Realization of knots and links in a spatial graph

For a graph G, let Γ be either the set Γ₂ of cycles of G or the set Γ₂ of pairs of disjoint cycles of G. Suppose that for each γ ε Γ, an embedding φ_γ : γ → S³ is given. A set {φγ \ γ ε Γ) is realizable if there is an embedding f : G → S³ such that the restriction map f\γ is ambient isotopic to φγ for any γ ε Γ. A graph is adaptable if any set {φ_γ \ γ ε Γ₁] is realizable. In this paper, we have the following three results: (1) For the complete graph K₅ on 5 vertices and the complete bipartite graph K_3,3 on 3 + 3 vertices, we give a necessary and sufficient condition for {φγ \ γ ε Γ₁} to be realizable in terms of the second coefficient of the Conway polynomial. (2) For a graph in the Petersen family, we give a necessary and sufficient condition for {φ_γ | γ ε Γ₂) to be realizable in terms of the linking number. (3) The set of non-adaptable graphs all of whose proper minors are adaptable contains eight specified planar graphs.