78,086 research outputs found
Visibility Graphs, Dismantlability, and the Cops and Robbers Game
We study versions of cop and robber pursuit-evasion games on the visibility
graphs of polygons, and inside polygons with straight and curved sides. Each
player has full information about the other player's location, players take
turns, and the robber is captured when the cop arrives at the same point as the
robber. In visibility graphs we show the cop can always win because visibility
graphs are dismantlable, which is interesting as one of the few results
relating visibility graphs to other known graph classes. We extend this to show
that the cop wins games in which players move along straight line segments
inside any polygon and, more generally, inside any simply connected planar
region with a reasonable boundary. Essentially, our problem is a type of
pursuit-evasion using the link metric rather than the Euclidean metric, and our
result provides an interesting class of infinite cop-win graphs.Comment: 23 page
The simplicity of planar networks
Shortest paths are not always simple. In planar networks, they can be very
different from those with the smallest number of turns - the simplest paths.
The statistical comparison of the lengths of the shortest and simplest paths
provides a non trivial and non local information about the spatial organization
of these graphs. We define the simplicity index as the average ratio of these
lengths and the simplicity profile characterizes the simplicity at different
scales. We measure these metrics on artificial (roads, highways, railways) and
natural networks (leaves, slime mould, insect wings) and show that there are
fundamental differences in the organization of urban and biological systems,
related to their function, navigation or distribution: straight lines are
organized hierarchically in biological cases, and have random lengths and
locations in urban systems. In the case of time evolving networks, the
simplicity is able to reveal important structural changes during their
evolution.Comment: 8 pages, 4 figure
Exact Geosedics and Shortest Paths on Polyhedral Surface
We present two algorithms for computing distances along a non-convex polyhedral surface. The first algorithm computes exact minimal-geodesic distances and the second algorithm combines these distances to compute exact shortest-path distances along the surface. Both algorithms have been extended to compute the exact minimalgeodesic paths and shortest paths. These algorithms have been implemented and validated on surfaces for which the correct solutions are known, in order to verify the accuracy and to measure the run-time performance, which is cubic or less for each algorithm. The exact-distance computations carried out by these algorithms are feasible for large-scale surfaces containing tens of thousands of vertices, and are a necessary component of near-isometric surface flattening methods that accurately transform curved manifolds into flat representations.National Institute for Biomedical Imaging and Bioengineering (R01 EB001550
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