# Infinite sequence of graphs

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.
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 $D_k$.