Minimum embedding of path designs into kite systems

Let G a simple graph and H be a subgraph of G, and let U⊆V. We say that a λ-fold H-design (U,C) of order u is embedded into a μ-fold G-design (V,B) of order u+w, if there is a injective function f:C→B such that B is a subgraph of f(B) for every B∈C. If f:C→B is bijective, the embedding is called exact. In this paper we solve the embedding problem and the exact embedding problem of a Pk(u,λ) into a KS(u,μ), with k=2,3,4.