**Polyhedra and Graphs**

Prepared by:

Joseph Malkevitch

Department of Mathematics and Computer Studies

York College (CUNY)

Jamaica, New York 11451

Email: malkevitch@york.cuny.edu (for additions, suggestions, and corrections)

1. For each of the graphs below:

a. Is the graph the vertex-edge graph of some convex polyhedron? Is the graph the vertex-edge graph of some non-convex polyhedron?

b. How many vertices does the graph have? How many edges? How many faces?

c. What are the valences of the vertices? What are the number of sides that the faces of the polyhedron (assuming that it is the graph of a polyhedron) have?

d. If the graph has crossings at point that are not vertices as it is drawn, redraw the graph in the plane so that such crossings are avoided.

e. For each graph which is already shown in a plane embedding, (or from your redrawing as above) draw an isomorphic plane graph with a face having a different number of sides as the infinite face.

f. Draw the dual of each of the graphs after you redraw them so that they have been embedded in the plane. Is the dual in each case a simple graph? If not, what was the problem.