The morphological operation imerode() suits your needs. I give you some hints, look at this plot after image erosion with an element of size 5 (which is approx the thickness of the line you want to remove):
A=rgb2gray(imread('Final Output.png'));
B=imerode(A,ones(5))
imagesc(B);
Thus, you can identify all connected components in the original image, and also in the eroded one. Then, the conn comp found in the original image that have large interseection with the ones in the eroded image can be stored.
