Abstract
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...
Highlights
Kernelization provides a rigorous framework within the paradigm of parameterized complexity to analyze preprocessing routines for various combinatorial problems
Of particular importance are polynomial kernels, where the function g is required to be a polynomial, that are interpreted as theoretical tractability of preprocessing for the considered problem Π
We focus on the vertex-deletion variant of Multiway Cut with undeletable terminals
Summary
Kernelization provides a rigorous framework within the paradigm of parameterized complexity to analyze preprocessing routines for various combinatorial problems. Odd Cycle Transversal and Vertex Multiway Cut, when restricted to planar graphs and parameterized by the solution size, admit deterministic polynomial kernels. The main technical result of [25] is a sparsification routine that, given a connected plane graph G with infinite face surrounded by a simple cycle ∂G, provides a subgraph of G of size polynomial in the length of ∂G that, for every A ⊆ V (∂G), preserves an optimal Steiner tree connecting A Both Odd Cycle Transversal and Vertex Multiway Cut in a plane graph G translate into Steiner forest-like questions in the overlay graph L(G) of G: a supergraph of G that has a vertex vf for every face of G, adjacent to every vertex of G incident with f. We first use known LP-based rules [4, 12, 14, 26] to reduce the number of terminals and neighbors of terminals to O(k) and use an argument based on outerplanarity layers to reduce the diameter
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