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

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

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.

K_{5}

K_{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.

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 cn^{2} edges, for some constant c > 0.

10 points.
Let P be a piece of a biconnected graph with respect to a
cycle C:
 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.
 Show that the graph obtained by adding P to C is
biconnected.