Ramified rectilinear polygons: coordinatization by dendrons

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 $l_1$-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 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 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.