Genus Distributions for Iterated Claws
We derive a recursion for the genus distributions of the graphs obtained by iteratively attaching a claw to the dipole $D_3$. The minimum genus of the graphs in this sequence grows arbitrarily large. The families of graphs whose genus distributions have been calculated previously are either planar or almost planar, or they can be obtained by iterative single-vertex or single-edge amalgamation of small graphs. A significant simplifying construction within this calculation achieves the effect of an amalgamation at three vertices with a single root vertex, rather than with multiple roots.