Your data matches 136 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000362
(load all 4 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([],2)
 => 0
([(0,1)],2)
 => ([],1)
 => 0
([],3)
 => ([],3)
 => 0
([(1,2)],3)
 => ([],2)
 => 0
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
([],4)
 => ([],4)
 => 0
([(2,3)],4)
 => ([],3)
 => 0
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([],5)
 => ([],5)
 => 0
([(3,4)],5)
 => ([],4)
 => 0
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(4,5)],6)
 => ([],5)
 => 0
Description
The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St000387
(load all 5 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St000387: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([],2)
 => 0
([(0,1)],2)
 => ([],1)
 => 0
([],3)
 => ([],3)
 => 0
([(1,2)],3)
 => ([],2)
 => 0
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
([],4)
 => ([],4)
 => 0
([(2,3)],4)
 => ([],3)
 => 0
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([],5)
 => ([],5)
 => 0
([(3,4)],5)
 => ([],4)
 => 0
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(4,5)],6)
 => ([],5)
 => 0
Description
The matching number of a graph. For a graph $G$, this is defined as the maximal size of a '''matching''' or '''independent edge set''' (a set of edges without common vertices) contained in $G$.
Matching statistic: St000985
(load all 3 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([],2)
 => 0
([(0,1)],2)
 => ([],1)
 => 0
([],3)
 => ([],3)
 => 0
([(1,2)],3)
 => ([],2)
 => 0
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
([],4)
 => ([],4)
 => 0
([(2,3)],4)
 => ([],3)
 => 0
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([],5)
 => ([],5)
 => 0
([(3,4)],5)
 => ([],4)
 => 0
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(4,5)],6)
 => ([],5)
 => 0
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
Matching statistic: St001349
(load all 2 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St001349: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([],2)
 => 0
([(0,1)],2)
 => ([],1)
 => 0
([],3)
 => ([],3)
 => 0
([(1,2)],3)
 => ([],2)
 => 0
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
([],4)
 => ([],4)
 => 0
([(2,3)],4)
 => ([],3)
 => 0
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([],5)
 => ([],5)
 => 0
([(3,4)],5)
 => ([],4)
 => 0
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(4,5)],6)
 => ([],5)
 => 0
Description
The number of different graphs obtained from the given graph by removing an edge.
Matching statistic: St001393
(load all 2 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([],2)
 => 0
([(0,1)],2)
 => ([],1)
 => 0
([],3)
 => ([],3)
 => 0
([(1,2)],3)
 => ([],2)
 => 0
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
([],4)
 => ([],4)
 => 0
([(2,3)],4)
 => ([],3)
 => 0
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([],5)
 => ([],5)
 => 0
([(3,4)],5)
 => ([],4)
 => 0
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(4,5)],6)
 => ([],5)
 => 0
Description
The induced matching number of a graph. An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St001459
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St001459: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([],2)
 => 0
([(0,1)],2)
 => ([],1)
 => 0
([],3)
 => ([],3)
 => 0
([(1,2)],3)
 => ([],2)
 => 0
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
([],4)
 => ([],4)
 => 0
([(2,3)],4)
 => ([],3)
 => 0
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([],5)
 => ([],5)
 => 0
([(3,4)],5)
 => ([],4)
 => 0
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(4,5)],6)
 => ([],5)
 => 0
Description
The number of zero columns in the nullspace of a graph.
Matching statistic: St001812
(load all 5 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([],2)
 => 0
([(0,1)],2)
 => ([],1)
 => 0
([],3)
 => ([],3)
 => 0
([(1,2)],3)
 => ([],2)
 => 0
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
([],4)
 => ([],4)
 => 0
([(2,3)],4)
 => ([],3)
 => 0
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
([],5)
 => ([],5)
 => 0
([(3,4)],5)
 => ([],4)
 => 0
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(4,5)],6)
 => ([],5)
 => 0
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 $n - 1$.
Matching statistic: St000172
(load all 5 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 1 = 0 + 1
([],2)
 => ([],2)
 => 1 = 0 + 1
([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
([],3)
 => ([],3)
 => 1 = 0 + 1
([(1,2)],3)
 => ([],2)
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1 = 0 + 1
([],4)
 => ([],4)
 => 1 = 0 + 1
([(2,3)],4)
 => ([],3)
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
([],5)
 => ([],5)
 => 1 = 0 + 1
([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(4,5)],6)
 => ([],5)
 => 1 = 0 + 1
Description
The Grundy number of a graph. The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St001116
(load all 5 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St001116: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 1 = 0 + 1
([],2)
 => ([],2)
 => 1 = 0 + 1
([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
([],3)
 => ([],3)
 => 1 = 0 + 1
([(1,2)],3)
 => ([],2)
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1 = 0 + 1
([],4)
 => ([],4)
 => 1 = 0 + 1
([(2,3)],4)
 => ([],3)
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
([],5)
 => ([],5)
 => 1 = 0 + 1
([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(4,5)],6)
 => ([],5)
 => 1 = 0 + 1
Description
The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.
Matching statistic: St001261
(load all 2 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 1 = 0 + 1
([],2)
 => ([],2)
 => 1 = 0 + 1
([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
([],3)
 => ([],3)
 => 1 = 0 + 1
([(1,2)],3)
 => ([],2)
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1 = 0 + 1
([],4)
 => ([],4)
 => 1 = 0 + 1
([(2,3)],4)
 => ([],3)
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
([],5)
 => ([],5)
 => 1 = 0 + 1
([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
([(4,5)],6)
 => ([],5)
 => 1 = 0 + 1
Description
The Castelnuovo-Mumford regularity of a graph.
The following 126 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001734The lettericity of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St001277The degeneracy of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001358The largest degree of a regular subgraph of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001689The number of celebrities in a graph. St001792The arboricity of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000258The burning number of a graph. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St000537The cutwidth of a graph. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001743The discrepancy of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000926The clique-coclique number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001716The 1-improper chromatic number of a graph. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000552The number of cut vertices of a graph. St000260The radius of a connected graph. St000259The diameter 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. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000379The number of Hamiltonian cycles in a graph. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St001570The minimal number of edges to add to make a graph Hamiltonian. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000699The toughness times the least common multiple of 1,.