Linear-vertex kernel for the problem of packing r-stars into a graph without long induced paths

Let integers r ≥ 2 and d ≥ 3 be fixed. Let G_d be the set of graphs with no induced path on d vertices. We study the problem of packing k vertex-disjoint copies of K_1,r (k ≥ 2) into a graph G from parameterized preprocessing, i.e., kernelization, point of view. We show that every graph G ∈ G_d can be reduced, in polynomial time, to a graph G′ ∈ G_d with O(k) vertices such that G has at least k vertex-disjoint copies of K_1,r if and only if G′ has. Such a result is known for arbitrary graphs G when r = 2 and we conjecture that it holds for every r ≥ 2.