Core (graph theory)

From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about graph homomorphisms. For the subgraph in which all vertices have high degree, see k-core. For the union of all maximum matchings, see Dulmage–Mendelsohn decomposition.

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 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.

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 and then the graphs and are necessarily homomorphically equivalent.

Computational complexity[edit]

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[edit]