Your data matches 43 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001307
(load all 3 compositions to match this statistic)
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
St001307: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
Description
The number of induced stars on four vertices in a graph.
Matching statistic: St000095
(load all 2 compositions to match this statistic)
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000095: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([],1)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([],1)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
Description
The number of triangles of a graph. A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.
Matching statistic: St001309
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001309: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
Description
The number of four-cliques in a graph.
Matching statistic: St001311
(load all 2 compositions to match this statistic)
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([],1)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([],1)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
Description
The cyclomatic number of a graph. This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Matching statistic: St001317
(load all 2 compositions to match this statistic)
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001317: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([],1)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([],1)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. A graph is a forest 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. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001572
(load all 2 compositions to match this statistic)
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001572: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([],1)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([],1)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
Description
The minimal number of edges to remove to make a graph bipartite.
Matching statistic: St001573
(load all 2 compositions to match this statistic)
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001573: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([],1)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([],1)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
Description
The minimal number of edges to remove to make a graph triangle-free.
Matching statistic: St001736
(load all 2 compositions to match this statistic)
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001736: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([],1)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([],1)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
Description
The total number of cycles in a graph. For example, the complete graph on four vertices has four triangles and three different four-cycles.
Matching statistic: St000450
(load all 2 compositions to match this statistic)
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000450: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
Description
The number of edges minus the number of vertices plus 2 of a graph. When G is connected and planar, this is also the number of its faces. When $G=(V,E)$ is a connected graph, this is its $k$-monochromatic index for $k>2$: for $2\leq k\leq |V|$, the $k$-monochromatic index of $G$ is the maximum number of edge colors allowed such that for each set $S$ of $k$ vertices, there exists a monochromatic tree in $G$ which contains all vertices from $S$. It is shown in [1] that for $k>2$, this is given by this statistic.
Matching statistic: St000482
(load all 3 compositions to match this statistic)
Mapping
Mp00011: Binary trees —to graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000482: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
Description
The (zero)-forcing number of a graph. This is the minimal number of vertices initially coloured black, such that eventually all vertices of the graph are coloured black when using the following rule: when $u$ is a black vertex of $G$, and exactly one neighbour $v$ of $u$ is white, then colour $v$ black.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000252The number of nodes of degree 3 of a binary tree. St001964The interval resolution global dimension of a poset. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001871The number of triconnected components of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001734The lettericity of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000327The number of cover relations in a poset. St000741The Colin de Verdière graph invariant. St001282The number of graphs with the same chromatic polynomial. St000455The second largest eigenvalue of a graph if it is integral. St001845The number of join irreducibles minus the rank of a lattice. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset.