[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Vertex Spanning Planar Laman Graphs in Triangulated Surfaces

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

We prove that every triangulation of either of the torus, projective plane and Klein bottle, contains a vertex-spanning planar Laman graph as a subcomplex. Invoking a result of Király, we conclude that every 1-skeleton of a triangulation of a surface of nonnegative Euler characteristic has a rigid realization in the plane using at most 26 locations for the vertices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. This connected sum may have at most two edges contained in no triangle face; however, putting back the triangle which is the intersection of those two discs yields a (nonplanar) pure complex which is strongly-connected and has the same graph, namely same 1-skeleton, hence the graph of this connected sum is 2-rigid.

  2. In case \(x_1=B\) and \(x_k=C\) both \(A'BC\) and \(A''BC\) are 3-cycles, and we chose to name \(A'=A\) rather than \(A''=A\).

  3. In a connected sum of discs along ABC, some of the vertices A, B, C may be singular, but this is irrelevant.

  4. A pinched disc at x is the quotient space of a disc where two distinct points are identified, and x is the resulted singular point. In our case the two identified points are on the boundary of the disc.

References

  1. Adiprasito, K., Nevo, E.: Rigidity with few locations. Israel J. Math. 240(2), 711–723 (2020)

    Article  MathSciNet  Google Scholar 

  2. Barnette, D.: Generating the triangulations of the projective plane. J. Combin. Theory Ser. B 33(3), 222–230 (1982)

    Article  MathSciNet  Google Scholar 

  3. Barnette, D.W., Edelson, A.: All orientable \(2\)-manifolds have finitely many minimal triangulations. Israel J. Math. 62(1), 90–98 (1988)

    Article  MathSciNet  Google Scholar 

  4. Barnette, D.W., Edelson, A.L.: All \(2\)-manifolds have finitely many minimal triangulations. Israel J. Math. 67(1), 123–128 (1989)

    Article  MathSciNet  Google Scholar 

  5. Connelly, R.: Rigidity. In: Handbook of Convex Geometry, Vol. A, pp. 223–271. North-Holland, Amsterdam (1993)

  6. Fekete, Zs., Jordán, T.: Rigid realizations of graphs on small grids. Comput. Geom. 32(3), 216–222 (2005)

  7. Fogelsanger, A.L.: The Generic Rigidity of Minimal Cycles. PhD thesis, Cornell University (1988)

  8. Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Mathematics, vol. 2. American Mathematical Society, Providence (1993)

  9. Kalai, G.: Rigidity and the lower bound theorem 1. Invent. Math. 88(1), 125–151 (1987)

    Article  MathSciNet  Google Scholar 

  10. Király, Cs.: Rigid realizations of graphs with few locations in the plane. Eur. J. Combin. 94, #103304 (2021)

  11. Lawrencenko, S., Negami, S.: Irreducible triangulations of the Klein bottle. J. Combin. Theory Ser. B 70(2), 265–291 (1997)

    Article  MathSciNet  Google Scholar 

  12. Lavrenchenko, S.A.: Irreducible triangulations of the torus. J. Soviet Math. 51(5), 2537–2543 (1990)

    Article  MathSciNet  Google Scholar 

  13. Nevo, E., Tarabykin, S.: Vertex spanning planar Laman graphs in triangulated surfaces. Sém. Lothar. Combin. 86B, # 43 (2022)

  14. Sitharam, M., St. John, A., Sidman, J. (eds.): Handbook of Geometric Constraint Systems Principles. Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton (2019)

  15. Sulanke, Th.: Generating irreducible triangulations of surfaces (2006). arXiv:math/0606687

  16. Sulanke, Th.: Note on the irreducible triangulations of the Klein bottle. J. Combin. Theory Ser. B 96(6), 964–972 (2006)

    Article  MathSciNet  Google Scholar 

  17. Whiteley, W.: Vertex splitting in isostatic frameworks. Struct. Topol. 16, 23–30 (1990)

    Google Scholar 

Download references

Acknowledgements

An extended abstract to this paper was presented at FPSAC2022 [13]. We thank the anonymous referees of both FPSAC2022 and of DCG for their helpful comments.

Author information

Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Eran Nevo & Simion Tarabykin

Authors
  1. Eran Nevo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simion Tarabykin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eran Nevo.

Additional information

Editor in Charge: János Pach

Dedicated to the memory of Eli Goodman.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Eran Nevo was partially supported by the Israel Science Foundation grants ISF-1695/15 and ISF-2480/20 and by ISF-BSF joint grant 2016288. Simion Tarabykin was partially supported by ISF grant 1695/15.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nevo, E., Tarabykin, S. Vertex Spanning Planar Laman Graphs in Triangulated Surfaces. Discrete Comput Geom 72, 912–927 (2024). https://doi.org/10.1007/s00454-023-00517-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-023-00517-w

Keywords

Mathematics Subject Classification

Search

Navigation