The art gallery theorem for simple polygons in terms of the number of reflex and convex vertices

We present an art gallery theorem for simple polygons having n vertices in terms of the number, r, of reflex vertices and the number, c  , of convex vertices (n=r+cn=r+c). Tight combinatorial bounds have previously been shown when 0⩽r⩽⌊c2⌋ and when r⩾5c−12r⩾5c−12. We give a lower bound construction that matches the ⌊n3⌋ sufficiency condition from the traditional art gallery theorem when ⌊c2⌋

► Art gallery theorem for simple polygons in terms of the number of reflex vertices (r), and convex vertices (c). ► Algorithm for generating lower bound constructions that match corresponding upper bounds for certain ranges of r and c is given. ► The lower bound constructions interpolate between previously known constructions for specific ranges of r and c.