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.
Melissaratos and Souvaine developed a linear time algorithm to find shortest path between two points in a splinegon @cite_24 . They introduced a trapezoid decomposition of the curved region that is a substitute to polygon triangulation in @math time. Using the trapezoid decomposition the shortest path in a splinegon can be computed in @math time with an algorithm similar to shortest path finding in a simple polygon. D.Z. Chen and H. Wang @cite_3 developed an @math algorithm to find shortest path in a splinegon with @math splines and @math vertices, where @math is the number of all common tangents inside the region and is in @math . The algorithm works for any arbitrary small @math . They also proposed an algorithm in @math time to find all common tangents in a splinegon.
@cite_24 @cite_3
2063244634 2022670034
The goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of efficient algorithms for certain geometric optimization problems involving simple polygons: computing optimum separators, maximum area or perimeter-inscribed triangles, a minimum area circumscribed concave quadrilateral, or a maximum area contained triangle. The structure for the algorithms presented is as follows: (a) decompose the initial problem into a low-degree polynomial number of optimization problems; (b) solve each individual subproblem in constant time using standard methods of calculus, basic methods of numerical analysis, or linear programming. These same optimization techniques can be applied to splinegons (curved polygons). First a decomposition technique for curved polygons is developed; this technique is substituted for triangulation in creating equally efficient curved versions of the algorithms for the shortest-path tree, ray-shooting, and two-point shortest path... In this paper, we study the problem of finding Euclidean shortest paths among curved obstacles in the plane. We model curved obstacles as splinegons. A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge, and each curved edge is assumed to be of O(1) complexity. Given in the plane two points s and t and a set of h pairwise disjoint splinegons with a total of n vertices, we present an algorithm that can compute a shortest path from s to t avoiding the splinegons in O(n+hlogeh+k) time for any e>0, where k is a parameter sensitive to the input splinegons and k=O(h2). If all splinegons are convex, a common tangent of two splinegons is "free" if it does not intersect the interior of any splingegon; our techniques yield an output sensitive algorithm for computing all free common tangents of the h splinegons in O(n+hlogh+k) time and O(n) working space, where k is the number of all free common tangents.