Ramified rectilinear polygons: coordinatization by dendrons
Hans-J\"urgen Bandelt, Victor Chepoi, David Eppstein

TL;DR
This paper introduces ramified rectilinear polygons, a generalization of simple rectilinear polygons, characterized by complex vertex links and coordinatized by dendrons, with applications in graph recognition and symmetry analysis.
Contribution
It characterizes ramified rectilinear polygons as rectangular complexes from cube-free median graphs and provides linear-time recognition algorithms.
Findings
Recognition of underlying graphs in linear time
Ramified rectilinear polygons can realize any finite automorphism group
Extension of rectilinear polygon theory to more complex vertex links
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 -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 4-cycles or paths of length at most 3. 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…
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