I learned this theorem in the graph theory textbook.

**Theorem** Every $ 2$ -connected plane graph can be embedded in the plane so that **any specified face is the exterior**.

`G=PlanarGraph[{1 <-> 2, 1 <-> 3, 1 <-> 4, 2 <-> 3, 3 <-> 4, 2 <-> 5, 5 <-> 6, 6 <-> 3}, VertexLabels -> All] `

In the above embedding of this graph, we know $ 1256341$ is boundary exterior face of $ G$ .

I don’t know if there is a way to make the triangle face $ \Delta_{134}$ outside.

The above is just an example. For the graph $ G$ , maybe I can change the layout of some points by **VertexCoordinates**. But for the large number of vertices, I don’t know if there is a good and unified way to **arbitrarily specify an external face** and give a good plane drawing.