Tree 3-Spanner Problem for Unweighted Graphs

Determine whether an unweighted graph admits a spanning tree with dilation 3, resolving the case left undecided between the known polynomial-time solvability for dilation at most 2 and NP-completeness for dilation at least 4.

Background

The paper discusses minimum dilation trees in unweighted graph metrics as related work. For dilation thresholds at most 2, a linear-time algorithm is known, whereas deciding whether the dilation is at least 4 is NP-complete. The authors explicitly identify the threshold-3 case as unresolved.

References

For $t=3$, the problem still remains open.

— A Sublinear Approximation Algorithm for Minimum Dilation Trees in the Plane  (2609.08990 - Berg et al., 8 Sep 2026) in Section 1, subsection “Other related work,” paragraph “Unweighted graphs”