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Séminaire Lotharingien de Combinatoire, 86B.43 (2022), 12 pp.

# Eran Nevo and Simion Tarabykin

# Vertex Spanning Planar Laman Graphs in Triangulated Surfaces

**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.

Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.

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