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Euclidean MST: Grid Size for Degree-5 Trees

Investigate whether the current double-exponential grid size bound for realizing every degree-5 tree as a Euclidean minimum spanning tree can be improved.

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Background

Trees of maximum degree five are known to be realizable as Euclidean minimum spanning trees, but existing constructions require double-exponential grid sizes.

The survey explicitly notes that it remains open whether this bound can be improved.

References

The maximum degree of $T$ has to be $6$, since every tree of maximum degree $5$ is a Euclidean Minimum Spanning Tree; the construction showing this requires a double-exponential grid; as far as we know it is open whether this can be improved.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in ProblemCG “Euclidean Minimum Spanning Tree” entry