
([(2,5),(3,5),(4,5)],6) => ([(1,2)],3) => ([(0,1)],2) => 2
([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,2)],3) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 2
([(2,5),(3,4)],6) => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,2),(3,5),(4,5)],6) => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(2,4),(2,5),(3,4),(3,5)],6) => ([(1,2)],3) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,2)],3) => ([(0,1)],2) => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(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) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,5),(1,4),(2,3)],6) => ([(0,5),(1,4),(2,3)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([],7) => ([],1) => ([],1) => 1
([(5,6)],7) => ([(1,2)],3) => ([(0,1)],2) => 2
([(4,6),(5,6)],7) => ([(1,2)],3) => ([(0,1)],2) => 2
([(3,6),(4,6),(5,6)],7) => ([(1,2)],3) => ([(0,1)],2) => 2
([(2,6),(3,6),(4,6),(5,6)],7) => ([(1,2)],3) => ([(0,1)],2) => 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(1,2)],3) => ([(0,1)],2) => 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1)],2) => ([(0,1)],2) => 2
([(3,6),(4,5)],7) => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(3,6),(4,5),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(2,3),(4,6),(5,6)],7) => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(2,6),(3,6),(4,5),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,2),(3,6),(4,6),(5,6)],7) => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2)],3) => ([(0,1)],2) => 2
([(1,6),(2,6),(3,5),(4,5)],7) => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2)],3) => ([(0,1)],2) => 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2)],3) => ([(0,1)],2) => 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,1)],2) => ([(0,1)],2) => 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
([(1,6),(2,5),(3,4)],7) => ([(1,6),(2,5),(3,4)],7) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(0,3),(1,2),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
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Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Map
connected vertex partitions
Description
Sends a graph to the lattice of its connected vertex partitions.
A connected vertex partition of a graph $G = (V,E)$ is a set partition of $V$ such that each part induced a connected subgraph of $G$. The connected vertex partitions of $G$ form a lattice under refinement. If $G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on $n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.
A connected vertex partition of a graph $G = (V,E)$ is a set partition of $V$ such that each part induced a connected subgraph of $G$. The connected vertex partitions of $G$ form a lattice under refinement. If $G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on $n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.
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.
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