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 Gd be the set of graphs with no induced path on d vertices. We study the problem of packing k vertex-disjoint copies of K1,r (k≥2) into a graph G from parameterized preprocessing, i.e., kernelization, point of view. We show that every graph G∈Gd can be reduced, in polynomial time, to a graph G'∈Gd with O(k) vertices such that G has at least k vertex-disjoint copies of K1,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.