In the mathematical field of

graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, a core is a notion that describes behavior of a graph with respect to graph homomorphisms.

Definition

Graph

C

is a core if every homomorphism

f:C \to C

is an isomorphism, that is it is a bijection of vertices of

C

. A core of a graph

G

is a graph

C

such that # There exists a homomorphism from

G

C

, # there exists a homomorphism from

C

G

, and #

C

is minimal with this property. Two graphs are said to be homomorphism equivalent or hom-equivalent if they have isomorphic cores.

Examples

* Any complete graph is a core. * A

cycle Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in soc ...

of odd length is a core. * A graph

G

is a core if and only if the core of

G

is equal to

G

. * Every two cycles of even length, and more generally every two bipartite graphs are hom-equivalent. The core of each of these graphs is the two-vertex complete graph ''K''₂. * By the Beckman–Quarles theorem, the infinite unit distance graph on all points of the Euclidean plane or of any higher-dimensional Euclidean space is a core.

Properties

Every finite graph has a core, which is determined uniquely, up to isomorphism. The core of a graph ''G'' is always an induced subgraph of ''G''. If

G \to H

and

H \to G

then the graphs

G

and

H

are necessarily homomorphically equivalent.

Computational complexity

It is NP-complete to test whether a graph has a homomorphism to a proper subgraph, and co-NP-complete to test whether a graph is its own core (i.e. whether no such homomorphism exists) .

References

* Godsil, Chris, and Royle, Gordon. ''Algebraic Graph Theory.'' Graduate Texts in Mathematics, Vol. 207. Springer-Verlag, New York, 2001. Chapter 6 section 2. *. *{{citation , last1 = Nešetřil , first1 = Jaroslav , author1-link = Jaroslav Nešetřil , last2 = Ossona de Mendez , first2 = Patrice , author2-link = Patrice Ossona de Mendez , contribution = Proposition 3.5 , doi = 10.1007/978-3-642-27875-4 , isbn = 978-3-642-27874-7 , location = Heidelberg , mr = 2920058 , page = 43 , publisher = Springer , series = Algorithms and Combinatorics , title = Sparsity: Graphs, Structures, and Algorithms , volume = 28 , year = 2012 . Graph theory objects