Graph homomorphisms and components of quotient graphs

We study how the number c(X) of components of a graph X can be expressed through the number and properties of the components of a quotient graph X/similar to. We partially rely on classic qualifications of graph homomorphisms such as locally constrained homomorphisms and on the concept of equitable partition and orbit partition. We introduce the new definitions of pseudo-covering homomorphism and of component equitable partition, exhibiting interesting inclusions among the various classes of considered homomorphisms. As a consequence, we find a procedure for computing c(X) when the projection on the quotient X/similar to is pseudo-covering. That procedure becomes particularly easy to handle when the partition corresponding to X/similar to is an orbit partition.