Two-sets cut-uncut on planar graphs

We study the following Two-Sets Cut-Uncut problem on planar graphs. Therein, one is given an undirected planar graph $G$ and two sets of vertices $S$ and $T$. The question is, what is the minimum number of edges to remove from $G$, such that we separate all of $S$ from all of $T$, while maintaining that every vertex in $S$, and respectively in $T$, stays in the same connected component. We show that this problem can be solved in time $2^{|S|+|T|} n^{O(1)}$ with a one-sided error randomized algorithm. Our algorithm implies a polynomial-time algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that Two-Sets Cut-Uncut remains fixed-parameter tractable even when parameterized by the number $r$ of faces in the plane graph covering the terminals $S \cup T$, by providing an algorithm of running time $4^{r + O(\sqrt r)} n^{O(1)}$.