Find lines in shape

I have a binary image and I'm looking for a robust way to find the lines in the shape and the topology (how the lines connect).

I have experimented in matlab (although what I'm asking for is which methods to use).

I've tried using skeletonization on the binary image and then used hough-transform, works sometimes but not a robust solution. I struggled with boundary disturbance.

Could anyone point me in a direction of which methods to use here (and in what order).