Compatible Triangulations and Point Partitions by Series-Triangular Graphs

TL;DR

This paper introduces series-triangular graph embeddings to partition point sets and improves bounds on Steiner points for compatible triangulations, generalizing to multiple point sets with linear Steiner point addition.

Contribution

It presents a new class of graph embeddings and applies them to enhance bounds on Steiner points for compatible triangulations, including the multi-set case.

Findings

Improved upper bounds on Steiner points for compatible triangulations.

Introduction of series-triangular graph embeddings for point set partitioning.

Linear Steiner point addition for multiple point sets.

Abstract

We introduce series-triangular graph embeddings and show how to partition point sets with them. This result is then used to improve the upper bound on the number of Steiner points needed to obtain compatible triangulations of point sets. The problem is generalized to finding compatible triangulations for more than two point sets and we show that such triangulations can be constructed with only a linear number of Steiner points added to each point set.