Wow - very inspiring! Thank you Vitaliy one more for this link! I think I need one of these lasers! Even though - lasers are hot, the jigsaw might melt:
img1 = Binarize@ColorNegate[Graphics[{EdgeForm[White], Black, jigsawPolygsRaw}, ImageSize -> Large]];
SkeletonTransform[img1] // Pruning[#, 40] & // InverseDistanceTransform