# Minimum vertex-separators under edge addition

I am trying to prove the following claim.

Let $$T$$ be a minimum $$st$$-separator in an undirected graph $$G$$, and let $$x \in T$$. Let $$S\neq T$$ be a minimal $$st$$-separator (i.e., not necessarily minimum), where $$C_s$$ and $$C_t$$ are the connected components of $$G-S$$ containing vertices $$s$$ and $$t$$ respectively. Suppose that $$x \in C_s$$, and let $$y\in C_t$$. Let $$G'$$ be the graph that results from $$G$$ by adding the edge $$(y,t)$$. Then there is a minimum $$st$$-separator $$T'$$ in $$G'$$ that contains $$x$$ (i.e., $$x\in T'$$).

The claim clearly holds if the vertex-connectivity of $$G'$$ is greater than the vertex-connectivity of $$G$$, because the connectivity can grow by at most one. I cannot prove that this holds in the case where the connectivity of $$G$$ and $$G'$$ is the same.