Separation problems and circular arc systems

  • Paul Fischer
  • Hans Ulrich Simon
Computational Geometry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 484)


We show that the problem of finding a smallest convex polygon which separates two given finite sets of points in the plane is a special case of the combinatorial problem of finding a minimum transversal of a circular arc system. We present an O(n log n) algorithm for the latter problem. We describe also a close relationship between visibility graphs and intersection graphs. It is furthermore shown that a smallest separating convex polygon is not greater than any separating arbitrary polygon or any separating planar subdivision. Moreover, we determine the number of stages needed for learning convex polygons from examples.


computational geometry separating polygons separating planar subdivisions visibility circular arc systems transversals intersection graphs computational learning number of stages 


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  1. [1]
    H. Edelsbrunner and F. P. Preparata. Minimum polygonal separation. Information and Computation, 77:218–232, 1988.Google Scholar
  2. [2]
    Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Volume Computer Science and Applied Mathematics of Monographs and Textbooks, Academic Press, New York, 1980.Google Scholar
  3. [3]
    Nathan Linial, Yishai Mansour, and Ronald L. Rivest. Results on learnability and the Vapnik-Chervonenkis dimension. In Proc. of the 29th IEEE Symp. on Foundations of Computer Science, pages 120–129, IEEE, IEEE Computer Society Press, Los Alamitos, CA, October 1988.Google Scholar
  4. [4]
    Nimrod Megiddo. On the complexity of polyhedral separability. Discrete Computational Geometry, 3:325–337, 1988.Google Scholar
  5. [5]
    L.G. Valiant. A theory of the learnable. Communications of the ACM, 27(11):1134–1142, 1984.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Paul Fischer
    • 1
  • Hans Ulrich Simon
    • 1
  1. 1.Universität DortmundDortmundWest-Germany

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