Embedding graphs into embedded graphs

TL;DR

This paper proves that determining whether a straight-line planar graph drawing can be approximated by an embedding within a fixed isotopy class is polynomial-time solvable, extending known results beyond cycles.

Contribution

It establishes polynomial-time algorithms for testing approximability by embedding for planar graphs with fixed embeddings, generalizing prior cycle-specific results.

Findings

Testing approximability by embedding is polynomial-time for fixed isotopy classes.

The result extends known cycle-specific tractability to broader planar graphs.

C-planarity with embedded pipes is tractable for fixed embeddings.

Abstract

A (possibly denerate) drawing of a graph $G$ in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a straight-line drawing of a planar graph $G$ in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of $G$ belongs to a fixed isotopy class. In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings. To the best of our knowledge an analogous result was previously known essentially only when $G$ is a cycle.