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Logarithmic edge-length ratios for planar drawings of 2-trees

  • Zahra Sabaei
  • Fatemeh Emami
  • Yasmin Tousinejad
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Abstract

For a planar graph G with at least one edge, let ρ(G) be the infimum of the ratio of the longest to the shortest edge length over all planar straight-line drawings of G. We prove that every n-vertex $2$-tree G satisfies ρ(G)≤4+5⌊log2(2n−3)⌋. This improves the known O(√n) upper bound and, together with the lower bound of Blažej, Fiala and Liotta, shows that the maximum of ρ(G) over n-vertex $2$-trees is Θ(log n). The proof adapts Biedl's recursive bar construction using nonintegral heights and applies a height-preserving straightening theorem.

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Cite

BibTeXRIS
BibTeX
@misc{sabaei2026logarithmic,
  title = {{Logarithmic edge-length ratios for planar drawings of 2-trees}},
  author = {Zahra Sabaei and Fatemeh Emami and Yasmin Tousinejad},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28030},
  eprint = {2610.28030},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28030},
}
RIS
TY  - UNPB
TI  - Logarithmic edge-length ratios for planar drawings of 2-trees
AU  - Zahra Sabaei
AU  - Fatemeh Emami
AU  - Yasmin Tousinejad
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.28030
UR  - https://arxiv.org/abs/2610.28030
AB  - For a planar graph G with at least one edge, let ρ(G) be the infimum of the ratio of the longest to the shortest edge length over all planar straight-line drawings of G. We prove that every n-vertex $2$-tree G satisfies ρ(G)≤4+5⌊log_2(2n−3)⌋. This improves the known O(√n) upper bound and, together with the lower bound of Blažej, Fiala and Liotta, shows that the maximum of ρ(G) over n-vertex $2$-trees is Θ(log n). The proof adapts Biedl's recursive bar construction using nonintegral heights and applies a height-preserving straightening theorem.
ER  -