Core (graph theory)

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 to C, • there exists a homomorphism from C to G, and • C is minimal with this property. Two graphs are homomorphically equivalent if and only if they have isomorphic cores. == Examples ==

• Any complete graph is a core. • A cycle 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 K2. • 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==