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

A core of a graph [math]\displaystyle{ G }[/math] is a graph [math]\displaystyle{ C }[/math] such that

1. There exists a homomorphism from [math]\displaystyle{ G }[/math] to [math]\displaystyle{ C }[/math],
2. there exists a homomorphism from [math]\displaystyle{ C }[/math] to [math]\displaystyle{ G }[/math], and
3. [math]\displaystyle{ C }[/math] 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 a core.
• A graph [math]\displaystyle{ G }[/math] is a core if and only if the core of [math]\displaystyle{ G }[/math] is equal to [math]\displaystyle{ G }[/math].
• 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 [math]\displaystyle{ G \to H }[/math] and [math]\displaystyle{ H \to G }[/math] then the graphs [math]\displaystyle{ G }[/math] and [math]\displaystyle{ H }[/math] 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).

References

• Godsil, Chris, and Royle, Gordon. Algebraic Graph Theory. Graduate Texts in Mathematics, Vol. 207. Springer-Verlag, New York, 2001. Chapter 6 section 2.
• "The core of a graph", Discrete Mathematics 109 (1-3): 117–126, 1992, doi:10.1016/0012-365X(92)90282-K .
• "Proposition 3.5", Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, 28, Heidelberg: Springer, 2012, p. 43, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7 .