# Prove: If in all subgraphs of $G$ there is a vertex of degree $<2$ then $G$ is a forest

I need help proving this:

Given a graph $G$, prove that if in all subgraphs of $G$ there is a vertex of degree less than $2$ ($1$ or $0$) then $G$ is a forest.

-
Where do you get stuck? Plug in the definition of forest, tree, subgraph. –  Hagen von Eitzen Jun 19 '13 at 16:38
Assume there is a cycle in $G$... –  Damian Sobota Jun 19 '13 at 16:39
Is it sufficient to prove that in all connected components of $G$ there is no cycle and that means it is a forest? –  TheNotMe Jun 19 '13 at 16:45
What can those connected components look like? Consider subgraphs with three vertices. –  dfeuer Jun 19 '13 at 16:48
Thank you so much for the hint, Damian Sobota. I proved it. –  TheNotMe Jun 19 '13 at 16:53