Total embedding distributions for two graph families obtained from the dipole D3

The distribution of the set of embeddings of a graph into orientable or non-orientable surfaces is called the total embedding distribution. Chen, Gross and Rieper [Discrete Math. 128(1994) 73-94.] first used the overlap matrix for calculating the total embedding distributions of necklaces, closed-end ladders and cobblestone paths. In this paper, also by using the overlap matrix, closed formulas of the total embedding distributions for two classes of graphs are given.