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arXiv · 1608.08892

Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

Nicolas Bonichon·Prosenjit Bose·Paz Carmi·Irina Kostitsyna·Anna Lubiw·Sander Verdonschot
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Abstract

A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is $x$- and $y$-monotone. Angle-monotone graphs are $\sqrt 2$-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-$\theta_6$-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex $s$ to any vertex $t$ whose length is within $1 + \sqrt 2$ times the Euclidean distance from $s$ to $t$. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.

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Nicolas Bonichon, Prosenjit Bose, Paz Carmi, Irina Kostitsyna, Anna Lubiw, Sander Verdonschot. 2016-08-31. Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition. https://arxiv.org/abs/1608.08892

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