A deterministic polynomial kernel for Odd Cycle Transversal and Vertex Multiway Cut in planar graphs

We show that Odd Cycle Transversal and Vertex Multiway Cut admit deterministic polynomial kernels when restricted to planar graphs and parameterized by the solution size. This answers a question of Saurabh. On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs (FOCS 2014). It differs from the previous work because it preserves the existence of low-cost subgraphs that are not necessarily Steiner trees in the original plane graph, but structures that turn into (supergraphs of) Steiner trees after adding all edges between pairs of vertices that lie on a common face. We also show connections between Vertex Multiway Cut and the Vertex Planarization problem, where the existence of a polynomial kernel remains an important open problem.