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

Let GG be a connected simple graph and let SGSG be the spectrum of integers vv for which there exists a GG-design of order vv. Put e={x,y}e={x,y}, with x∈V(G)x∈V(G) and y∉V(G)y∉V(G). Denote by G+eG+e the graph having vertex set V(G)∪{y}V(G)∪{y} and edge set E(G)∪{e}E(G)∪{e}. Let (X,D)(X,D) be a (G+e)(G+e)-design. We say that two GG-designs (Vi,Bi)(Vi,Bi), i=1,2i=1,2, are exactly embedded into (X,D)(X,D) if X=V1∪V2X=V1∪V2, |V1∩V2|=0|V1∩V2|=0 and there is a bijective mapping f:B1∪B2→Df:B1∪B2→D such that BB is a subgraph of f(B), for every B∈B1∪B2B∈B1∪B2. We give necessary and sufficient conditions so that two GG-designs can be exactly embedded into a (G+e)(G+e)-design. We also consider the following two problems: (1) determine the pairs {v1,v2}⊆SG{v1,v2}⊆SG for which any two nontrivial GG-designs (Vi,Bi)(Vi,Bi), |Vi|=vi|Vi|=vi, i=1,2i=1,2, can be exactly embedded into a (G+e)(G+e)-design; (2) determine the pairs {v1,v2}⊆SG{v1,v2}⊆SG for which there exists a (G+e)(G+e)-design of order v1+v2v1+v2 exactly embedding two nontrivial GG-designs (Vi,Bi)(Vi,Bi), |Vi|=vi|Vi|=vi, i=1,2i=1,2. We study these problems for BIBDs, cycle systems, cube systems, path designs and star designs.