- FindStat

Identifier

Mp00154: Graphs —core⟶ Graphs
St001337: Graphs ⟶ ℤ

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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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q$

$F_{4} = 11\ q$

$F_{5} = 33\ q + q^{2}$

$F_{6} = 152\ q + 4\ q^{2}$

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].

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].