The $st$-Planar Edge Completion Problem is Fixed-Parameter Tractable

TL;DR

This paper proves that determining whether a biconnected planar digraph can be augmented to an $st$-planar graph by adding a limited set of edges is fixed-parameter tractable, despite the problem's NP-completeness.

Contribution

The paper establishes fixed-parameter tractability for the $st$-planar edge completion problem when parameterized by the number of added edges.

Findings

The problem is NP-complete in general.

Fixed-parameter tractability is achieved with respect to the size of added edges.

Provides an algorithmic approach for the augmentation problem.

Abstract

The problem of deciding whether a biconnected planar digraph $G=(V,E)$ can be augmented to become an $st$-planar graph by adding a set of oriented edges $E' \subseteq V \times V$ is known to be NP-complete. We show that the problem is fixed-parameter tractable when parameterized by the size of the set $E'$.