Art Galleries and Mobile Guards: Revisiting O'Rourke’s Proof

O'Rourke (1983) proved that every n-vertex polygon, with n ≥ 4, can be guarded by ⌊ n/4⌋ edges or diagonals - a variant of Chvátal’s theorem for sufficiency of ⌊ n/3⌋ vertices. We present a short proof for a somewhat stronger result that allows us to impose some constraints on the guards. We prove that for every given subset V of vertices, the polygon can be guarded by ⌊(n+2|V|)/4⌋ edges or diagonals that include at least one edge or diagonal incident to every vertex of V. This bound is the best achievable given the constraint for V. Our proof is by induction and suggests a simple linear-time algorithm after triangulating the polygon. The sufficiency of ⌊4⌋ guards is a special case of the new result where V is the empty set.