**Review Examination II (Spring 2006)**

Geometric Structures, Mathematics 244 (Part II)

prepared by:

Joseph Malkevitch

Department of Mathematics and Computing

York College (CUNY)

Jamaica, New York 11451

1. For each of the polygons below:

a. List all of the ears.

b. Find the visibility polygons of vertices 1 and 4.

c. Find the minimum number of guards to guard the polygon.

d. Triangulate the polygon in two different ways, and find the smallest guard set associated with a 3-coloring of the triangulated graph. How do these guard sets compare in size with a minimum size guard set?

e. List all of the reflex vertices of each polygon.

f. How many vertices does the convex hull of each of these polygons have?

g. What is the difference between visibility and clear visibility?

h. How many edges must be added to the polygon to triangulate it?

i. How many triangles are there in the triangulation?

j. What is the angle sum of the interior angles of the polygon?

k. Find the smallest number of convex pieces that you can decompose the polygon into.

2. a. For the graphs below compute the values of v_{i} and p_{i}.

b. Find the vertex coloring number for each of the graphs.

c. Find the face coloring number for each of the graphs.

d. Verify Euler's formula for each of the graphs.

3. a. Draw a graph which is planar but not plane.

b. Draw a 3-valent plane graph which is not the graph of some 3-polytope.

c. Give an exact statement of Steinitz's Theorem

d. Draw a 4-valent 3-polytopal graph.

e. Draw a 5-valent 3-polytopal graph.

f. Draw graphs of the five Regular or Platonic polyhedra.