Given an undirected graph G and a set A \subseteq 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) \setminus A. An A-path is called an (A,\ell)-path if the length of the path is exactly \ell. In the {\sc (A, \ell)-Path Packing} problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, \ell)-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) + \ell. 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(vc^2) vertices.

翻译：给定一个无向图G和一个顶点子集A ⊆ V(G)，一条A路径是指G中一条以A中两个不同顶点为起点和终点、且中间顶点属于V(G) \ A的路径。若一条A路径的长度恰好为ℓ，则称其为(A,ℓ)-路径。在{\sc (A, ℓ)-路径打包}问题（ALPP）中，我们需要判定图G中是否存在k条顶点不相交的(A,ℓ)-路径。本文从结构参数角度研究该问题。我们证明当以参数组合“到路径的距离（dtp）+ |A|”作为参数时，ALPP是W[1]-难的。此外，我们考虑了参数组合“到簇的距离（cvd）+ |A|”以及“到簇的距离（cvd）+ ℓ”。针对这两种参数组合，我们给出了固定参数可解（FPT）算法。最后，我们以顶点覆盖数（vc）作为参数，构造了一个具有O(vc²)个顶点的核。