Graph theory: definiton of the crown of a graph
Yes, it implies that $|I| \geq |H|$ but it does not imply $I$ contains only vertices of degree 1. The edges joining $H$ to $I$ contain an H-covering matching, but there can be many more edges. See, for instance, D. Marx' slides on crown reduction: http://www.cs.bme.hu/~dmarx/papers/marx-agape09-slides-print.pdf