We define Akβ-moves for embeddings of a finite graph into the 3-sphere for
each natural number k. Let Akβ-equivalence denote an equivalence relation
generated by Akβ-moves and ambient isotopy. Akβ-equivalence implies
Akβ1β-equivalence. Let F be an Akβ1β-equivalence class of the
embeddings of a finite graph into the 3-sphere. Let G be the quotient
set of F under Akβ-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 Akβ-equivalent. Akβ-move is a generalization of
Ckβ-move defined by K. Habiro. Habiro showed that two oriented knots are the
same up to Ckβ-move and ambient isotopy if and only if any Vassiliev
invariant of order β€kβ1 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 Ckβ-equivalent.Comment: LaTeX, 18 pages with figures, to appear in Pacific Journal of
Mathematic