 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

The <complete bipartite graph> $K_{2,3}$ is connected and has five vertices and six edges. It is a <planar graph>, as in the right drawing above. <Euler formula for a connected planar graph> therefore gives
$$
\boxed{f=2-v+e=2-5+6=3.}
$$
This includes the unbounded exterior face.

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