Planar graph with triangular faces in which the link of every vertex bounds a face What possible connected planar graphs are there that satisfy the following properties?


*

*Every face (including unbounded face) is triangular, i.e. is bounded by exactly 3 edges.

*For every vertex, the set of faces not incident to that vertex is either singleton or empty.


So far I can only think of two genuinely distinct such graphs, namely $K_3$ and $K_4$. I'm hoping that these are the only possibilities.

If you allow loops (as well as both sides of an edge counting as distinct sides of the same face), there's this degenerate case:

Otherwise there's only $K_3$ and $K_4$. When at least two faces meet at each vertex, any two vertices must share a face, and because the faces are triangular, this means that they are neighbors. So the graph is complete! But a complete graph with more than $4$ vertices is not planar.
