Abstract
A geometric graph in the plane is angle-monotone of width \(\gamma \) if every pair of vertices is connected by an angle-monotone path of width \(\gamma \), a path such that the angles of any two edges in the path differ by at most \(\gamma \). Angle-monotone graphs have good spanning properties.
We prove that every point set in the plane admits an angle-monotone graph of width \(90^\circ \), hence with spanning ratio \(\sqrt{2}\), and a subquadratic number of edges. This answers an open question posed by Dehkordi, Frati and Gudmundsson.
We show how to construct, for any point set of size n and any angle \(\alpha \), \(0< \alpha < 45^\circ \), an angle-monotone graph of width \((90^\circ +\alpha )\) with \(O(\frac{n}{\alpha })\) edges. Furthermore, we give a local routing algorithm to find angle-monotone paths of width \((90^\circ +\alpha )\) in these graphs. The routing ratio, which is the ratio of path length to Euclidean distance, is at most \(1/\cos (45^\circ + \frac{\alpha }{2})\), i.e., ranging from \(\sqrt{2} \approx 1.414\) to 2.613. For the special case \(\alpha = 30^\circ \), we obtain the \(\varTheta _6\)-graph and our routing algorithm achieves the known routing ratio 2 while finding angle-monotone paths of width \(120^\circ \).
This work is partially supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Lubiw, A., Mondal, D. (2018). Construction and Local Routing for Angle-Monotone Graphs. In: Brandstädt, A., Köhler, E., Meer, K. (eds) Graph-Theoretic Concepts in Computer Science. WG 2018. Lecture Notes in Computer Science(), vol 11159. Springer, Cham. https://doi.org/10.1007/978-3-030-00256-5_29
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