Proof that a graph isn't hamiltonian

Question

I can't do that. I have this graph. I think that it doesn't contain hamiltonian cycle.

I have no idea how I can proof it. I can't find proper subset $S$ to do it with Chvatal-Erdos theorem. I can't use Ore and Dirac theorem because, I can't conclude that "a graph is not hamiltonian because it doesn't meet Ore & Dirac conditions" - it's not true, so actually I don't know any other theorems I could use. I will really appreciate any kind of help, hint, anything.

Answer 1

Each of the edges incident on the degree $2$ vertices must be used in any Hamilton cycle. Then, in any Hamilton cycle, the top vertex has degree at least $3$, which is impossible...