Tutte Embeddings of Tetrahedral Meshes

TL;DR

This paper extends Tutte's planar embedding theorem to 3D tetrahedral meshes, providing conditions under which such meshes can be embedded based on convexity and graph minor restrictions.

Contribution

It generalizes Tutte's theorem from planar graphs to tetrahedral meshes, introducing new conditions involving graph minors for 3D embeddings.

Findings

Embedding is guaranteed if the mesh's graph contains no K6 or K_{3,3,1} minors.

Outer polyhedron convexity ensures proper embedding.

Interior vertices are convex combinations of neighbors under specified conditions.

Abstract

Tutte's embedding theorem states that every 3-connected graph without a $K_5$ or $K_{3,3}$ minor (i.e. a planar graph) is embedded in the plane if the outer face is in convex position and the interior vertices are convex combinations of their neighbors. We show that this result extends to simply connected tetrahedral meshes in a natural way: for the tetrahedral mesh to be embedded if the outer polyhedron is in convex position and the interior vertices are convex combination of their neighbors it is sufficient (but not necessary) that the graph of the tetrahedral mesh contains no $K_6$ and no $K_{3,3,1}$, and all triangles incident on three boundary vertices are boundary triangles.