On the parameterized complexity of lineal topologies (depth-first spanning trees) with many or few leaves

Benjamin Bergougnoux, Nello Blaser, Michael R. Fellows, Petr A. Golovach, Frances A. Rosamond, Emmanuel Sam

Abstract: This paper considers four problems with possible applications in network design: Given a graph G with |G|=n<math><mo stretchy="false" is="true">|</mo><mi is="true">G</mi><mo stretchy="false" is="true">|</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">n</mi></math> and an integer k≥0<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">0</mn></math>, does G have a DFS tree with (i) ≤k leaves, (ii) ≥k leaves, (iii) ≤n−k<math><mo is="true">≤</mo><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">k</mi></math> leaves, and (iv) ≥n−k<math><mo is="true">≥</mo><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">k</mi></math> leaves? We show that all four problems are NP-hard. When parameterized by k, we prove that while (i) is para-NP-hard and (ii) is W[1]-hard, both (iii) and (iv) admit polynomial kernels with O(k3)<math><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> vertices, implying FPT algorithms running in kO(k)⋅nO(1)<math><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></mrow></msup><mo is="true">⋅</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math> time. Our polynomial kernels are based on a O(k)<math><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></math>-sized vertex cover structure associated with the solution of these problems. As a byproduct, we obtain polynomial kernels for these problems parameterized by the vertex cover number of the input graph.