On the Parameterized Tractability of Packing Vertex-Disjoint A-Paths with Length Constraints
Abstract
Given an undirected graph G and a set A ⊂eq V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G) A. An A-path is called an (A,)-path if the length of the path is exactly . In the (A, )-Path Packing problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, )-paths in G or not. We pursue this problem with respect to structural parameters. We prove that ALPP is W[1]-hard when it is parameterized by the combined parameter distance to path (dtp) and |A|. In addition, we consider the combined parameters distance to cluster (cvd) + |A| and distance to cluster (cvd) + . For both these combined parameters, we provide FPT algorithms. Finally, we consider the vertex cover number (vc) as the parameter and provide a kernel with O(vc2) vertices.