Graph homomorphism

Definitions

A graph homomorphism f from a graph G = ( V , E ) to a graph G ′ = ( V ′ , E ′ ) , written f : G → G ′ , is a mapping f : V → V ′ from the vertex set of G to the vertex set of G ′ such that { u , v } ∈ E implies { f ( u ) , f ( v ) } ∈ E ′ .

The above definition is extended to directed graphs. Then, for a homomorphism f : G → G ′ , ( f ( u ) , f ( v ) ) is an arc of G ′ if ( u , v ) is an arc of G .

If there exists a homomorphism f : G → H we shall write G → H , and G ↛ H otherwise. If G → H , G is said to be homomorphic to H or H -colourable.

If the homomorphism f : G → G ′ is a bijection whose inverse function is also a graph homomorphism, then f is a graph isomorphism.

Two graphs G and G ′ are homomorphically equivalent if G → G ′ and G ′ → G .

A retract of a graph G is a subgraph H of G such that there exists a homomorphism r : G → H , called retraction with r ( x ) = x for any vertex x of H . A core is a graph which does not retract to a proper subgraph. Any graph is homomorphically equivalent to a unique core.

Properties

The composition of homomorphisms are homomorphisms.

Graph homomorphism preserves connectedness.

The tensor product of graphs is the category-theoretic product for the category of graphs and graph homomorphisms.

Connection to coloring and girth

A graph coloring is an assignment of one of k colors to each vertex of a graph G so that the endpoints of each edge have different colors, for some number k. Any coloring corresponds to a homomorphism f : G → K k from G to a complete graph K_k: the vertices of K_k correspond to the colors of G, and f maps each vertex of G with color c to the vertex of K_k that corresponds to c. This is a valid homomorphism because the endpoints of each edge of G are mapped to distinct vertices of K_k, and every two distinct vertices of K_k are connected by an edge, so every edge in G is mapped to an adjacent pair of vertices in K_k. Conversely if f is a homomorphism from G to K_k, then one can color G by using the same color for two vertices in G whenever they are both mapped to the same vertex in K_k. Because K_k has no edges that connect a vertex to itself, it is not possible for two adjacent vertices in G to both be mapped to the same vertex in K_k, so this gives a valid coloring. That is, G has a k-coloring if and only if it has a homomorphism to K_k.

If there are two homomorphisms H → G → K k , then their composition H → K k is also a homomorphism. In other words, if a graph G can be colored with k colors, and there is a homomorphism H → G , then H can also be k-colored. Therefore, whenever a homomorphism H → G exists, the chromatic number of H is less than or equal to the chromatic number of G.

Homomorphisms can also be used very similarly to characterize the odd girth of a graph G, the length of its shortest odd-length cycle. The odd girth is, equivalently, the smallest odd number g for which there exists a homomorphism C g → G . For this reason, if G → H , then the odd girth of G is greater than or equal to the corresponding invariant of H.

Complexity

The associated decision problem, i.e. deciding whether there exists a homomorphism from one graph to another, is NP-complete. Determining whether there is an isomorphism between two graphs is also an important problem in computational complexity theory; see graph isomorphism problem.