Abstract
We study planar straight-line drawings of graphs that minimize the ratio between the length of the longest and the shortest edge. We answer a question of Lazard et al. [Theor. Comput. Sci. 770 (2019), 88–94] and, for any given constant r, we provide a 2-tree which does not admit a planar straight-line drawing with a ratio bounded by r. When the ratio is restricted to adjacent edges only, we prove that any 2-tree admits a planar straight-line drawing whose edge-length ratio is at most \(4 + \varepsilon \) for any arbitrarily small \(\varepsilon > 0\), hence the upper bound on the local edge-length ratio of partial 2-trees is 4.
The research was initiated during workshop Homonolo 2018. Research partially supported by MIUR, the Italian Ministry of Education, University and Research, under Grant 20174LF3T8 AHeAD: efficient Algorithms for HArnessing networked Data. V. Blažej acknowledges the support of the OP VVV MEYS funded project CZ.02.1.01/0.0/0.0/16_019/0000765 “Research Center for Informatics”. This work was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS20/208/OHK3/3T/18. The work of J. Fiala was supported by the grant 19-17314J of the GA ČR.
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We thank all three reviewers for their positive comments and careful review, which helped improve our contribution.
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Blažej, V., Fiala, J., Liotta, G. (2020). On the Edge-Length Ratio of 2-Trees. In: Auber, D., Valtr, P. (eds) Graph Drawing and Network Visualization. GD 2020. Lecture Notes in Computer Science(), vol 12590. Springer, Cham. https://doi.org/10.1007/978-3-030-68766-3_7
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