Prove about Hamiltonian I need to prove if graph $G$ contains a cut vertex, then it is not Hamiltonian. I get that it wouldn't be Hamiltonian because to be Hamiltonian you have to have a Hamiltonian cycle and if you have a cut vertex there's no possible way to have a cycle including that vertex. Is it true argument ?

Basically, yes. If you remove the cut vertex, the graph falls into disconnected pieces. But any Hamiltonian cycle may be converted to a Hamiltonian path (in a different graph) by removing any single vertex; remove the cut vertex and we get a disconnected graph, which cannot have a Hamiltonian path.
