Findng paths between nodes in binary image

I am trying to find the paths connecting nodes in a binary image as well as node coordinates.

In this context, I call node to pixel connecting 3 or more paths.

I have tried to find functions or easy procedures to get this output.

This procedure is of course depending on the connectivity.

This image with 4-connectivity has been processed by searching for all pixels having 3 or more neighbors (nodes) and then following "step-by-step" each "exit" path.

But the process is a bit combersome.

Is there a Matlab easiest way to do it?

Thanks

[Lateef Adewale Kareem]

if you know the points coordinates, use delaunay triangulation

[Luis Rosety]

Thanks for your reply but what I need is the branch or path connecting the nodes as seen in the image

They are not straight lines as you can see.