Identifier
Values
[1] => ([],1) => ([],1) => 1
[1,1] => ([(0,1)],2) => ([],1) => 1
[2] => ([],2) => ([],2) => 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([],1) => 1
[1,2] => ([(1,2)],3) => ([],2) => 1
[2,1] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 3
[3] => ([],3) => ([],3) => 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],1) => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([],2) => 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(1,2)],3) => 3
[1,3] => ([(2,3)],4) => ([],3) => 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],1) => 1
[2,2] => ([(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => 3
[3,1] => ([(0,3),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 3
[4] => ([],4) => ([],4) => 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],1) => 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],2) => 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 3
[1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([],3) => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],1) => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,3),(2,3)],4) => 3
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(1,3),(2,3)],4) => 3
[1,4] => ([(3,4)],5) => ([],4) => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],1) => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],2) => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 3
[2,3] => ([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 3
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],1) => 1
[3,2] => ([(1,4),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[5] => ([],5) => ([],5) => 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],1) => 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 3
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],3) => 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],1) => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 3
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 3
[1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([],4) => 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],1) => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 3
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => ([(2,4),(3,4)],5) => 3
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],1) => 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,4),(3,4)],5) => 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[1,5] => ([(4,5)],6) => ([],5) => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],1) => 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],3) => 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],1) => 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 3
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 3
[2,4] => ([(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => 3
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],1) => 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 3
[3,3] => ([(2,5),(3,5),(4,5)],6) => ([(2,5),(3,5),(4,5)],6) => 3
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],1) => 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 3
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[6] => ([],6) => ([],6) => 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],2) => 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(1,2)],3) => 3
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],3) => 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,3),(2,3)],4) => 3
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,3),(1,3),(2,3)],4) => 3
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],4) => 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],2) => 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(1,2)],3) => 3
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,4),(3,4)],5) => 3
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(2,4),(3,4)],5) => 3
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([],5) => 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],2) => 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(1,2)],3) => 3
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],3) => 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,3),(2,3)],4) => 3
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,3),(1,3),(2,3)],4) => 3
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => ([(3,5),(4,5)],6) => 3
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],2) => 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(1,2)],3) => 3
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,5),(3,5),(4,5)],6) => 3
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,6] => ([(5,6)],7) => ([],6) => 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],2) => 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(1,2)],3) => 3
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],3) => 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],1) => 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,3),(2,3)],4) => 3
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Description
The detour number of a graph.
This is the number of vertices in a longest induced path in a graph.
Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.
This is the number of vertices in a longest induced path in a graph.
Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.
Map
block-cut tree
Description
Sends a graph to its block-cut tree.
The block-cut tree has a vertex for each block and for each cut-vertex of the given graph, and there is an edge for each pair of block and cut-vertex that belongs to that block. A block is a maximal biconnected (or 2-vertex connected) subgraph. A cut-vertex is a vertex whose removal increases the number of connected components.
The block-cut tree has a vertex for each block and for each cut-vertex of the given graph, and there is an edge for each pair of block and cut-vertex that belongs to that block. A block is a maximal biconnected (or 2-vertex connected) subgraph. A cut-vertex is a vertex whose removal increases the number of connected components.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
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