Minimum embedding of a P4-design into a balanced incomplete block design of index lambda

Let H be a subgraph of G. An H-design (V, C) of order v and index μ is embedded into a G-design (X,B) of order v + w and index λ if μ ≤ λ, V ⊆ X and there is an injective mapping f : C → B such that B is subgraph of f (B) for every B ∈ C. For every pair of positive integers v, λ, (except when λ = 3 and v = 30, 34, 42, 46, 54, 58, 66 or λ = 5 and v = 19) we determine the minimum value of w such that there exists a balanced incomplete block design of order v+w, index λ and block-size 4 which embeds a P4-design of order v and index μ = 1 (P4 denotes the path of length 3). AMS classification: 05B05.