Exact embedding of two G-designs into a (G+e)-design

Let G be a connected simple graph and let SG be the spectrum of integers v for which there exists a G-design of order v. Put e = {x, y}, with x ∈ V (G) and y 6∈ V (G). Denote by G + e the graph having vertex set V (G) ∪ {y} and edge set E(G) ∪ {e}. Let (X,D) be a (G+e)-design. We say that two G-designs (Vi,Bi), i = 1, 2, are exactly embedded into (X,D) if X = V1 ∪ V2, |V1 ∩ V2| = 0 and there is a bijective mapping f : B1∪B2 → D such that B is a subgraph of f (B), for every B ∈ B1 ∪ B2. We give necessary and sufficient conditions so that two G-designs can be exactly embedded into a (G + e)-design. We also consider the following two problems: 1) determine the pairs {v1, v2} ⊆ SG for which any two nontrivial G-designs (Vi,Bi), |Vi| = vi i = 1, 2, can be exactly embedded into a (G+ e)-design; 2) determine the pairs {v1, v2} ⊆ SG for which there exists a (G+e)-design of order v1 + v2 exactly embedding two nontrivial G-designs (Vi,Bi), |Vi| = vi, i = 1, 2. We study these problems for BIBDs, cycle systems, cube systems, path designs and star designs.