Genus distributions of graphs under self-edge-amalgamations

We investigate the well-known problem of counting graph imbeddings on all oriented surfaces with a focus on graphs that are obtained by pasting together two root-edges of another base graph. We require that the partitioned genus distribution of the base graph with respect to these root-edges be known and that both root-edges have two 2-valent endpoints. We derive general formulas for calculating the genus distributions of graphs that can be obtained either by self-co-amalgamating or by self-contra-amalgamating a base graph whose partitioned genus distribution is already known. We see how these general formulas provide a unified approach to calculating genus distributions of many new graph families, such as co-pasted and contra-pasted closed chains of copies of the triangular prism graph, as well as graph families like circular and Möbius ladders with previously known solutions to the genus distribution problem.