Basketball and a fun waste of time

It's a great time for my basketball teams.  The Wash. U. men just won the Division III national championship.  The Texas A&M men are leading no. 2 seed UCLA at halftime and would meet a low seed in the next round.  The A&M women (notice the "whoops" in the URL!) are a no. 2 seed and are cruising so far.  The North Carolina men are top-ranked and favored to win it all; the women are ranked second in the country.  (My favorite NBA team recently won 22 straight games, but who cares about the NBA in March?)

I'm not much for online games, but Untangle is really fun.  As a computer scientist would see it, you're basically given an undirected graph and are asked to construct a proof that it's planar.  I finally got to the EVIL level and beat it; can you?  What strategies work for you?