Alternating Paths along Orthogonal Segments

It was shown recently that the segment endpoint visibility graph Vis(S) of any set S of n disjoint line segments in the plane admits an alternating path of length Θ(logn), and this bound is best possible apart from a constant factor. This paper focuses on the variant of the problem where S is a set of n disjoint axis-parallel line segments. We show that the length of a longest alternating path in the worst case is $\Theta(\sqrt{n})$. We also present an O(n2.5) time algorithm to find an alternating path of length $\Omega(\sqrt{n})$. Finally, we consider sets of axis-parallel segments where the extensions of no two segments meet in the free space $\mathbb{E}^2 \setminus \cup S$, and show that in that case all the segments can be included in a common alternating path.