Construct a quadrangulation with connectivity 2 and minimum degree 3

Question

A quadrangulation is a simple graph(map) where every face is quadrilateral.

It is not difficult to find that minimum degree of any quadrangulation is no more than $3$. Furthermore, connectivity of any quadrangulation is between $2$ and $3$.

I want to find some quadrangulations with connectivity $2$. It is easy to find when the minimum degree is $2$. For example: $K_{2,n-2}$.

However I have not found a quadrangulation with minimum degree $3$ and connectivity $2$.

Answer 1

You can use SageMath with plantri, a tool written by Brinkmann et. al to generate all quadrangulations up to isomorphism with the desired properties as follows

for n in range(1, 15):
    gen = graphs.quadrangulations(n, minimum_degree=3, minimum_connectivity=2)
    for g in gen:
        if g.vertex_connectivity() == 2:
            g.show()

Some printed graph look as follows