Regular Labelings and Geometric Structures

TL;DR

This paper explores how regular labelings of planar graphs can describe various geometric structures like triangulations and polyhedra, highlighting their combinatorial properties and applications in algorithm design.

Contribution

It surveys the connections between different types of regular labelings and their role in developing efficient geometric algorithms.

Findings

Regular labelings characterize grid triangulations, rectangular subdivisions, and orthogonal polyhedra.

These labelings facilitate the design of efficient algorithms for geometric problems.

The paper highlights analogies and connections among different geometric structures via combinatorial labelings.

Abstract

Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or near-maximal planar graph. We briefly survey the connections and analogies between these three kinds of labelings, and their uses in designing efficient geometric algorithms.