A Graph-Coloring Result and Its Consequences For Polygon-Guarding Problems

The following graph-coloring result is proved: let $G$ be a 2-connected, bipartite, and plane graph. Then one can triangulate $G$ in such a way that the resulting graph is 3-colorable. Such a triangulation can be computed in $O(n^2)$ time. This result implies several new upper bounds for polygon guarding problems, including the first nontrivial upper bound for the rectilinear prison yard problem. (1) $\lfloor{n}/{3}\rfloor$ vertex guards are sufficient to watch the interior of a rectilinear polygon with holes. (2) $\lfloor{5n}/{12}\rfloor +3$ vertex guards ($\lfloor{n+4}/{3}\rfloor $ point guards) are sufficient to simultaneously watch both the interior and exterior of a rectilinear polygon. Moreover, a new lower bound of $\lfloor{5n}/{16}\rfloor$ vertex guards for the rectilinear prison yard problem is shown and proved to be asymptotically tight for the class of orthoconvex polygons.