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 [Pilipczuk et al., ACM Trans. Algorithms, 14 (2018), 53]. 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.


  1. planar graphs
  2. polynomial kernel
  3. odd cycle transversal
  4. vertex multiway cut

MSC codes

  1. 68Q25
  2. 68W40
  3. 68R10
  4. 05C10
  5. 05C85

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Information & Authors


Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 2387 - 2429
ISSN (online): 1095-7146


Submitted: 17 July 2020
Accepted: 7 June 2021
Published online: 18 October 2021


  1. planar graphs
  2. polynomial kernel
  3. odd cycle transversal
  4. vertex multiway cut

MSC codes

  1. 68Q25
  2. 68W40
  3. 68R10
  4. 05C10
  5. 05C85



Funding Information

Fundacja na rzecz Nauki Polskiej https://doi.org/10.13039/501100001870

Funding Information

Nederlandse Organisatie voor Wetenschappelijk Onderzoek https://doi.org/10.13039/501100003246

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