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

TL;DR

This paper studies angle-monotone graphs, providing recognition algorithms, exploring their properties, and developing a local routing algorithm with bounded stretch factor, along with establishing lower bounds for routing.

Contribution

It introduces a polynomial time recognition algorithm for angle-monotone graphs, proves the half-θ6 graph is generalized angle-monotone, and presents a local routing algorithm with a specific stretch factor.

Findings

Recognition algorithm for angle-monotone graphs

Half-θ6 graph is generalized angle-monotone

Routing algorithm with stretch factor 1 + √2

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.