Question

Show that g(5)= 1. That is, show that all pentagons can be guarded using one point. (Hint:Show that there are either 0, 1, or 2 vertices with an interior angle greater than 180 degrees and that in each case, one guard suffices.

Short answer

It is Proved that g(5)=1 shows that all Pentagons can be guarded using one point.

Step-by-step solution

Given Information

Pentagons

Related Definitions

x covers or sees a point y, if the entire line segment xy lies in the simple polygon P or n its boundary.

Guarding set of a simple polygon P is every point y in P, there exists a point z in the guarding set that sees y.

G(P)= minimum value of G(P) of points required in the guarding set of P

g(n)= maximum values of G(P) over all simple polygons P with n vertices.

Proving that g(3)=1 and g(4)=1 by showing that all triangles can be guarded using one point

We follow the hint,

Because the measure of the interior angles of the pentagon is total 540, there can not be as many as three interior angles of measure more than 180 (reflex angles).

If there are no reflex angles, then the pentagon is convex, and a guard placed at any vertex can see all points.

If there is one reflex angle, then the pentagon must look essentially like figure (a) below, and a guard at a vertex v can see all points.

If there are two reflex angles, then they can be adjacent or non-adjacent (figures (b) and (c)); in either case a guard at vertex v can see all points (In figure (c), choose the reflex vertex closer to the bottom side).

Thus, all Pentagons, one guard suffices, g(5)=1.

Hence proved ,that g(5)=1 shows that all Pentagons can be guarded using one point.