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Ramified Rectilinear Polygons: Coordinatization by Dendrons

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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.

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Notes

  1. More precisely, according to Whitehead’s definition of an elementary contraction in a simplicial complex [41, p. 247], this step can be represented as a pair of elementary contractions in a triangulation of |G| constructed by splitting each quadrilateral of |G| arbitrarily into two triangles.

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Acknowledgments

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

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Authors and Affiliations

  1. Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146, Hamburg, Germany

    Hans-Jürgen Bandelt

  2. Laboratoire d’Informatique Fondamentale, Université d’ Aix-Marseille and CNRS, Faculté des Sciences de Luminy, 13288, Marseille Cedex 9, France

    Victor Chepoi

  3. Computer Science Department, University of California, Irvine, Irvine, CA, 92697-3435, USA

    David Eppstein

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  1. Hans-Jürgen Bandelt
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Correspondence to David Eppstein.

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Bandelt, HJ., Chepoi, V. & Eppstein, D. Ramified Rectilinear Polygons: Coordinatization by Dendrons. Discrete Comput Geom 54, 771–797 (2015). https://doi.org/10.1007/s00454-015-9743-5

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