Complexity and parameterized algorithms for Cograph Editing

Yunlong Liu, Jianxin Wang, Jiong Guo, Jianer Chen

Abstract: Cograph Editing is to find for a given 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> a set of at most k<math><mi is="true">k</mi></math> edge additions and deletions that transform G<math><mi is="true">G</mi></math> into a cograph. The computational complexity of this problem was open in the past. In this paper, we first show that this problem is NP-hard by a reduction from Exact 3-Cover. Subsequently, we present a parameterized algorithm based on a refined search tree technique with a running time of O(4.612k+|V|4.5)<math><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><mn is="true">4.61</mn><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup><mo is="true">+</mo><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">V</mi><mo is="true">|</mo></mrow></mrow><mrow is="true"><mn is="true">4.5</mn></mrow></msup><mo is="true">)</mo></mrow></math>, which improves the trivial algorithm of running time O(6k+|V|4.5)<math><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">6</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup><mo is="true">+</mo><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">V</mi><mo is="true">|</mo></mrow></mrow><mrow is="true"><mn is="true">4.5</mn></mrow></msup><mo is="true">)</mo></mrow></math>.