The spectrum of and with any

Let ( X , B ) be a ( λ K v , G 1 ) -design and G 2 a subgraph of G 1 . Define sets B ( G 2 ) and D ( G 1 ∖ G 2 ) as follows: for each block B ∈ B , partition B into copies of G 2 and G 1 ∖ G 2 and place the copy of G 2 in B ( G 2 ) and the edges belonging to the copy of G 1 ∖ G 2 in D ( G 1 ∖ G 2 ) . If the edges belonging to D ( G 1 ∖ G 2 ) can be assembled into a collection D ( G 2 ) of copies of G 2 , then ( X , B ( G 2 ) ∪ D ( G 2 ) ) is a ( λ K v , G 2 ) -design, called a metamorphosis of the ( λ K v , G 1 ) -design ( X , B ) . For brevity we denote such ( λ K v , G 1 ) -design ( X , B ) with a metamorphosis into ( λ K v , G 2 ) -design ( X , B ( G 2 ) ∪ D ( G 2 ) ) by ( λ K v , G 1 > G 2 ) -design. Let Meta ( G 1 > G 2 , λ ) denote the set of all integers v such that there exists a ( λ K v , G 1 > G 2 ) -design. In this paper we completely determine the set Meta ( K 3 + e > P 4 , λ ) or Meta ( K 3 + e > H 4 , λ ) when the admissible conditions are satisfied, for any λ .