## ICS 163 - Graph Algorithms Homework 6, 50 Points Due: Wednesday, March 12, 2003

1. 10 points. Use Kuratowski's Theorem to prove that the following graph, called Petersen's graph, is nonplanar.

2. 10 points. Show how to draw each of the following graphs on the surface of a torus (i.e., a "donut" with a hole in the middle) so that no two two edges cross.
1. K5
2. K3,3
3. 10 points. Draw a biconnected planar graph G with 10 vertices that has a separating cycle C with 6 vertices such that there are 5 pieces with respect to C that induce an interlacement graph with 6 edges.
4. 10 points. Give an example of a biconnected graph G and a separating cycle C in G such that the interlacement graph for the pieces of G with respect to C has at least cn2 edges, for some constant c > 0.
5. 10 points. Let P be a piece of a biconnected graph with respect to a cycle C:
1. Show that if P has at least one vertex, the number of edges of P is great than or equal to the number of attachments of P.
2. Show that the graph obtained by adding P to C is biconnected.