Let $P$ be a plane, and a simple closed curve $l$ on $P$ is separating if $P-l$ is not connected.
My question is as follows.
Question Let $G$ be a planar graph with connectivity $k$, and $\phi(G)$ be a plane embedding of $G$. If $ S:=\{u_1,u_2,\dots u_k\}$ is a $k$-separating set of $G$. Then $\phi(G)$ admits a simple separating closed curve $l$ on $P$ intersecting only $u_1,u_2 ,\dots,$ and $u_k$ which separates $P$ so that each component of $P-l$ contains at least one component of $G − S$.
The above question is observed from some specific examples below, in conjunction with the proof of Lemma 11 in a paper (Noguchi, K., Suzuki, Y. Relationship Among Triangulations, Quadrangulations and Optimal 1-Planar Graphs. Graphs and Combinatorics 31, 1965–1972 (2015)). But I don't know how to prove it rigorously.
Note: The three red dotted lines $l_i$ for $i\in \{1,2,3\}$ in the figures above represent the Jordan closed curves satisfying the conditions in the qustion, separately.
PS: Recently I have been reading some paper on planar graphs or 1-plane graphs, and I see that Jordan curves often play a key role for proving the connectivity of graphs. I wonder if there are some systematic papers or books that summarize examples of the role of Jordan curves in graph theory.
In general, you won't get a cut that contains each point in $S$, but only a subset. For the proof, take one component $C$ of $G \setminus S$, and let $S' \subseteq S$ be the points of $S$ adjacent to some vertex in $C$. Consider the subgraph induced by $C \cup S'$: its border defines a curve, which can be slightly enlarged to be a simple closed curve intersecting $\phi(G)$ only on $S'$.