Fractal squares with finitely many connected components^*

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 ⩾ 2 connected components, are added in addition.