Identifier
Values
([],1) => ([],1) => ([],1) => 0
([],2) => ([],1) => ([],1) => 0
([(0,1)],2) => ([(0,1)],2) => ([(0,1)],2) => 0
([],3) => ([],1) => ([],1) => 0
([(1,2)],3) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,2),(1,2)],3) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([],4) => ([],1) => ([],1) => 0
([(2,3)],4) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,3),(2,3)],4) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,3),(1,2)],4) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(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) => 0
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([],5) => ([],1) => ([],1) => 0
([(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,4),(2,3)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(0,4),(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) => 0
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(0,4),(1,3),(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) => 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),(0,3),(1,4),(2,4),(3,4)],5) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 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),(0,3),(1,4),(2,4),(3,4)],5) => 0
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,1),(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) => 0
([(0,3),(1,2),(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) => 0
([(0,3),(0,4),(1,2),(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) => 0
([(0,1),(0,4),(1,3),(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) => 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),(0,3),(1,4),(2,4),(3,4)],5) => 0
([(0,4),(1,2),(1,3),(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) => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 0
([(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) => 0
([],6) => ([],1) => ([],1) => 0
([(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(2,5),(3,4)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,5),(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) => 0
([(0,5),(1,5),(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) => 0
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(1,5),(2,4),(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) => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(0,5),(1,5),(2,4),(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) => 0
([(0,5),(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) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(0,5),(1,4),(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) => 0
([(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) => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(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) => 0
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,2),(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) => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,4),(2,3),(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) => 0
([(0,1),(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) => 0
([(0,4),(1,5),(2,3),(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) => 0
([(1,4),(1,5),(2,3),(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) => 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),(0,3),(1,4),(2,4),(3,4)],5) => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(1,2),(1,5),(2,4),(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) => 0
([(1,5),(2,3),(2,4),(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) => 0
([(0,5),(1,4),(2,3),(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) => 0
([(0,5),(1,2),(1,5),(2,4),(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) => 0
([(0,5),(1,3),(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) => 0
([(1,4),(1,5),(2,3),(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) => 0
([(0,5),(1,4),(1,5),(2,3),(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) => 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 0
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Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
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
core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
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