Abstract
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/\sim . 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/\sim is pseudo-covering. That procedure becomes particularly easy to handle when the partition corresponding to X/\sim is an orbit partition.
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