Alternating Paths along Axis-Parallel Segments

It is shown that for a set S of n pairwise disjoint axis-parallel line segments in the plane there is a simple alternating path of length Ω( √ n). This bound is best possible in the worst case. In the special case that the n pairwise disjoint axis-parallel line segments are protruded (that is, if the intersection point of the lines through every two nonparallel segments is not visible from both segments), there is a simple alternating path of length n.