Abstract
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(log n), 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(n 2.5) 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.
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Benczúr, A.A., Förster, J., Király, Z.: Dilworth’s theorem and its application for path systems of a cycle – implementation and analysis. In: Nešetřil, J. (ed.) ESA 1999. LNCS, vol. 1643, pp. 498–509. Springer, Heidelberg (1999)
Bespamyatnikh, S.: Computing homotopic shortest paths in the plane. In: Proc. 14th ACM-SIAM Symp. Discrete Algorithms, Baltimore, MD, pp. 609–617 (2003)
Demaine, E.D., O’Rourke, J.: Open Problems from CCCG 1999. In: Proc. 11th Canadian Conf. on Comput. Geom., Vancouver, BC (1999)
Dilworth, R.: A decomposition theorem for partially ordered sets. Ann. of Maths. 51, 161–166 (1950)
Dumitrescu, A., Tóth, G.: Ramsey-type results for unions of comparability graphs. Graphs and Combinatorics 18, 245–251 (2002)
Ford Jr., L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)
Hopcroft, J.E., Karp, R.M.: An n 5/2 algorithm for maximum matching in bipartite graphs. SIAM J. Cornput. 2, 225–231 (1973)
Hoffmann, M., Tóth, C.D.: Alternating paths through disjoint line segments. Inform. Proc. Letts. (to appear)
Hoffmann, M., Tóth, C.D.: Segment endpoint visibility graphs are Hamiltonian. Comput. Geom. Theory Appl. 26(1) (2003)
Larman, D.G., Matoušek, J., Pach, J., Törőcsik, J.: A Ramsey-type result for planar convex sets. Bulletin of the London Mathematical Society 26, 132–136 (1994)
O’Rourke, J.: Visibility. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, ch. 25, pp. 467–480. CRC Press, Boca Raton (1997)
O’Rourke, J., Rippel, J.: Two segment classes with Hamiltonian visibility graphs. Comput. Geom. Theory Appl. 4, 209–218 (1994)
Overmars, M.H., Welzl, E.: New methods for computing visibility graphs. In: Proc. 4th ACM Symp. Comput. Geom. (Urbana-Champaign, IL, 1988), pp. 164–171
Tamassia, R., Tollis, I.G.: A unified approach to visibility representations of planar graphs. Discrete Comput. Geom. 1, 321–341 (1986)
Tóth, C.D.: Illumination in the presence of opaque line segments in the plane. Comput. Geom. Theory Appl. 21, 193–204 (2002)
Tóth, G.: Note on geometric graphs. J. Combin. Theory, Ser. A 89, 126–132 (2000)
Urabe, M., Watanabe, M.: On a counterexample to a conjecture of Mirzaian. Comput. Geom. Theory Appl. 2, 51–53 (1992)
Urrutia, J.: Algunos problemas abiertos (in Spanish). In: Actas de los IX Encuentros de Geometría Computacional, Girona (2001)
Urrutia, J.: Open problems in computational geometry. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 4–11. Springer, Heidelberg (2002)
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Tóth, C.D. (2003). Alternating Paths along Orthogonal Segments. In: Dehne, F., Sack, JR., Smid, M. (eds) Algorithms and Data Structures. WADS 2003. Lecture Notes in Computer Science, vol 2748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45078-8_34
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DOI: https://doi.org/10.1007/978-3-540-45078-8_34
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