Improved linear problem kernel for planar connected dominating set

Weizhong Luo, Jianxin Wang, Qilong Feng, Jiong Guo, Jianer Chen

Abstract: In this paper, we study the Planar Connected Dominating Set problem, which, given a planar graph  G=(V,E)<math><mi is="true">G</mi><mo is="true">=</mo><mrow is="true"><mo is="true">(</mo><mi is="true">V</mi><mo is="true">,</mo><mi is="true">E</mi><mo is="true">)</mo></mrow></math> and a non-negative integer  k<math><mi is="true">k</mi></math>, asks for a subset  D⊆V<math><mi is="true">D</mi><mo is="true">⊆</mo><mi is="true">V</mi></math> with  ∣D∣≤k<math><mo is="true">∣</mo><mi is="true">D</mi><mo is="true">∣</mo><mo is="true">≤</mo><mi is="true">k</mi></math> such that  D<math><mi is="true">D</mi></math> forms a dominating set of  G<math><mi is="true">G</mi></math> and induces a connected graph. Answering an open question posed at the 2nd Workshop on Kernelization (WorKer 2010), we provide a kernelization algorithm for this problem, leading to a problem kernel with at most  130k<math><mn is="true">130</mn><mi is="true">k</mi></math> vertices, improving the previously best upper bound on the kernel size. To this end, we incorporate a vertex coloring technique with data reduction rules and introduce a type distinction of regions into the region decomposition framework, which allows a refined analysis of the region size.