Images
([],0) => ([],0)
([],1) => ([],1)
([],2) => ([],1)
([(0,1)],2) => ([(0,1)],2)
([],3) => ([],1)
([(1,2)],3) => ([(0,1)],2)
([(0,2),(1,2)],3) => ([(0,1)],2)
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3)
([],4) => ([],1)
([(2,3)],4) => ([(0,1)],2)
([(1,3),(2,3)],4) => ([(0,1)],2)
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2)
([(0,3),(1,2)],4) => ([(0,1)],2)
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2)
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3)
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3)
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2)
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3)
([(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)
([],5) => ([],1)
([(3,4)],5) => ([(0,1)],2)
([(2,4),(3,4)],5) => ([(0,1)],2)
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
([(1,4),(2,3)],5) => ([(0,1)],2)
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2)
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2)
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2)
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2)
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2)
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(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)
([(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)
([(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)
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
([(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)
([(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)
([],6) => ([],1)
([(4,5)],6) => ([(0,1)],2)
([(3,5),(4,5)],6) => ([(0,1)],2)
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
([(2,5),(3,4)],6) => ([(0,1)],2)
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2)
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2)
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2)
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2)
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2)
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2)
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2)
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2)
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2)
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2)
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2)
([(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)
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2)
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2)
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2)
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2)
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
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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].
References
[1] wikipedia:Core_(graph_theory)
[2] Cameron, Peter J. "Graph homomorphisms." Combinatorics Study Group Notes (2006): 1-7. http://www.maths.qmul.ac.uk/~pjc/csgnotes/hom1.pdf
[2] Cameron, Peter J. "Graph homomorphisms." Combinatorics Study Group Notes (2006): 1-7. http://www.maths.qmul.ac.uk/~pjc/csgnotes/hom1.pdf
Properties
idempotent
Sage code
def mapping(G):
if G.num_verts() < 7:
V = G.has_homomorphism_to(G, core=True).values()
return G.subgraph(vertices=V)
else:
raise ValueError("Graph too big for this map")
Weight
26
Created
Sep 04, 2020 at 08:59 by Martin Rubey
Updated
Sep 04, 2020 at 08:59 by Martin Rubey
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