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Vertex Spanning Planar Laman Graphs in Triangulated Surfaces

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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.

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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.

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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.

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

    Eran Nevo & Simion Tarabykin

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  1. Eran Nevo
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  2. Simion Tarabykin
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Correspondence to Eran Nevo.

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Dedicated to the memory of Eli Goodman.

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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.

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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

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