# 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.
Is my construction ok? Also which procedure gives us the sequence shown on the article?

• Please choose a more informative title, and reference the article properly.
– Raphael
Nov 17, 2017 at 15:54
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– D.W.
Nov 17, 2017 at 20:09

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 $D_k$.