I converted polygon layer to line layer using "Polygons to Lines" tool. Then I used "Split" function of the Networks plugin. What I need to do next is to remove identical lines that were shared boundary of the touching polygons. I've tried using MMQGIS plugin and v.clean (break,snap,rmline,rmdupl) but nothing works. You can see more details in attached image.
Edit: you can download shapefile here: https://drive.google.com/file/d/0B25RJIaeA41SVmNOUndWWGNlWU0/view?usp=sharing