Core (graph theory)


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 is a core if every homomorphism is an isomorphism, that is it is a bijection of vertices of.
A core of a graph is a graph such that
  1. There exists a homomorphism from to,
  2. there exists a homomorphism from to, and
  3. is minimal with this property.
Two graphs are said to be homomorphism equivalent or hom-equivalent if they have isomorphic cores.

Examples

Every 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 and then the graphs and 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 .