How should I handle lines?

I’ve imported the bird svg & created the circles. I’d like to cut out the shape of the bird without cutting the points at the beak & tail. Those points I’d like to just make a line on the circle edge or maybe shade them in. So basically cut out the shape but keep the two points touchging the circle intact. How do I join the lines,then set the power level for each cut/line.
Thanks in advance.