  In this paper, we present an effective method to characterize completely when
a disconnected fractal square has only finitely many connected components. Our
method is to establish some graph structures on fractal squares to reveal the
evolution of the connectedness during their geometric iterated construction. We
also prove that every fractal square contains either finitely or uncountably
many connected components. A few examples, including the construction of
fractal squares with exactly $m\geqslant 2$ connected components, are added in
addition.
