The problem concerns the construction of a polygon, and a placement of guards within the polygon, such that all edges of the polygon are visible but there remains an invisible interior region. In the applets linked from this page, large points represent vertex guards and small points represent unguarded vertices. The yellow areas are those visible to the guards.

This problem can be solved with a 6-vertex polygon. It is not possible with any polygon having fewer vertices.

3-Vertex Polygons: All 3-vertex polygons are convex, and in a convex polygon all points are visible with only 1 guard. Therefore no convex polygon (and by extension, no 3-vertex polygon) can pose a solution to this problem.
4-Vertex Polygons: There is only one type of non-convex 4-vertex polygon, that which has a V-shape. The applet shows a V-shaped 4-gon with 1 guard. It is clear that no matter how the points are moved, 1 guard is insufficient to cover the edges (assuming that the reflex angle is maintained), and 2 guards would inevitably cover all points. Therefore there is no 4-gon that poses a solution to this problem.
5-Vertex Polygons: A 5-vertex polygon with 1 reflex angle will be some variation on the 4-vertex V-shape, and for this purpose can be considered the same as the 4-vertex version. So, we are limited to considering 5-vertex polygons with 2 reflex angles. Some variations within this constraint are possible, but in all cases 1 or 2 guards are insufficent to cover the egdes, and 3 guards cover all points. Therefore there is no 5-gon that poses a solution to the problem.