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 $C$ is a core if every homomorphism $f:C\to C$ 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

There exists a homomorphism from $G$ to $C$ ,
there exists a homomorphism from $C$ to $G$ , and
$C$ 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 $G\to H$ and $H\to G$ 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

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:10.1016/0012-365X(92)90282-K, MR 1192374.
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:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058.