Values
([],1) => ([],1) => ([],0) => 1
([],2) => ([],1) => ([],0) => 1
([(0,1)],2) => ([(0,1)],2) => ([],1) => 2
([],3) => ([],1) => ([],0) => 1
([(1,2)],3) => ([(0,1)],2) => ([],1) => 2
([(0,2),(1,2)],3) => ([(0,1)],2) => ([],1) => 2
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([],4) => ([],1) => ([],0) => 1
([(2,3)],4) => ([(0,1)],2) => ([],1) => 2
([(1,3),(2,3)],4) => ([(0,1)],2) => ([],1) => 2
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => ([],1) => 2
([(0,3),(1,2)],4) => ([(0,1)],2) => ([],1) => 2
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => ([],1) => 2
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => ([],1) => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([],5) => ([],1) => ([],0) => 1
([(3,4)],5) => ([(0,1)],2) => ([],1) => 2
([(2,4),(3,4)],5) => ([(0,1)],2) => ([],1) => 2
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([],1) => 2
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([],1) => 2
([(1,4),(2,3)],5) => ([(0,1)],2) => ([],1) => 2
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([],1) => 2
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => ([],1) => 2
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([],1) => 2
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([],1) => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => ([],1) => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([],1) => 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),(1,2)],3) => 2
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => ([],1) => 2
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(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),(1,2)],3) => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(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),(1,2)],3) => 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),(1,2)],3) => 2
([],6) => ([],1) => ([],0) => 1
([(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(2,5),(3,4)],6) => ([(0,1)],2) => ([],1) => 2
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],1) => 2
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => ([],1) => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(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),(1,2)],3) => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],1) => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],1) => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],1) => 2
([(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),(1,2)],3) => 2
([(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),(1,2)],3) => 2
([(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),(1,2)],3) => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],1) => 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),(1,2)],3) => 2
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => ([],1) => 2
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => ([],1) => 2
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(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),(1,2)],3) => 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([(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),(1,2)],3) => 2
([(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),(1,2)],3) => 2
([(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),(1,2)],3) => 2
([(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),(1,2)],3) => 2
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Description
The minimal order of a graph which is not an induced subgraph of the given graph.
For example, the graph with two isolated vertices is not an induced subgraph of the complete graph on three vertices.
By contrast, the minimal number of vertices of a graph which is not a subgraph of a graph is one plus the clique number St000097The order of the largest clique of the graph..
For example, the graph with two isolated vertices is not an induced subgraph of the complete graph on three vertices.
By contrast, the minimal number of vertices of a graph which is not a subgraph of a graph is one plus the clique number St000097The order of the largest clique of the graph..
Map
line graph
Description
The line graph of a graph.
Let $G$ be a graph with edge set $E$. Then its line graph is the graph with vertex set $E$, such that two vertices $e$ and $f$ are adjacent if and only if they are incident to a common vertex in $G$.
Let $G$ be a graph with edge set $E$. Then its line graph is the graph with vertex set $E$, such that two vertices $e$ and $f$ are adjacent if and only if they are incident to a common vertex in $G$.
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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