HELP: Merge overlapping vectors leaving one cut line

Aloha All,

I need to make a file for cutting with a laser cutter that will save the most material. When I combined the bottom two left most shapes with pathfinder the results look like the top most shape and all seems perfect. However, the Laser Cutter cuts the internal lines twice resulting in over burn and frankly  a lot of time wasted. Please someone explain me how to get the topmost shape with single vector line segments as the dividers. This will save a lot of wood and trees

Thanks in advance

FWIW: This is where I'm going with this request