293 research outputs found
Shortest Paths Avoiding Forbidden Subpaths
In this paper we study a variant of the shortest path problem in graphs:
given a weighted graph G and vertices s and t, and given a set X of forbidden
paths in G, find a shortest s-t path P such that no path in X is a subpath of
P. Path P is allowed to repeat vertices and edges. We call each path in X an
exception, and our desired path a shortest exception-avoiding path. We
formulate a new version of the problem where the algorithm has no a priori
knowledge of X, and finds out about an exception x in X only when a path
containing x fails. This situation arises in computing shortest paths in
optical networks. We give an algorithm that finds a shortest exception avoiding
path in time polynomial in |G| and |X|. The main idea is to run Dijkstra's
algorithm incrementally after replicating vertices when an exception is
discovered.Comment: 12 pages, 2 figures. Fixed a few typos, rephrased a few sentences,
and used the STACS styl
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
Enumerating Foldings and Unfoldings between Polygons and Polytopes
We pose and answer several questions concerning the number of ways to fold a
polygon to a polytope, and how many polytopes can be obtained from one polygon;
and the analogous questions for unfolding polytopes to polygons. Our answers
are, roughly: exponentially many, or nondenumerably infinite.Comment: 12 pages; 10 figures; 10 references. Revision of version in
Proceedings of the Japan Conference on Discrete and Computational Geometry,
Tokyo, Nov. 2000, pp. 9-12. See also cs.CG/000701
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