# Upward Book Embeddings of st-Graphs

## TL;DR

This paper investigates upward book embeddings of $st$-graphs, proving NP-completeness for $k \\geq 3$ and providing polynomial algorithms for specific cases with $k=2$, along with families of graphs that always admit such embeddings.

## Contribution

It establishes NP-completeness for testing $k$-page upward book embeddings for $k \\geq 3$ and offers polynomial algorithms for $k=2$ in certain graph classes, plus identifying graph families always admitting 2UBEs.

## Key findings

- NP-completeness for $k \\geq 3$
- Polynomial algorithms for planar $st$-graphs with branchwidth and face structure
- Families of plane $st$-graphs always admitting 2UBEs

## Abstract

We study $k$-page upward book embeddings ($k$UBEs) of $st$-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on $k$ pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a $k$UBE is NP-complete for $k\geq 3$. A hardness result for this problem was previously known only for $k = 6$ [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on $k=2$. On the algorithmic side, we present polynomial-time algorithms for testing the existence of $2$UBEs of planar $st$-graphs with branchwidth $\beta$ and of plane $st$-graphs whose faces have a special structure. These algorithms run in $O(f(\beta)\cdot n+n^3)$ time and $O(n)$ time, respectively, where $f$ is a singly-exponential function on $\beta$. Moreover, on the combinatorial side, we present two notable families of plane $st$-graphs that always admit an embedding-preserving $2$UBE.