  We introduce a new method for producing both maximal green and reddening
sequences of quivers. The method, called component preserving mutations,
generalizes the notion of direct sums of quivers and can be used as a tool to
both recover known reddening sequences as well as find reddening sequences that
were previously unknown. We use the method to produce and recover maximal green
sequences for many bipartite recurrent quivers that show up in the study of
periodicity of $T$-systems and $Y$-systems. Additionally, we show how our
method relates to the dominance phenomenon recently considered by Reading.
Given a maximal green sequence produced by our method, this relation to
dominance gives a maximal green sequence for infinitely many other quivers.
Other applications of this new methodology are explored including computing of
quantum dilogarithm identities and determining minimal length maximal green
sequences.
