An addition to art galleries with interior walls

Abstract

Consider an art gallery formed by a polygon onn vertices withm pairs of vertices joined by interior diagonals, the interior walls. Each interior wall has an arbitrarily placed, arbitrarily small doorway. It is shown in [5] that the minimum number of guards that suffice to guard all art galleries withn vertices andm interior walls is min⌊(2n − 3)/3], ⌊(2m +n)/3⌋, ⌊(2n +m − 2)/4⌋. The proofs for the first two bounds lead to linear-time guard placement algorithms, while this is not known for the third case. We present an alternative short proof for the third upper bound ⌊(2n +m − 2)/4⌋ that also leads to a linear-time guard placement algorithm.