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:

- Tutte, W. T. (1963), "How to draw a graph", Proceedings of the London Mathematical Society, 13: 743–767,doi:10.1112/plms/s3-13.1.743, MR 0158387.

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