Art galleries with interior walls

Consider an art gallery formed by a polygon on n vertices with m pairs of vertices joined by interior diagonals, the interior walls. Each interior wall has an arbitrarily placed, arbitrarily small doorway. We show that the minimum number of guards that suffice to guard all art galleries with n vertices and m interior walls is min {[(2n - 3)/3], [(2m + n - 2)/4], [(2m + n)/3]}. If we restrict ourselves to galleries with convex rooms of size at least r, the answer improves to min{m, [(n + m)/r]}. The proofs lead to linear time guard placement algorithms in most cases.