Core (graph theory)

{{about|graph homomorphisms|the subgraph in which all vertices have high degree|k-core|the union of all maximum matchings|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 $C$ 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 a core.
A graph $G$ is a core if and only if the core of $G$ is equal to $G$ .
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 $G \to H$ 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) {{harv|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.
{{citation

| last1 = Hell | first1 = Pavol | author1-link = Pavol Hell

| last2 = Nešetřil | first2 = Jaroslav | author2-link = Jaroslav Nešetřil

| doi = 10.1016/0012-365X(92)90282-K

| issue = 1-3

| journal = Discrete Mathematics

| mr = 1192374

| pages = 117–126

| title = The core of a graph

| volume = 109

| year = 1992| doi-access = free

}}.

{{citation

| last1 = Nešetřil | first1 = Jaroslav | author1-link = Jaroslav Nešetřil

| last2 = Ossona de Mendez | first2 = Patrice | author2-link = Patrice Ossona de Mendez

| contribution = Proposition 3.5

| doi = 10.1007/978-3-642-27875-4

| isbn = 978-3-642-27874-7

| location = Heidelberg

| mr = 2920058

| page = 43

| publisher = Springer

| series = Algorithms and Combinatorics

| title = Sparsity: Graphs, Structures, and Algorithms

| volume = 28

| year = 2012 }}.

Category:Graph theory objects