Computational complexity and approximation algorithms are reported for a
problem of stabbing a set of straight line segments with the least 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. Close
geometric problems arise in network security applications. We give strong
NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel
graphs and other subgraphs (which are often used in network design) for r∈[dmin,ηdmax] and some constant η where dmax and
dmin are Euclidean lengths of the longest and shortest graph edges
respectively. Fast O(∣E∣log∣E∣)-time O(1)-approximation algorithm is
proposed within the class of straight line drawings of planar graphs for which
the inequality r≥ηdmax holds uniformly for some constant
η>0, i.e. when lengths of edges of G are uniformly bounded from above by
some linear function of r.Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International
Conference on Analysis of Images, Social Networks and Texts (AIST-2017