Infinite sequence of graphs http://jgaa.info/accepted/2011/HasheminezhadMcKayReeves2011.15.3.pdf
 Hello I stumbled upon this paper. On page $4/20$ the figure shows an infinite sequence of $5-$regular, connected planar graphs. But they do not define it rigorously. The first graph of the sequence is clearly the icosahedron, and I feel that the next ones are constructed by "gluing" copies of $D_1$ but the only way I thought of constructing $D_k$ from $D_{k-1}$ and an icosahedron, would be removing a vertex from the border of its outer face. Then we remove a vertex from the border of the outer face of a copy of icosahedron and then we connected the vertices of degree $4.$ But this would leave us with less vertices than the graphs shown there.
 Is my construction ok? Also which procedure gives us the sequence shown on the article?
 thanks in advance.
Take the graph $D_1$, and cut the bolded edge in the middle, so that each of its two halves has degree 3 (including the bolded edge). Take $k$ copies of this, and paste them along bolded edges in a cyclic fashion. You get $D_k$.