Identifier
Values
([],1) => ([],1) => ([(0,1)],2) => 2
([],2) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,1)],2) => ([],1) => ([(0,1)],2) => 2
([],3) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,2)],3) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,2),(1,2)],3) => ([],1) => ([(0,1)],2) => 2
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([],4) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
([(2,3)],4) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,3),(2,3)],4) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,3),(1,3),(2,3)],4) => ([],1) => ([(0,1)],2) => 2
([(0,3),(1,2)],4) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,3),(1,2),(2,3)],4) => ([],1) => ([(0,1)],2) => 2
([(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(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) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(3,4)],5) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
([(2,4),(3,4)],5) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,4),(2,4),(3,4)],5) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,4),(1,4),(2,4),(3,4)],5) => ([],1) => ([(0,1)],2) => 2
([(1,4),(2,3)],5) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,4),(2,3),(3,4)],5) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,1),(2,4),(3,4)],5) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,4),(1,4),(2,3),(3,4)],5) => ([],1) => ([(0,1)],2) => 2
([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,4),(1,3),(2,3),(2,4)],5) => ([],1) => ([(0,1)],2) => 2
([(0,1),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 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) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(3,5),(4,5)],6) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
([(2,5),(3,5),(4,5)],6) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,5),(2,5),(3,5),(4,5)],6) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([],1) => ([(0,1)],2) => 2
([(2,5),(3,4)],6) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
([(2,5),(3,4),(4,5)],6) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,2),(3,5),(4,5)],6) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,5),(2,5),(3,4),(4,5)],6) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,1),(2,5),(3,5),(4,5)],6) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([],1) => ([(0,1)],2) => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(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),(1,2),(1,3),(2,3)],4) => 2
([(0,5),(1,5),(2,4),(3,4)],6) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([],1) => ([(0,1)],2) => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([],1) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 3
([(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),(1,2),(1,3),(2,3)],4) => 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,5),(1,4),(2,3)],6) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,5),(2,4),(3,4),(3,5)],6) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,1),(2,5),(3,4),(4,5)],6) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([],1) => ([(0,1)],2) => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(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),(1,2),(1,3),(2,3)],4) => 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 3
([(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),(1,2),(1,3),(2,3)],4) => 2
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([],1) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(3,6),(4,6),(5,6)],7) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
([(2,6),(3,6),(4,6),(5,6)],7) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([],1) => ([(0,1)],2) => 2
([(3,6),(4,5),(5,6)],7) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
([(2,3),(4,6),(5,6)],7) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
([(2,6),(3,6),(4,5),(5,6)],7) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,2),(3,6),(4,6),(5,6)],7) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([],1) => ([(0,1)],2) => 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([(1,6),(2,6),(3,5),(4,5)],7) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 3
([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => ([],2) => ([(0,2),(1,2)],3) => 3
([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([],2) => ([(0,2),(1,2)],3) => 3
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([],1) => ([(0,1)],2) => 2
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([],1) => ([(0,1)],2) => 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 3
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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
cone
Description
The cone of a graph.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.
Map
delete endpoints
Description
Sends a graph to a maximal subgraph with no endpoints.
An endpoint of a graph is a vertex of degree one. Given an arbitrary graph, this map repeatedly searches for an endpoint and deletes it, until no endpoint remains. The result does not depend on the order of endpoints chosen, up to isomorphism. The map preserves the number of connected components. For a connected graph with at least one cycle, this map returns the 2-core.
An endpoint of a graph is a vertex of degree one. Given an arbitrary graph, this map repeatedly searches for an endpoint and deletes it, until no endpoint remains. The result does not depend on the order of endpoints chosen, up to isomorphism. The map preserves the number of connected components. For a connected graph with at least one cycle, this map returns the 2-core.
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