# Core (graph theory)

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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

- There exists a homomorphism from to ,
- there exists a homomorphism from to , and
- 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
*K*_{2}.

## 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 and then the graphs and are necessarily hom-equivalent.

## References

- Godsil, Chris, and Royle, Gordon.
*Algebraic Graph Theory.*Graduate Texts in Mathematics, Vol. 207. Springer-Verlag, New York, 2001. Chapter 6 section 2. - {{#invoke:citation/CS1|citation

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