Genus distribution of graph amalgamations: self-pasting at root-vertices

Counting the number of imbeddings in various surfaces of each of the graphs in an interesting family is an ongoing topic in topological graph theory. Our special focus here is on a family of closed chains of copies of a given graph. We derive double-root partials for open chains of copies of a given graph, and we then apply a self-amalgamation theorem, thereby obtaining genus distributions for a sequence of closed chains of copies of that graph. We use recombinant strands of face-boundary walks, and we further develop the use of multiple production rules in deriving partitioned genus distributions.