Values
([],1) => ([],1) => 2
([],2) => ([],1) => 2
([(0,1)],2) => ([(0,1)],2) => 3
([],3) => ([],1) => 2
([(1,2)],3) => ([(0,1)],2) => 3
([(0,2),(1,2)],3) => ([(0,1)],2) => 3
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 4
([],4) => ([],1) => 2
([(2,3)],4) => ([(0,1)],2) => 3
([(1,3),(2,3)],4) => ([(0,1)],2) => 3
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => 3
([(0,3),(1,2)],4) => ([(0,1)],2) => 3
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => 3
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 4
([(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) => 5
([],5) => ([],1) => 2
([(3,4)],5) => ([(0,1)],2) => 3
([(2,4),(3,4)],5) => ([(0,1)],2) => 3
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 3
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 3
([(1,4),(2,3)],5) => ([(0,1)],2) => 3
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => 3
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => 3
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => 3
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => 3
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 11
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 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) => 5
([(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) => 5
([(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) => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 4
([(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) => 5
([(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) => 6
([],6) => ([],1) => 2
([(4,5)],6) => ([(0,1)],2) => 3
([(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(2,5),(3,4)],6) => ([(0,1)],2) => 3
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 3
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 3
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 3
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 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) => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 3
([(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) => 4
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => 3
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => 3
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 3
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 11
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 4
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 11
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Description
The number of independent sets of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is St000093The cardinality of a maximal independent set of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is St000093The cardinality of a maximal independent set of vertices of a graph.
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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