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In this paper, we study two classes of planar self-similar fractals$T_\varepsilon$ with a shifting parameter $\varepsilon$. The first one is aclass of self-similar tiles by shifting $x$-coordinates of some digits. We givea detailed discussion on the disk-likeness ({\it i.e., the property of being atopological disk}) in terms of $\varepsilon$. We also prove that$T_\varepsilon$ determines a quasi-periodic tiling if and only if $\varepsilon$is rational. The second one is a class of self-similar sets by shiftingdiagonal digits. We give a necessary and sufficient condition for$T_\varepsilon$ to be connected.

Topological properties of self-similar fractals with one parameter