Art Galleries with Interior Walls

Abstract

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 arbi- trarily 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 \(\min\{ \lfloor (2n-3)/3\rfloor\) , \(\lfloor (2n+m-2)/4\rfloor, \lfloor (2m+n)/3\rfloor\}.\) If we restrict ourselves to galleries with convex rooms of size at least r , the answer improves to \(\min\{m,\lfloor{(n+m)/r}\rfloor.\}\) The proofs lead to linear time guard placement algorithms in most cases.