Realization of knots and links in a spatial graph

For a graph G, let Γ be either the set Γ1 of cycles of G or the set Γ2 of pairs of disjoint cycles of G. Suppose that for each γ∈Γ, an embedding φγ :γ→S3 is given. A set {φγ∣γ∈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∈Γ. A graph is adaptable if any set {φγ∣γ∈Γ1} is realizable. In this paper, we have the following three results: r the complete graph K5 on 5 vertices and the complete bipartite graph K3,3 on 3+3 vertices, we give a necessary and sufficient condition for {φγ∣γ∈Γ1} to be realizable in terms of the second coefficient of the Conway polynomial. r a graph in the Petersen family, we give a necessary and sufficient condition for {φγ∣γ∈Γ2} to be realizable in terms of the linking number. e set of non-adaptable graphs all of whose proper minors are adaptable contains eight specified planar graphs.