Angles of planar triangular graphs

We give a characterization of all the planar drawings of a triangular graph through a system of equations and inequalities relating its angles, solving a problem that is explicitly mentioned as open by several authors; we also discuss minimalit y properties of the charact erization. The characterization can be used: (1) to decide in linear time whether a given distribution of angles between the edges of a planar triangular graph can result in a planar drawing; (2) to tackle the problem of maximizing the minimum angle of the drawing of a planar triangular graph by studying the solution-space of a non-linear optimization problem; (3) to give a characterization of the planar drawings of a triconnected graph through a system of equations and inequalities relating its angles; (4) to give a characterization of Delaunay triangulations through a system of equations and inequalities relating its angles; (5) to give a characterization of all the planar drawings of a triangular graph through a system of equations and inequalities relating the length of its edges; in turn, this result allows to give a new characterization of the disc packing represent ations of planar triangular graphs.

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