Identifier
Values
[2] => [1,1] => ([(0,1)],2) => ([],1) => 0
[1,2] => [1,2] => ([(1,2)],3) => ([],1) => 0
[2,1] => [2,1] => ([(0,2),(1,2)],3) => ([(0,1)],2) => 0
[3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 0
[1,1,2] => [1,3] => ([(2,3)],4) => ([],1) => 0
[1,2,1] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1)],2) => 0
[1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 0
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 0
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 1
[4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[1,1,1,2] => [1,4] => ([(3,4)],5) => ([],1) => 0
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1)],2) => 0
[1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 0
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 0
[1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 1
[1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,1,2] => [1,5] => ([(4,5)],6) => ([],1) => 0
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1)],2) => 0
[1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 0
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 0
[1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 1
[1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(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) => 0
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,1,1,2] => [1,6] => ([(5,6)],7) => ([],1) => 0
[1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1)],2) => 0
[1,1,1,1,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,2)],3) => 0
[1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,2)],3) => 0
[1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 1
[1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(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) => 0
[1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(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) => 0
search for individual values
searching the database for the individual values of this statistic
Description
The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph.
A graph is chordal if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges and $(a,b)$ 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
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.
Map
conjugate
Description
The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.
Map
line graph
Description
The line graph of a graph.
Let $G$ be a graph with edge set $E$. Then its line graph is the graph with vertex set $E$, such that two vertices $e$ and $f$ are adjacent if and only if they are incident to a common vertex in $G$.