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 P k ( u , λ ) into a K S ( u , μ ) , with k = 2 , 3 , 4 .