We define Ak-moves for
embeddings of a finite graph into the 3-sphere for each natural number k. Let
Ak-equivalence denote an equivalence relation generated by Ak-moves and ambient
isotopy. Ak-equivalence implies Ak−1-equivalence. Let ℱ be an Ak−1-equivalence class
of the embeddings of a finite graph into the 3-sphere. Let 𝒢 be the quotient set of ℱ
under Ak-equivalence. We show that the set 𝒢 forms an abelian group under a certain
geometric operation. We define finite type invariants on ℱ of order (n;k). And we
show that if any finite type invariant of order (1;k) takes the same value on two
elements of ℱ, then they are Ak-equivalent. Ak-move is a generalization of Ck-move
defined by K. Habiro. Habiro showed that two oriented knots are the same up
to Ck-move and ambient isotopy if and only if any Vassiliev invariant of
order ≤ k − 1 takes the same value on them. The ‘if’ part does not hold for
two-component links. Our result gives a sufficient condition for spatial graphs to be
Ck-equivalent.
