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.

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 homomorphically equivalent if and only if they have isomorphic cores.

• 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.

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.

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).