Art gallery theorems for guarded guards

We prove two art gallery theorems in which the guards must guard one another in addition to the gallery. A set G of points (the guards) in a simple closed polygon (the art gallery) is a guarded guard set provided (i) every point in the polygon is visible to some point in G; and (ii) every point in G is visible to some other point in G. We prove that a polygon with n sides always has a guarded guard set of cardinality (3n− 1)/7 and that this bound is sharp (n 5); our result corrects an erroneous formula in the literature. We also use a coloring argument to give an entirely new proof that the corresponding sharp function for orthogonal polygons is n/3 for n 6; this result was originally established by induction by Hernández-Peñalver.  2003 Elsevier B.V. All rights reserved.