Genus distribution of graph amalgamations: Pasting when one root has arbitrary degree

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 to allow one of the two roots to have arbitrarily high valence.

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