Identifier
Values
0 => 0 => [2] => ([],2) => 2
1 => 1 => [1,1] => ([(0,1)],2) => 2
00 => 00 => [3] => ([],3) => 2
01 => 10 => [1,2] => ([(1,2)],3) => 3
10 => 01 => [2,1] => ([(0,2),(1,2)],3) => 3
11 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 2
000 => 000 => [4] => ([],4) => 2
001 => 100 => [1,3] => ([(2,3)],4) => 3
010 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 3
011 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
100 => 010 => [2,2] => ([(1,3),(2,3)],4) => 3
101 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3
110 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
111 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
0000 => 0000 => [5] => ([],5) => 2
0001 => 1000 => [1,4] => ([(3,4)],5) => 3
0010 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
0011 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 4
0100 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5) => 3
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 4
0110 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
0111 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
1000 => 0100 => [2,3] => ([(2,4),(3,4)],5) => 3
1001 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
1010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
1011 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
1100 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
1101 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
1110 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
1111 => 1111 => [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) => 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..
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
rotate back-to-front
Description
The rotation of a binary word, last letter first.
This is the word obtained by moving the last letter to the beginnig.
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.