Values
([],1) => ([],1) => 1
([],2) => ([],1) => 1
([(0,1)],2) => ([(0,1)],2) => 1
([],3) => ([],1) => 1
([(1,2)],3) => ([(0,1)],2) => 1
([(0,2),(1,2)],3) => ([(0,1)],2) => 1
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 1
([],4) => ([],1) => 1
([(2,3)],4) => ([(0,1)],2) => 1
([(1,3),(2,3)],4) => ([(0,1)],2) => 1
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => 1
([(0,3),(1,2)],4) => ([(0,1)],2) => 1
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => 1
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
([],5) => ([],1) => 1
([(3,4)],5) => ([(0,1)],2) => 1
([(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(1,4),(2,3)],5) => ([(0,1)],2) => 1
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => 1
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => 1
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => 1
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
([],6) => ([],1) => 1
([(4,5)],6) => ([(0,1)],2) => 1
([(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(2,5),(3,4)],6) => ([(0,1)],2) => 1
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => 1
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
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Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Map
core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
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