New kernels for several problems on planar graphs

Guanlan Tan, Qilong Feng, Beilin Zhuo, Neng Huang, Jianxin Wang

Abstract: In this paper, we study the kernelization of the Induced Matching problem on planar graphs, the Parameterized Planar 4-Cycle Transversal problem and the Parameterized Planar Edge-Disjoint 4-Cycle Packing problem. For the Induced Matching problem on planar graphs, based on the Gallai-Edmonds decomposition structure, a kernel of size 26k is presented, which improves the previous best result 28k. For the Parameterized Planar 4-Cycle Transversal problem, by partitioning the vertices in a given instance into several parts and analyzing the size of each part independently, a kernel with at most 51k−22<math><mn is="true">51</mn><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">22</mn></math> vertices is obtained, which improves the previous best result 74k. Based on the kernelization process of the Parameterized Planar 4-Cycle Transversal problem, a kernel of size 51k−22<math><mn is="true">51</mn><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">22</mn></math> can also be obtained for the Parameterized Planar Edge-Disjoint 4-Cycle Packing problem, which improves the previous best result 96k.