One simple way to do this type of task is morphological opening. Here is an example.
% Read and binarize your image
I = imread('yourImage.png');
BW = imbinarize(rgb2gray(I));
% Apply morphological opening to remove 1-pixel-width vertical line
se = strel('line',2,0);
BW2 = imopen(BW,se);
The result is as follows (left: original / right: after applying morphological opening)