Connectedness of the basin of attraction for rational maps

We prove some results concerning degree of a rational map on the immediate basin $B(s)$ of an attracting fixed point $s$. In particular, if $B(s)$ contains all but two critical points or values counted with multiplicity, then the entire basin of attraction is connected. For any number $k \geq 3$ we give examples of rational maps with disconnected basin of an attracting fixed point such that there are exactly $k$ critical points outside the immediate basin of attraction.