Sum of variables in graph optimum

Let $s(W) = \displaystyle \sum_{v\in W} x_v$ be the sum over a subset $W$. $s(V)$ is the total sum of an optimal solution. As we also have an optimal solution by changing all the sign, we can suppose that $s(V)\geq 0$. Suppose by contradiction that we have $s(V)>0$. Lets choose a vertex $v$ with negative value (If all vertices have a positive value and $|V|>1|$, we can change the sign of the smallest value while keeping a valid solution with the same objective value and $s(V)$ still positive). If for every admissible subset $W$, $v\in W$, $s(W)$ is strictly more than $-1$, then we can decrease the value of $v$. So the solution is optimal only if there exists an admissible $W$, $v\in W$ with $s(W) = -1$. However, $V\setminus W$ is also an admissible subset. So we have $s(V) = s(W) + s(V\setminus W)$, and $s(V\setminus w) = s(V) - s(W) > 1$, which is a contradiction. So $s(V) = 0$