The parametric complexity of graph diameter augmentation

Yong Gao, Donovan R. Hare, James Nastos

Abstract: The diameter of a graph is the maximum distance between any pair of vertices in the graph. The Diameter-t<math><mi is="true">t</mi></math>Augmentation problem takes as input a 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 positive integer k<math><mi is="true">k</mi></math> and asks whether there exists a set E2<math><msub is="true"><mrow is="true"><mi is="true">E</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math> of at most k<math><mi is="true">k</mi></math> new edges so that the graph G2=(V,E∪E2)<math><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><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><msub is="true"><mrow is="true"><mi is="true">E</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mo is="true">)</mo></mrow></math> has diameter t<math><mi is="true">t</mi></math>. This problem is NP-hard (Schoone et al. 1987) [10], even in the t=2<math><mi is="true">t</mi><mo is="true">=</mo><mn is="true">2</mn></math> case (Li et al. 1992) [7]. We give a parameterized reduction from Dominating Set to Diameter-t<math><mi is="true">t</mi></math>Augmentation to prove that Diameter-t<math><mi is="true">t</mi></math>Augmentation is W[2]<math><mi is="true">W</mi><mrow is="true"><mo is="true">[</mo><mn is="true">2</mn><mo is="true">]</mo></mrow></math>-hard for every t<math><mi is="true">t</mi></math>.