In the mathematical field of graph theory, 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 Two graphs are said to be homomorphism equivalent or hom-equivalent if they have isomorphic cores.

Examples

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).