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

@article{Brckner2020AnSE, title={An SPQR-Tree-Like Embedding Representation for Level Planarity}, author={Guido Br{\"u}ckner and Ignaz Rutter}, journal={ArXiv}, year={2020}, volume={abs/2009.12309} }

An SPQR-tree is a data structure that efficiently represents all planar embeddings of a biconnected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to some given set of constraints. We develop an SPQR-tree-like data structure that represents all levelplanar embeddings of a biconnected level graph with a single source, called the LP-tree, and give a simple algorithm to compute it in linear time. Moreover, we…

## References

SHOWING 1-10 OF 43 REFERENCES

An SPQR-Tree-Like Embedding Representation for Upward Planarity

- Computer ScienceGraph Drawing
- 2019

The usefulness of the UP-tree is demonstrated by solving the upward planar embedding extension problem for biconnected single-source directed graphs.

On-Line Graph Algorithms with SPQR-Trees

- Computer ScienceICALP
- 1990

We present the SPQR-tree, a versatile data structure that represents the decomposition of a biconnected graph with respect to its triconnected components, and show its application to a variety of…

Inserting an Edge into a Planar Graph

- Computer Science, MathematicsSODA '01
- 2001

Surprisingly, a conceptually simple linear time algorithm based on SPQR-trees, that is able to find a solution with the minimum number of crossings is found.

Testing planarity of partially embedded graphs

- Mathematics, Computer ScienceSODA '10
- 2010

This work shows that the planarity question remains polynomial-time solvable, and implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently.

A Linear Algorithm for Embedding Planar Graphs Using PQ-Trees

- Computer ScienceJ. Comput. Syst. Sci.
- 1985

On-line maintenance of triconnected components with SPQR-trees

- MathematicsAlgorithmica
- 2005

AnO(n)-space data structure that supports insertions of vertices and edges, and queries of the type “Are there three vertex-disjoint paths between verticesv1 andv2?” is presented.

A Linear Time Implementation of SPQR-Trees

- Computer ScienceGraph Drawing
- 2000

The relationship between SPQR-trees and triconnected components is described and the incorrectness of the Hopcroft and Tarjan algorithm is shown and the resulting algorithm is applied to the computation of SPQRs.

A Linear Time Algorithm for Constructing Maximally Symmetric Straight Line Drawings of Triconnected Planar Graphs

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2006

It is shown that an algorithm of Fontet can be used to find an embedding in the plane with the maximum number of symmetries, and a new algorithm for finding a straight line drawing that achieves that maximum is presented.

Finding a Minimum-depth Embedding of a Planar Graph in O(n4) Time

- Computer ScienceAlgorithmica
- 2009

An O(n4)-time algorithm for computing an embedding of G with minimum depth is presented, which improves on the best previous bound by an O( nlog n) factor and improves the bounds of several algorithms that require the computation of a minimum-depth embedding.

On Embedding a Graph in the Grid with the Minimum Number of Bends

- Computer Science, MathematicsSIAM J. Comput.
- 1987

An algorithm is presented that computes a region preserving grid embedding with the minimum number of bends in edges with use of network flow techniques, and runs in time $O(n^2 \log n)$, where n is the number of vertices of the graph.