Two-sets cut-uncut on planar graphs

05/02/2023

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by   Matthias Bentert, et al.

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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 ∪ T, by providing an algorithm of running time 4^r + O(√(r)) n^O(1).