  In this paper, we introduce an equivalence relation on the set of local moves
and classify local moves, called the extended $ST$-moves, up to the
equivalence. Moreover, by inducing a binary relation on the set of equivalence
classes of local moves, we show that an extended $ST$-move realizes the
crossing change or the $SH(2)$-move. In addition, for any oriented knot and two
extended $ST$-moves, we disscus the magnitude relation between the unknotting
numbers of the knot via the moves, and show that there is an extended $ST$-move
except $SH$-moves so that the knot can be transformed into the trivial knot by
the single extended $ST$-move. Finally, we provide some examples of $ST$-moves
with the binary relation.
