# On the Upward Book Thickness Problem: Combinatorial and Complexity Results

@inproceedings{Bhore2021OnTU, title={On the Upward Book Thickness Problem: Combinatorial and Complexity Results}, author={Sujoy Kumar Bhore and Giordano Da Lozzo and Fabrizio Montecchiani and Martin N{\"o}llenburg}, booktitle={International Symposium Graph Drawing and Network Visualization}, year={2021} }

A long-standing conjecture by Heath, Pemmaraju, and Trenk states that the upward book thickness of outerplanar DAGs is bounded above by a constant. In this paper, we show that the conjecture holds for subfamilies of upward outerplanar graphs, namely those whose underlying graph is an internally-triangulated outerpath or a cactus, and those whose biconnected components are st-outerplanar graphs. On the complexity side, it is known that deciding whether a graph has upward book thickness k is NP…

## 3 Citations

### Directed Acyclic Outerplanar Graphs Have Constant Stack Number

- MathematicsArXiv
- 2022

The stack number of a directed acyclic graph G is the minimum k for which there is a topological ordering of G and a k -coloring of the edges such that no two edges of the same color cross, i.e.,…

### Recognizing DAGs with Page-Number 2 is NP-complete

- MathematicsGD
- 2022

The page-number of a directed acyclic graph (a DAG, for short) is the minimum k for which the DAG has a topological order and a k -coloring of its edges such that no two edges of the same color…

### Parameterized Algorithms for Queue Layouts

- Mathematics, Computer ScienceGD
- 2020

This paper presents two fixed-parameter tractable algorithms that exploit structural properties of graphs to compute optimal queue layouts and uses a more restrictive parameter, the vertex cover number, to solve the problem for arbitrary graphs.

## References

SHOWING 1-10 OF 44 REFERENCES

### A Sublinear Bound on the Page Number of Upward Planar Graphs

- Mathematics, Computer ScienceSODA
- 2022

It is proved that the page number of every upward planar graph is bounded in terms of its width, as well as its height, and it is combined with the best known lower bound to improve the asymptotic improvement over the trivial O ( n ) upper bound.

### The Book Thickness of 1-Planar Graphs is Constant

- MathematicsAlgorithmica
- 2016

It is proved that every 1-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages.

### Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages

- MathematicsSoCG
- 2020

It is proved that this family of graphs has bounded book thickness, and as a corollary, the first constant upper bound for the book thickness of optimal 2-planar graphs is obtained.

### Parameterized Algorithms for Book Embedding Problems

- MathematicsGD
- 2019

It is shown that Fixed-Order book Thickness and Book Thickness are fixed-parameter tractable parameterized by the vertex cover number of the graph and that fixed- parameters tractable by the pathwidth of the vertex order.

### Graph Treewidth and Geometric Thickness Parameters

- MathematicsGD
- 2005

It is shown that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting, and that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉.

### Upward Planarity Testing in Practice

- Computer Science, MathematicsACM J. Exp. Algorithmics
- 2015

Two fundamentally different approaches based on the seemingly novel concept of ordered embeddings and on the concept of a Hanani-Tutte-type characterization of monotone drawings are proposed and it is shown that the SAT formulations outperform all known approaches for graphs with up to 400 edges.

### Book Embeddability of Series–Parallel Digraphs

- Computer Science, MathematicsAlgorithmica
- 2005

The equivalence between upward two-page book embeddable and upward point-set embeddability with at most one bend per edge on any given set of points is proved.

### Upward Book Embeddings of st-Graphs

- Mathematics, Computer ScienceSoCG
- 2019

It is shown that testing whether a graph admits aUBE is NP-complete for k-page upward 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.

### Advancements on SEFE and Partitioned Book Embedding problems

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2015