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+c). Tight combinatorial bounds have previously been shown when 0≤r≤⌊c2⌋ and when r≥5c-12. We give a lower bound construction that matches the ⌊n3⌋ sufficiency condition from the traditional art gallery theorem when ⌊c2⌋<r<5c-12, thereby providing tight combinatorial bounds for all r and c≥3.