Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001303
(load all 2 compositions to match this statistic)
Mapping
St001303: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 3
([],3)
 => 1
([(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => 5
([(0,1),(0,2),(1,2)],3)
 => 7
([],4)
 => 1
([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => 5
([(0,3),(1,3),(2,3)],4)
 => 9
([(0,3),(1,2)],4)
 => 9
([(0,3),(1,2),(2,3)],4)
 => 9
([(1,2),(1,3),(2,3)],4)
 => 7
([(0,3),(1,2),(1,3),(2,3)],4)
 => 11
([(0,2),(0,3),(1,2),(1,3)],4)
 => 11
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 13
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 15
([],5)
 => 1
([(3,4)],5)
 => 3
([(2,4),(3,4)],5)
 => 5
([(1,4),(2,4),(3,4)],5)
 => 9
([(0,4),(1,4),(2,4),(3,4)],5)
 => 17
([(1,4),(2,3)],5)
 => 9
([(1,4),(2,3),(3,4)],5)
 => 9
([(0,1),(2,4),(3,4)],5)
 => 15
([(2,3),(2,4),(3,4)],5)
 => 7
([(0,4),(1,4),(2,3),(3,4)],5)
 => 15
([(1,4),(2,3),(2,4),(3,4)],5)
 => 11
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 19
([(1,3),(1,4),(2,3),(2,4)],5)
 => 11
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 19
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 13
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 17
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 21
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 23
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 25
([(0,4),(1,3),(2,3),(2,4)],5)
 => 17
([(0,1),(2,3),(2,4),(3,4)],5)
 => 21
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 21
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 25
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 21
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 23
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 25
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 21
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 15
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 23
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 27
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 25
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 27
Description
The number of dominating sets of vertices of a graph. This is, the number of subsets of vertices such that every vertex is either in this subset or adjacent to an element therein [1].
Matching statistic: St001794
(load all 2 compositions to match this statistic)
Mapping
Mp00203: Graphs —cone⟶ Graphs
St001794: Graphs ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([(0,1)],2)
 => 1
([],2)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 7
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 9
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 9
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 9
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 11
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 11
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 13
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 15
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 17
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 9
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 15
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 15
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 11
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 19
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 11
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 19
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 13
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 17
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 21
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 23
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 25
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 17
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 21
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 21
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 25
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 21
([(0,1),(0,4),(1,3),(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)
 => 23
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 25
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 21
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 15
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 23
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 27
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 25
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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),(4,5)],6)
 => 27
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {21,33,33,35,35,35,39,39,45,49}
Description
Half the number of sets of vertices in a graph which are dominating and non-blocking. A set of vertices $U$ in a graph is dominating, if every vertex not in $U$ is adjacent to a vertex in $U$. A set of vertices $U$ in a graph is non-blocking, if every vertex in $U$ is adjacent to a vertex not in $U$. Therefore, a set of vertices is non-blocking if and only if its complement is dominating. In particular, if a set of vertices is dominating and non-blocking, so is its complement.