Twosets cutuncut on planar graphs
Abstract
We study the following TwoSets CutUncut 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 onesided error randomized algorithm. Our algorithm implies a polynomialtime algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that TwoSets CutUncut remains fixedparameter 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 $4^{r + O(\sqrt r)} n^{O(1)}$.
 Publication:

arXiv eprints
 Pub Date:
 May 2023
 DOI:
 10.48550/arXiv.2305.01314
 arXiv:
 arXiv:2305.01314
 Bibcode:
 2023arXiv230501314B
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 22 pages, 5 figures