  We introduce the notion of balanced strong shift equivalence between square
nonnegative integer matrices, and show that two finite graphs with no sinks are
one-sided eventually conjugate if and only if their adjacency matrices are
conjugate to balanced strong shift equivalent matrices. Moreover, we show that
such graphs are eventually conjugate if and only if one can be reached by the
other via a sequence of out-splits and balanced in-splits; the latter move
being a variation of the classical in-split move introduced by Williams in his
study of shifts of finite type. We also relate one-sided eventual conjugacies
to certain block maps on the finite paths of the graphs. These
characterizations emphasize that eventual conjugacy is the one-sided analog of
two-sided conjugacy.
