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.


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.


  • 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.


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.


  • Godsil, Chris, and Royle, Gordon. Algebraic Graph Theory. Graduate Texts in Mathematics, Vol. 207. Springer-Verlag, New York, 2001. Chapter 6 section 2.
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