Plane graphs with straight edges whose bounded faces are acute triangles (Q1400958)

Let \(T_n\) denote a graph obtained as a triangulation of an \(n\)-gon in the plane. A cycle of \(T_n\) is called an enclosing cycle if at least one vertex lies inside the cycle. In this paper it is proved that a \(T_n\) admits a straight-line embedding in the plane whose bounded faces are all acute triangles if and only if \(T_n\) has no enclosing cycle of length at most 4. Those \(T_n\) that admit straight-line embeddings in the plane without obtuse triangle are also characterized.