In the Project Euler problem

Tom and Jerry, it defines Tom graphs. A cat named Tom and a mouse named Jerry play a game on a graph G. Each vertex of G is a mouse hole. Jerry hides in one of the holes. Then, Tom tries to catch Jerry by picking a hole. Then, Jerry moves to an adjacent hole. Then, Tom picks another hole. They keep doing this, where Jerry moves to an adjacent hole and Tom picks a hole. A Tom graph is a graph that Tom can always catch Jerry by following a sequence of holes.

Example: This is a Tom graph.

1---2---3

Tom can do the sequence 2, 2 to guarantee catching Jerry.

1. Prove that all Tom graphs are cycle-free.

2. Find the smallest cycle-free graph that is not a Tom graph.