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Discrete & Computational Geometry

, Volume 54, Issue 4, pp 771–797 | Cite as

Ramified Rectilinear Polygons: Coordinatization by Dendrons

  • Hans-Jürgen Bandelt
  • Victor Chepoi
  • David Eppstein
Article
  • 95 Downloads

Abstract

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic \(l_1\)-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either four-cycles or paths of length at most three. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group \(D_4\)), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.

Keywords

Median graphs Rectilinear polygons Cartesian products Trees 

Notes

Acknowledgments

We would like to thank anonymous referees for careful reading of the first version and several corrections.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Hans-Jürgen Bandelt
    • 1
  • Victor Chepoi
    • 2
  • David Eppstein
    • 3
  1. 1.Department of MathematicsUniversity of HamburgHamburgGermany
  2. 2.Laboratoire d’Informatique FondamentaleUniversité d’ Aix-Marseille and CNRS, Faculté des Sciences de LuminyMarseille Cedex 9France
  3. 3.Computer Science DepartmentUniversity of California, IrvineIrvineUSA

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