Girish Mahajan (Editor)

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.

Contents

Definition

Graph C is a core if every homomorphism f : C → 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 → H and H → 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).

    References

    Core (graph theory) Wikipedia
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