Before revision: Fast constant factor approximation algorithms are devised for an NP- and W 1 -hard problem of intersecting a  set of n straight line segments  with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without  edge crossings.
After revision: Fast constant factor approximation algorithms are devised for an NP- and W 1 -hard problem of intersecting a set of n straight line segments with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without proper edge crossings.