# Complexity of Finding Non-Planar Rectilinear Drawings of Graphs

@inproceedings{Manuch2010ComplexityOF, title={Complexity of Finding Non-Planar Rectilinear Drawings of Graphs}, author={J{\'a}n Manuch and Murray D. Patterson and Sheung-Hung Poon and Chris Thachuk}, booktitle={Graph Drawing}, year={2010} }

We study the complexity of the problem of finding non-planar rectilinear drawings of graphs. This problem is known to be NP-complete. We consider natural restrictions of this problem where constraints are placed on the possible orientations of edges. In particular, we show that if each edge has prescribed direction "left", "right", "down" or "up", the problem of finding a rectilinear drawing is polynomial, while finding such a drawing with the minimum area is NP-complete. When assigned…

## 10 Citations

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## References

SHOWING 1-10 OF 16 REFERENCES

### On Rectilinear Drawing of Graphs

- Mathematics, Computer ScienceGraph Drawing
- 2009

This paper presents a linear time fixed-parameter tractable algorithm to test whether a degree-4 graph has a rectilinear drawing, and shows that the problem is NP-hard for the graphs that consist of 4-cycle blocks connected by single edges, as well as the case where each vertex has degree 2 or 4.

### On the Computational Complexity of Upward and Rectilinear Planarity Testing

- Computer ScienceSIAM J. Comput.
- 2001

It is shown that upward planarity testing and rectilinear planar testing are NP-complete problems and that it is NP-hard to approximate the minimum number of bends in a planar orthogonal drawing of an n-vertex graph with an $O(n^{1-\epsilon})$ error for any $\ep silon > 0$.

### The Topology of Bendless Three-Dimensional Orthogonal Graph Drawing

- MathematicsGraph Drawing
- 2008

It is NP-complete to recognize xyz graphs, but it is shown how to do this in time O, under which bipartiteness of the graph is equivalent to orientability of the map.

### Orthogonal Drawings of Plane Graphs Without Bends

- MathematicsJ. Graph Algorithms Appl.
- 2001

A necessary and sufficient condition for a plane graph G of the maximum degree three to have an orthogonal drawing without bends is obtained and a linear-time algorithm to find such a drawing of G if it exists is given.

### Drawing graphs in the plane with high resolution

- MathematicsProceedings [1990] 31st Annual Symposium on Foundations of Computer Science
- 1990

It is shown that the problem of deciding if R=2 pi /d for a graph is NP-hard for d=4, and a counting argument is used to show that R=O(log d/d/sup 2/) for many graphs.

### No-bend Orthogonal Drawings of Series-Parallel Graphs

- MathematicsGraph Drawing
- 2005

This paper gives a linear-time algorithm to examine whether a series-parallel graph G of the maximum degree three has a no-bend orthogonal drawing and to find one if G has.

### Drawing graphs with right angle crossings

- Computer ScienceTheor. Comput. Sci.
- 2009

This paper studies the interplay between number of bends per edge and total number of edges in RAC drawings to establish upper and lower bounds on these quantities.

### Rectilinear Graphs and their Embeddings

- Mathematics, Computer ScienceSIAM J. Comput.
- 1985

The embedding problem for a class of graphs called rectilinear graphs is discussed. These graphs have applications in many VLSI Layout Problems. An interesting topological characterization of these…