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Identifier

St001303: Graphs ⟶ ℤ

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

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 29

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 31

([],6) => 1

([(4,5)],6) => 3

([(3,5),(4,5)],6) => 5

([(2,5),(3,5),(4,5)],6) => 9

([(1,5),(2,5),(3,5),(4,5)],6) => 17

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 33

([(2,5),(3,4)],6) => 9

([(2,5),(3,4),(4,5)],6) => 9

([(1,2),(3,5),(4,5)],6) => 15

([(3,4),(3,5),(4,5)],6) => 7

([(1,5),(2,5),(3,4),(4,5)],6) => 15

([(0,1),(2,5),(3,5),(4,5)],6) => 27

([(2,5),(3,4),(3,5),(4,5)],6) => 11

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 27

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 19

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 35

([(2,4),(2,5),(3,4),(3,5)],6) => 11

([(0,5),(1,5),(2,4),(3,4)],6) => 25

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 19

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 29

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 13

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 17

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 25

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 33

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 21

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 29

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 37

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 23

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 33

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 39

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 25

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 33

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 41

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 47

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 49

([(0,5),(1,4),(2,3)],6) => 27

([(1,5),(2,4),(3,4),(3,5)],6) => 17

([(0,1),(2,5),(3,4),(4,5)],6) => 27

([(1,2),(3,4),(3,5),(4,5)],6) => 21

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 27

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 21

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 33

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 33

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 25

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 41

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 21

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 31

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 23

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 35

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$F_{1} = q$

$F_{2} = q + q^{3}$

$F_{3} = q + q^{3} + q^{5} + q^{7}$

$F_{4} = q + q^{3} + q^{5} + q^{7} + 3\ q^{9} + 2\ q^{11} + q^{13} + q^{15}$

$F_{5} = q + q^{3} + q^{5} + q^{7} + 3\ q^{9} + 2\ q^{11} + q^{13} + 3\ q^{15} + 3\ q^{17} + 2\ q^{19} + 5\ q^{21} + 3\ q^{23} + 4\ q^{25} + 2\ q^{27} + q^{29} + q^{31}$

$F_{6} = q + q^{3} + q^{5} + q^{7} + 3\ q^{9} + 2\ q^{11} + q^{13} + 3\ q^{15} + 3\ q^{17} + 2\ q^{19} + 5\ q^{21} + 3\ q^{23} + 6\ q^{25} + 8\ q^{27} + 3\ q^{29} + 3\ q^{31} + 9\ q^{33} + 8\ q^{35} + 4\ q^{37} + 11\ q^{39} + 9\ q^{41} + 7\ q^{43} + 11\ q^{45} + 7\ q^{47} + 13\ q^{49} + 8\ q^{51} + 8\ q^{53} + 6\ q^{55} + 5\ q^{57} + 2\ q^{59} + q^{61} + q^{63}$

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].

References

[1] wikipedia:Dominating_set

Code

def is_dominating(G,V):
    return set(G.vertices()) == set(V).union(*[G.neighbors(v) for v in V])

def statistic(G):
    return sum( 1 for V in Subsets(G.vertices()) if is_dominating(G, V) )

Created

Dec 10, 2018 at 14:51 by Christian Stump

Updated

Dec 10, 2018 at 15:27 by Christian Stump