Minimum embedding of a λ-fold P3-design into a λ-fold kite system for any index λ

Let G′ be a subgraph of a simple finite graph G and U be a subset of a set V . We say that a λ-fold G′ -design (U, C) of order u is embedded into a µfold G-design (V, B) of order u+w, µ ≥ λ, if there is an injective function f : C → B such that C is a subgraph of f(C) for every C ∈ C. The mapping f is called the embedding of (U, C) into (V, B). If w attains the minimum possible value, then f is a minimum embedding. In this paper a complete solution is given to the problem of determining a minimum embedding of a λ-fold P_3-design into a λ-fold kite system (λ = µ) for any index λ.