Graph homomorphisms and components of quotient graphs

TL;DR

This paper explores how the number of components in a graph relates to the components of its quotient graph, introducing new homomorphism concepts and providing a method to compute components efficiently.

Contribution

It introduces pseudo-covering homomorphisms and component equitable partitions, advancing the understanding of graph component enumeration via quotient graphs.

Findings

Provides a procedure to compute the number of components using pseudo-covering homomorphisms.

Identifies conditions under which the computation simplifies, especially with orbit partitions.

Establishes inclusion relations among classes of graph homomorphisms.

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.