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This paper concerns counting the imbeddings of a graph in a surface. In the first installment of our current work, we showed how to calculate the genus distribution of an iterated amalgamation of copies of a graph whose genus distribution is already known and is further analyzed into a partitioned genus distribution (which is defined for a double-rooted graph). Our methods were restricted there to the case with two 2-valent roots. In this sequel we substantially extend the method in order todoi:10.26493/1855-3974.111.e89 fatcat:sq7qh4h53jej5mr7itz6x2tsai