  We define $A_k$-moves for embeddings of a finite graph into the 3-sphere for
each natural number $k$. Let $A_k$-equivalence denote an equivalence relation
generated by $A_k$-moves and ambient isotopy. $A_k$-equivalence implies
$A_{k-1}$-equivalence. Let ${\cal F}$ be an $A_{k-1}$-equivalence class of the
embeddings of a finite graph into the 3-sphere. Let ${\cal G}$ be the quotient
set of ${\cal F}$ under $A_k$-equivalence. We show that the set ${\cal G}$
forms an abelian group under a certain geometric operation. We define finite
type invariants on ${\cal F}$ 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 ${\cal
F}$, then they are $A_k$-equivalent. $A_k$-move is a generalization of
$C_k$-move defined by K. Habiro. Habiro showed that two oriented knots are the
same up to $C_k$-move and ambient isotopy if and only if any Vassiliev
invariant of order $\leq k-1$ 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 $C_k$-equivalent.
