Emanation graphs of grade k , introduced by Hamedmohseni, Rahmati, and Mondal, are plane spanners made by shooting 2^{k+1}  rays from each given point, where the shorter rays stop the longer ones upon collision. The collision points are the Steiner points of the spanner. We introduce a method of simplification for emanation graphs of grade k= 2 , which makes it a competent spanner for many possible use cases such as network visualizationand geometric routing. In particular, the simplification reduces the number of Steiner points by half and also significantly decreases the total)\approx 3.41. We improve this upper bound to 10 \approx 3.162 and show this to be tight, i.e., there exist emanation graphs with spanning ratio 10. We also explore emanation graphs of grade two, which may have twice the } number of edges , without increasing the spanning ratio. Exact methods of simplification along with mathematical proofs on properties of the simplified graph is provided. We compare simplified emanation graphs against Shewchuk's constrained Delaunay triangulations on both synthetic and real-life datasets. Our experimental results reveal that the simplified emanation graphs outperform constrained Delaunay triangulations in common quality measures (e.g., edge count, angular resolution, average degree, total edge length) while maintain a comparable spanning ratio and Steiner point count.
An emanation graph of grade k on a set of points is a plane spanner made by shooting 2^{k+1} equally spaced rays from each point, where the shorter rays stop the longer ones upon collision. The collision points are the Steiner points of the spanner. Emanation graphs of grade one were studied by Mondal and Nachmanson in the context of network visualization. They proved that the spanning ratio of such a graph is bounded by ( 2 +\sqrt{2)\approx 3.41. We improve this upper bound to 10 \approx 3.162 and show this to be tight, i.e., there exist emanation graphs with spanning ratio 10. We also explore emanation graphs of grade two, which may have twice the } number of edges compared to grade one graphs. We introduce a method of simplification that makes it an interesting geometric graph to be used for network visualization and geometric routing. In particular, we compare simplified emanation graphs against Shewchuk's constrained Delaunay triangulations on both synthetic and real-life datasets. Our experimental results reveal that the simplified emanation graphs outperform constrained Delaunay triangulations in common quality measures (e.g., edge count, angular resolution, average degree, total edge length) while maintain a comparable spanning ratio and Steiner point count.
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Emanation graphs
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We introduce a method of simplification for emanation
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one were studied by Mondal and Nachmanson in the context of network visualization. They proved that the spanning ratio of such a graph is bounded by (
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, which makes it a competent spanner for many possible use cases such as network visualizationand geometric routing. In particular, the simplification reduces the number of Steiner points by half and also significantly decreases the total
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