In the mathematical field of graph theory, a core is a notion that describes behavior of a graph with respect to graph homomorphisms.

Graph ${\displaystyle C}$ is a core if every homomorphism ${\displaystyle f:C\to C}$ is an isomorphism, that is it is a bijection of vertices of ${\displaystyle C}$.

A core of a graph ${\displaystyle G}$ is a graph ${\displaystyle C}$ such that

Two graphs are said to be homomorphism equivalent or hom-equivalent if they have isomorphic cores.

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 ${\displaystyle G\to H}$ and ${\displaystyle H\to G}$ then the graphs ${\displaystyle G}$ and ${\displaystyle H}$ are necessarily homomorphically equivalent.

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) (Hell & Nešetřil 1992).