The Doyen-Wilson theorem for kite systems

Let G be a simple graph. A G-design of order n is a pair (X, B) where B is a collection of subgraphs (blocks), each isomorphic to G, which partitions the edge set of the complete undirected graph K_n with vertex set X. The G-design (X_1,B_1) is said to be embedded in the G-design (X_2,B_2) provided X_2 and B_2 containe X_1 and B_2, respectively ((X_1,B_1) is also said a subdesign of (X_2,B_2)). Let N(G) denote the set of integers n such that there exists a G-design of order n. A natural question to ask is: given n,m in N(G), with m > n, and a G-design (X, B) of order n, does exists a G-design of order m containing (X, B) as subdesign? Doyen and Wilson were the first to pose this problem for G = K_3 (Steiner triple systems) and in 1973 they showed that given n,m in N(K_3) then any Steiner triple system of order n can be embedded in a Steiner triple system of order m if and only if m > 2n + 1 or m = n. Over the years, any such problem has come to be called a "Doyen-Wilson problem". Here a complete solution of the Doyen-Wilson problem is given for kite systems.