PreprintCC BY 4.0arXiv
Realizing graph automorphisms as geometric symmetries of straight line drawings
Abstract
Given a finite graph G, we characterize the automorphisms and automorphism groups of G that can be realized as geometric symmetries and symmetry groups of straight line drawings of G in the plane. In doing so, we correct a previous version of this characterization from the literature. As an application, we show that the graphs recently constructed by the first two authors to disprove the Polycirculant Conjecture, which have orders 16,464k for each positive integer k, are the first known vertex-transitive graphs with no non-trivial realizable geometric symmetries. We also construct infinite families of graphs that exhibit all possible kinds of geometric symmetry.
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BibTeX
@misc{freedman2026realizing,
title = {{Realizing graph automorphisms as geometric symmetries of straight line drawings}},
author = {Saul D. Freedman and Melissa Lee and Dragan Marušič and Tomaž Pisanski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36394},
eprint = {2610.36394},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36394},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Realizing graph automorphisms as geometric symmetries of straight line drawings
AU - Saul D. Freedman
AU - Melissa Lee
AU - Dragan Marušič
AU - Tomaž Pisanski
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36394
UR - https://arxiv.org/abs/2610.36394
AB - Given a finite graph G, we characterize the automorphisms and automorphism groups of G that can be realized as geometric symmetries and symmetry groups of straight line drawings of G in the plane. In doing so, we correct a previous version of this characterization from the literature. As an application, we show that the graphs recently constructed by the first two authors to disprove the Polycirculant Conjecture, which have orders 16,464^k for each positive integer k, are the first known vertex-transitive graphs with no non-trivial realizable geometric symmetries. We also construct infinite families of graphs that exhibit all possible kinds of geometric symmetry.
ER -