Nodally 3-connected planar graphs and convex combination mappings

TL;DR

This paper characterizes nodally 3-connected planar graphs and extends known results on convex combination mappings, showing that such mappings are embeddings for these graphs and more generally for any planar graph admitting a convex embedding.

Contribution

It provides a simple characterization of nodally 3-connected planar graphs and generalizes embedding results to all planar graphs with convex embeddings.

Findings

Nodally 3-connected planar graphs are characterized simply.

Convex combination mappings are embeddings for these graphs.

The results extend to any planar graph with a convex embedding.

Abstract

A convex combination mapping of a planar graph is a plane mapping in which the external vertices are mapped to the corners of a convex polygon and every internal vertex is a proper weighted average of its neighbours. If a planar graph is nodally 3-connected or triangulated then every such mapping is an embedding (Tutte, Floater). We give a simple characterisation of nodally 3-connected planar graphs, and generalise the above result to any planar graph which admits any convex embedding.