Identifier
Values
[1] => ([],1) => ([],2) => 0
[1,2] => ([],2) => ([],3) => 0
[2,1] => ([(0,1)],2) => ([(1,2)],3) => 1
[1,2,3] => ([],3) => ([],4) => 0
[1,3,2] => ([(1,2)],3) => ([(2,3)],4) => 1
[2,1,3] => ([(1,2)],3) => ([(2,3)],4) => 1
[2,3,1] => ([(0,2),(1,2)],3) => ([(1,3),(2,3)],4) => 1
[3,1,2] => ([(0,2),(1,2)],3) => ([(1,3),(2,3)],4) => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3) => ([(1,2),(1,3),(2,3)],4) => 2
[1,2,3,4] => ([],4) => ([],5) => 0
[1,2,4,3] => ([(2,3)],4) => ([(3,4)],5) => 1
[1,3,2,4] => ([(2,3)],4) => ([(3,4)],5) => 1
[1,3,4,2] => ([(1,3),(2,3)],4) => ([(2,4),(3,4)],5) => 1
[1,4,2,3] => ([(1,3),(2,3)],4) => ([(2,4),(3,4)],5) => 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => ([(2,3),(2,4),(3,4)],5) => 2
[2,1,3,4] => ([(2,3)],4) => ([(3,4)],5) => 1
[2,1,4,3] => ([(0,3),(1,2)],4) => ([(1,4),(2,3)],5) => 2
[2,3,1,4] => ([(1,3),(2,3)],4) => ([(2,4),(3,4)],5) => 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => ([(1,4),(2,4),(3,4)],5) => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => ([(1,4),(2,3),(3,4)],5) => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[3,1,2,4] => ([(1,3),(2,3)],4) => ([(2,4),(3,4)],5) => 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4) => ([(1,4),(2,3),(3,4)],5) => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => ([(2,3),(2,4),(3,4)],5) => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(1,3),(1,4),(2,3),(2,4)],5) => 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => ([(1,4),(2,4),(3,4)],5) => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,2,3,4,5] => ([],5) => ([],6) => 0
[1,2,3,5,4] => ([(3,4)],5) => ([(4,5)],6) => 1
[1,2,4,3,5] => ([(3,4)],5) => ([(4,5)],6) => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5) => ([(3,5),(4,5)],6) => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5) => ([(3,5),(4,5)],6) => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5) => ([(3,4),(3,5),(4,5)],6) => 2
[1,3,2,4,5] => ([(3,4)],5) => ([(4,5)],6) => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(2,5),(3,4)],6) => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5) => ([(3,5),(4,5)],6) => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => ([(2,5),(3,5),(4,5)],6) => 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => ([(2,5),(3,4),(4,5)],6) => 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,4,2,3,5] => ([(2,4),(3,4)],5) => ([(3,5),(4,5)],6) => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5) => ([(2,5),(3,4),(4,5)],6) => 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5) => ([(3,4),(3,5),(4,5)],6) => 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(2,4),(2,5),(3,4),(3,5)],6) => 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5) => ([(2,5),(3,5),(4,5)],6) => 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,1,3,4,5] => ([(3,4)],5) => ([(4,5)],6) => 1
[2,1,3,5,4] => ([(1,4),(2,3)],5) => ([(2,5),(3,4)],6) => 2
[2,1,4,3,5] => ([(1,4),(2,3)],5) => ([(2,5),(3,4)],6) => 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5) => ([(1,2),(3,5),(4,5)],6) => 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5) => ([(1,2),(3,5),(4,5)],6) => 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5) => ([(1,2),(3,4),(3,5),(4,5)],6) => 3
[2,3,1,4,5] => ([(2,4),(3,4)],5) => ([(3,5),(4,5)],6) => 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5) => ([(1,2),(3,5),(4,5)],6) => 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5) => ([(2,5),(3,5),(4,5)],6) => 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(1,5),(2,5),(3,5),(4,5)],6) => 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(1,5),(2,5),(3,4),(4,5)],6) => 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5) => ([(2,5),(3,4),(4,5)],6) => 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(1,5),(2,5),(3,4),(4,5)],6) => 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 3
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,2,4,5] => ([(2,4),(3,4)],5) => ([(3,5),(4,5)],6) => 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5) => ([(1,2),(3,5),(4,5)],6) => 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5) => ([(2,5),(3,4),(4,5)],6) => 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(1,5),(2,5),(3,4),(4,5)],6) => 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => ([(3,4),(3,5),(4,5)],6) => 2
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5) => ([(1,2),(3,4),(3,5),(4,5)],6) => 3
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(2,4),(2,5),(3,4),(3,5)],6) => 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 2
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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
vertex addition
Description
Adds a disconnected vertex to a graph.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
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