Core (graph theory)

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This article is about graph homomorphisms. For the subgraph in which all vertices have high degree, see k-core.

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

Definition[edit]

Graph G is a core if every homomorphism f:G \to G is an isomorphism, that is it is a bijection of vertices of G.

A core of a graph H is a graph G such that

  1. There exists a homomorphism from H to G,
  2. there exists a homomorphism from G to H, and
  3. G is minimal with this property.

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

Examples[edit]

  • 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[edit]

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 hom-equivalent.

References[edit]