It is well-known that planar graphs have minimum degree at most 5. Furthermore, we know any planar graph with minimum degree exactly 5 has at least 12 vertices. The icosahedron graph is the unique 12-vertex planar graph with minimum degree exactly 5. Here's an interesting related paper.
- Brinkmann G, McKay B D. Construction of planar triangulations with minimum degree 5[J]. Discrete mathematics, 2005, 301(2-3): 147-163.
It is not difficult to see that the icosahedron is 3-connected. Then the two questions below make sense.
- what is the smallest number of vertices among planar graphs with vertex connectivity $1$ of minimum degree of $5$ ?
- what is the smallest number of vertices among planar graphs with vertex connectivity $2$ of minimum degree of $5$ ?
For the first question, my guess is $23$. As in the figure below, the planar graph with 23 vertices consists of two icosahedron graphs glued together at a single vertex.
For the second question, my guess is $22$. As in the figure below, the left graph consists of two icosahedron graphs glued together with two vertices. The graph on the right is obtained by deleting an edge between two glued vertices of the left graph.
![![enter image description here][2]][2](https://i.stack.imgur.com/xAmVF.png)
But I lacked the ideas to prove these possibly correct facts.