Core (graph theory)

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

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

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

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

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.