Drawing Planar Graphs with a Prescribed Inner Face

TL;DR

This paper characterizes when a plane graph with a cycle mapped to a convex polygon can be extended to a planar straight-line drawing, providing necessary and sufficient conditions and a linear-time algorithm for testing and constructing such drawings.

Contribution

It offers a complete characterization and an efficient linear-time algorithm for extending partial drawings of plane graphs to full planar straight-line drawings.

Findings

Characterization of extendability conditions for planar graph drawings

Linear-time algorithm for testing and constructing extensions

Conditions proven to be both necessary and sufficient

Abstract

Given a plane graph $G$ (i.e., a planar graph with a fixed planar embedding) and a simple cycle $C$ in $G$ whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar straight-line drawing of $G$. We characterize when this is possible in terms of simple necessary conditions, which we prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time.