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

We prove the following graph coloring result Let G be a connected bipartite planar graph Then one can triangulate G in such a way that the resulting graph is colorable This result implies several new upper bounds for guarding problems including the rst non trivial upper bound for the rectilinear Prison Yard Problem n vertex guards are su cient to watch the interior of a rectilinear polygon with holes n vertex guards resp n point guards are su cient to watch simultaneously both the interior and exterior of a rectilinear polygon Moreover we show a new lower bound of n vertex guards for the rectilinear Prison Yard Problem and prove it to be asymptotically tight for the class of orthoconvex polygons Institut f ur Informatik Freie Universit at Berlin Takustr D Berlin ho mann tcs fu berlin de kriegel tcs fu berlin de Both authors have been partially supported by the ESPRIT Basic Research Action project ALCOM II and the Wissenschaftler Integrationsprogramm Berlin