Approximate realizations in 2D and under restricted graph classes

Investigate the complexity of approximate realizations in the plane, where each edge length may stretch by at most a multiplicative factor α>1, including under restrictions to globally noncrossing graphs, matchstick graphs, unit‑distance graphs, and {1,2}‑distance graphs; extend or adapt Saxe’s 1D hardness results to these 2D and restricted‑class settings.

Background

Approximate (stretch‑bounded) realizations are extensively studied in metric embedding, and Saxe proved strong NP‑completeness for distinguishing certain stretch bounds in one dimension using small integer edge lengths.

The paper suggests exploring analogous questions in two dimensions and under noncrossing or fixed‑length restrictions, which would link classical metric embedding hardness with geometric graph realization constraints.

References

Table 1 settles most problems in this area, but a few interesting open problems remain. What about embedding into two dimensions? What about when restricted to globally noncrossing graphs, matchstick graphs, graphs with unit edge lengths, or graphs with edge lengths in {1,2}? (Saxe's proof uses edge lengths in {1,2,3,4}.)

— Who Needs Crossings?: Noncrossing Linkages are Universal, and Deciding (Global) Rigidity is Hard  (2510.17737 - Abel et al., 20 Oct 2025) in Section 7, Open Problems