Exact embedding of two -designs into a -design

Let G be a connected simple graph and let S G 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 ∉ 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 ( V i , B i ) , i = 1 , 2 , are exactly embedded into ( X , D ) if X = V 1 ∪ V 2 , | V 1 ∩ V 2 | = 0 and there is a bijective mapping f : B 1 ∪ B 2 → D such that B is a subgraph of f ( B ) , for every B ∈ B 1 ∪ B 2 . 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 { v 1 , v 2 } ⊆ S G for which any two nontrivial G -designs ( V i , B i ) , | V i | = v i , i = 1 , 2 , can be exactly embedded into a ( G + e ) -design; (2) determine the pairs { v 1 , v 2 } ⊆ S G for which there exists a ( G + e ) -design of order v 1 + v 2 exactly embedding two nontrivial G -designs ( V i , B i ) , | V i | = v i , i = 1 , 2 . We study these problems for BIBDs, cycle systems, cube systems, path designs and star designs.