3-coloring Arrangements of Line Segments with 4 Slopes Is Hard

Abstract In a paper first appeared at SODA '07, Eppstein proved that testing the 3-colorability of arrangements of line segments is an NP-complete problem. However, if the slopes of the segments are limited to three different values, a 3-coloring can be trivially obtained by assigning the same color to all the segments having the same slope. We thus study the complexity of testing the 3-colorability of arrangements of line segments, or equivalently of their intersection graphs, that are restricted to have a constant number s > 3 of slopes, and prove that this remains NP-complete even for s = 4 , which is hence tight. More in general, we prove that k-coloring arrangements of line segments is NP-complete even if the segments have at most k + 1 slopes. Since the problem of computing a k-coloring of an arrangement of line segments is equivalent to computing the constrained geometric thickness of a straight-line drawing of a graph in the plane, our result extends to this problem.