Vertex spanning planar Laman graphs in triangulated surfaces

TL;DR

This paper proves that certain triangulated surfaces contain vertex-spanning planar Laman graphs, leading to results on rigid realizations of surface triangulations in the plane with limited vertex locations.

Contribution

It establishes the existence of vertex-spanning planar Laman graphs in triangulations of specific surfaces and derives bounds on rigid realizations with limited vertex placements.

Findings

Triangulations of the torus, projective plane, and Klein bottle contain vertex-spanning planar Laman graphs.

Every triangulation of a surface with nonnegative Euler characteristic admits a rigid planar realization with at most 26 vertices.

The results connect topological surface properties with rigidity and graph embedding theories.

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 Kir{\'a}ly, 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.