Core (graph theory)

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 C {\displaystyle C} is a core if every homomorphism f : C → C {\displaystyle f:C\to C} is an isomorphism, that is it is a bijection of vertices of C {\displaystyle C}.

A core of a graph G {\displaystyle G} is a graph C {\displaystyle C} such that

1. There exists a homomorphism from G {\displaystyle G} to C {\displaystyle C},
2. there exists a homomorphism from C {\displaystyle C} to G {\displaystyle G}, and
3. C {\displaystyle C} is minimal with this property.

Two graphs are homomorphically equivalent if and only if they have isomorphic cores.

Examples

• Any complete graph is a core.
• A cycle of odd length is a core.
• A graph G {\displaystyle G} is a core if and only if the core of G {\displaystyle G} is equal to G {\displaystyle G}.
• 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.
• By the Beckman–Quarles theorem, the infinite unit distance graph on all points of the Euclidean plane or of any higher-dimensional Euclidean space is a core.

Properties

Every finite 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 {\displaystyle G\to H} and H → G {\displaystyle H\to G} then the graphs G {\displaystyle G} and H {\displaystyle 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).

• Godsil, Chris, and Royle, Gordon. Algebraic Graph Theory. Graduate Texts in Mathematics, Vol. 207. Springer-Verlag, New York, 2001. Chapter 6 section 2.
• Hell, Pavol; Nešetřil, Jaroslav (1992), "The core of a graph", Discrete Mathematics, 109 (1–3): 117–126, doi:, MR.
• Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), "Proposition 3.5", Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Heidelberg: Springer, p. 43, doi:, ISBN 978-3-642-27874-7, MR.