The spectrum of Meta(K-3 + e > P-4, lambda) and Meta(K-3 + e > H-4, lambda) with any lambda

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 {set minus} G_2) as follows: for each block B ∈ B, partition B into copies of G_2 and G_1 {set minus} G_2 and place the copy of G_2 in B (G_2) and the edges belonging to the copy of G_1 {set minus} G_2 in D (G_1 {set minus} G_2). If the edges belonging to D (G_1 {set minus} 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, G1 > G2)-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 (K3 + e > P_4, λ) or Meta (K3 + e > H_4, λ) when the admissible conditions are satisfied, for any λ.