Rounding for Vertex Cover

Recall the LP-rounding algorithm for weighted Vertex Cover:

\begin{aligned} \text{Minimize} \quad & \sum_{i=1}^{n} w(v_i)\,x_i \\ \text{subject to} \quad & x_i + x_j \ge 1 \quad \forall (v_i,v_j) \in E \\ & 0 \le x_i \le 1 \quad \forall i \end{aligned}

Then return $C = \{v_i \in V : x_i \ge 1/2\}$ .

Part A

Suppose we instead include $$v_i$$ when $x_i \ge 1/3$ . What happens?

Part B

Suppose we instead include $$v_i$$ when $x_i \ge 2/3$ . What happens?

Part C

Consider this alternative rounding rule: for each edge $$(v_i,v_j)$$ , include the endpoint with larger LP value (break ties toward larger index). Equivalently,
$C := \{ v_i : \exists (v_i,v_j)\in E \text{ with } (x_i > x_j) \text{ or } (x_i = x_j \text{ and } i > j)\}$ .

Which statement is true?