# An SPQR-Tree-Like Embedding Representation for Upward Planarity

## TL;DR

This paper introduces the UP-tree, a new data structure that efficiently represents all upward planar embeddings of biconnected single-source directed graphs, aiding in upward planarity testing and extension problems.

## Contribution

The paper develops the UP-tree, a novel structure analogous to SPQR-trees, specifically designed for upward planarity representations in directed graphs.

## Key findings

- The UP-tree compactly encodes all upward planar embeddings.
- It enables efficient solutions to the upward planar embedding extension problem.
- The structure enhances understanding of upward planarity in biconnected graphs.

## Abstract

The SPQR-tree is a data structure that compactly represents all planar embeddings of a biconnected planar graph. It plays a key role in constrained planarity testing. We develop a similar data structure, called the UP-tree, that compactly represents all upward planar embeddings of a biconnected single-source directed graph. We demonstrate the usefulness of the UP-tree by solving the upward planar embedding extension problem for biconnected single-source directed graphs.