Identifier
Values
0 => 0 => ([(0,1)],2) => ([],2) => 0
1 => 1 => ([(0,1)],2) => ([],2) => 0
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => 0
01 => 00 => ([(0,2),(2,1)],3) => ([],3) => 0
10 => 11 => ([(0,2),(2,1)],3) => ([],3) => 0
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => 0
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(2,5),(3,4)],6) => 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 0
010 => 000 => ([(0,3),(2,1),(3,2)],4) => ([],4) => 0
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 0
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 0
101 => 111 => ([(0,3),(2,1),(3,2)],4) => ([],4) => 0
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 0
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(2,5),(3,4)],6) => 1
0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 0
1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 0
01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 0
10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 0
010101 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([],7) => 0
101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([],7) => 0
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searching the database for the individual values of this statistic
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Description
The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph.
A graph is a split graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge and $(b,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Map
poset of factors
Description
The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that $u < v$ if and only if $u$ is a factor of $v$.
Map
incomparability graph
Description
The incomparability graph of a poset.
Map
alternating inverse
Description
Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.