I think there are no such graphs. 

It was shown by Sumner and Las Vergnas (you can find the references here: http://mathsci.kaist.ac.kr/~sangil/pdf/2009claw.pdf) that a claw-free connected graph has a perfect matching (assuming even number of vertices, of course!).  An intersection of three bridges is clearly a claw.

Cubic graphs without a perfect matching and a vertex incident to three bridges
I've been GMing for a couple of sessions now and I'm still struggling at making encounters or situations more difficult. I will describe dangerous situations and encounters in fiction but the mechanics make it too easy for my players to succeed.

For instance no matter how big I describe the Troll, if the fighter of the party charges him and swing his sword, roll+2 for Hack and slash, he'll easily hit the troll. Almost 40% chance to hit without consequences.
