# Core (graph theory) Information

https://en.wikipedia.org/wiki/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 $C$ 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

1. There exists a homomorphism from $G$ to $C$ ,
2. there exists a homomorphism from $C$ to $G$ , and
3. $C$ 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 of odd length is its own core.
• 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.

## Properties

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 $G\to H$ 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) ( Hell & Nešetřil 1992).