# Can Tutte embedding be guaranteed that each face is convex?

In graph drawing and geometric graph theory, a Tutte embedding of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbors' positions. Tutte embedding

But I don’t know if this means that such embedding when first we fix any arbitrary outer face (may be convex) can guarantee that each face of is convex.

Edit: : Narrate more clearly, is any internal face convex in Tutte ebemdding?

For example, will the following non-convex face $$f_1$$ appear in Tutte embedding embedding? If it exists, is there a way to make each face convex?

References:

• $w$ will be placed midway along the line connecting $e$ and $z$, so the face will never appear as you have drawn it. Apr 13, 2021 at 3:30
• You might also be interested in Steinitz's theorem. Apr 13, 2021 at 3:32
• @GordonRoyle Thank you very much for your answers. I already understand what you mean, for this example, there really is no such drawing in Tutte embemdding. This is just an example of mine. Maybe not good. I don’t know if any non-convex face can also be excluded for the general situation. Apr 13, 2021 at 3:36
• @SamHopkins Steinitz's theorem seems to only say that there is a correspondence between a 3-connected plane graph and a convex polyhedron, but I am not sure that given an any convex(may be not) external face, it can guarantee that there is a plane straight-line drawing where anyinternal face is all convex. Maybe I don't understand it well. Apr 13, 2021 at 3:46
• Tutte's method works with any face on the outside drawn as any convex polygon. It doesn't matter which face you choose. But this assumes 3-connectivity. Without 3-connectivity there might not be any Tutte drawing at all. For a simple example, suppose there is a cut-vertex -- then some face has a vertex appearing twice so it is impossible to draw it convex. Apr 13, 2021 at 4:27