3-Colorability of Pseudo-Triangulations

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain classes of plane graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations (a generalization of triangulations) and prove NP-completeness for this class. The complexity status does not change if the maximum face-degree is bounded to four, or pointed pseudo-triangulations with maximum face degree five are treated. As a complementary result, we show that for pointed pseudo-triangulations with maximum face-degree four, a 3-coloring always exists and can be found in linear time.