The Weighted Barycenter Drawing Recognition Problem

TL;DR

This paper investigates recognizing weighted barycenter drawings of triconnected planar graphs, providing an arithmetic characterization and polynomial recognition algorithms for specific graph classes like Halin and cubic graphs.

Contribution

It introduces an arithmetic characterization for weighted barycenter drawings and develops polynomial time recognition algorithms for certain classes of graphs.

Findings

Positive recognition results for Halin graphs

Polynomial time recognition algorithm for cubic graphs

Arithmetic characterization based on faces of the drawing

Abstract

We consider the question of whether a given graph drawing $\Gamma$ of a triconnected planar graph $G$ is a weighted barycenter drawing. We answer the question with an elegant arithmetic characterisation using the faces of $\Gamma$. This leads to positive answers when the graph is a Halin graph, and to a polynomial time recognition algorithm when the graph is cubic.