Identifier
-
Mp00061:
Permutations
—to increasing tree⟶
Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001812: Graphs ⟶ ℤ
Values
[1] => [.,.] => ([],1) => ([(0,1)],2) => 1
[1,2] => [.,[.,.]] => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 2
[2,1] => [[.,.],.] => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 2
[1,2,3] => [.,[.,[.,.]]] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,3,2] => [.,[[.,.],.]] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[2,1,3] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[2,3,1] => [[.,[.,.]],.] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[3,1,2] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[3,2,1] => [[[.,.],.],.] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,2,4,3] => [.,[.,[[.,.],.]]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,3,2,4] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,3,4,2] => [.,[[.,[.,.]],.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,4,2,3] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,4,3,2] => [.,[[[.,.],.],.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,1,3,4] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,1,4,3] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,3,1,4] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,3,4,1] => [[.,[.,[.,.]]],.] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,4,1,3] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,4,3,1] => [[.,[[.,.],.]],.] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,1,2,4] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,1,4,2] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,2,1,4] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,2,4,1] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[3,4,1,2] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,4,2,1] => [[[.,[.,.]],.],.] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,1,2,3] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,1,3,2] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,2,1,3] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,2,3,1] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[4,3,1,2] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,3,2,1] => [[[[.,.],.],.],.] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,5,2,4,1] => [[[.,[.,.]],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,1,5,2,3] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,2,3,5,1] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,2,5,1,3] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,2,5,3,1] => [[[.,.],[[.,.],.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,3,2,5,1] => [[[[.,.],.],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,3,5,2,1] => [[[[.,.],[.,.]],.],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,5,2,3,1] => [[[.,[.,.]],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,1,3,2,4] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,1,4,2,3] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,2,3,1,4] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,2,3,4,1] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,2,4,3,1] => [[[.,.],[[.,.],.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,3,2,4,1] => [[[[.,.],.],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,3,4,2,1] => [[[[.,.],[.,.]],.],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,4,2,3,1] => [[[[.,.],.],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
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Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
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
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.
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