# Extending Upward Planar Graph Drawings

## TL;DR

This paper investigates the computational complexity of extending partial upward planar graph drawings, proving NP-completeness in general but offering efficient algorithms for specific graph classes like upward st-graphs and directed paths or cycles.

## Contribution

It establishes NP-completeness for the general problem and provides improved algorithms for upward st-graphs and polynomial solutions for special cases.

## Key findings

- NP-complete in general case
- O(n log n) algorithm for upward st-graphs
- Polynomial-time solution for directed paths and cycles with unique y-coordinates

## Abstract

In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes in input an upward planar drawing $\Gamma_H$ of a subgraph $H$ of a directed graph $G$ and asks whether $\Gamma_H$ can be extended to an upward planar drawing of $G$. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing. We show the following results. First, we prove that the Upward Planarity Extension problem is NP-complete, even if $G$ has a prescribed upward embedding, the vertex set of $H$ coincides with the one of $G$, and $H$ contains no edge. Second, we show that the Upward Planarity Extension problem can be solved in $O(n \log n)$ time if $G$ is an $n$-vertex upward planar $st$-graph. This result improves upon a known $O(n^2)$-time algorithm, which however applies to all $n$-vertex single-source upward planar graphs. Finally, we show how to solve in polynomial time a surprisingly difficult version of the Upward Planarity Extension problem, in which $G$ is a directed path or cycle with a prescribed upward embedding, $H$ contains no edges, and no two vertices share the same $y$-coordinate in $\Gamma_H$.