# Testing bipartiteness of geometric intersection graphs

@article{Eppstein2004TestingBO, title={Testing bipartiteness of geometric intersection graphs}, author={David Eppstein}, journal={ACM Trans. Algorithms}, year={2004}, volume={5}, pages={15:1-15:35} }

We show how to test the bipartiteness of an intersection graph of n line segments or simple polygons in the plane, or of balls in Rd, in time O(n log n). More generally we find subquadratic algorithms for connectivity and bipartiteness testing of intersection graphs of a broad class of geometric objects. For unit balls in Rd, connectivity testing has equivalent randomized complexity to construction of Euclidean minimum spanning trees, and for line segments in the plane connectivity testing has…

## 25 Citations

### Spanners for Geometric Intersection Graphs

- Mathematics, Computer ScienceWADS
- 2007

This paper presents the first algorithm for constructing spanners of ball graphs with support for fast distance queries, and shows that the spanner construction has complexity almost equivalent to the construction of a Euclidean minimum spanning tree.

### Shortest paths in intersection graphs of unit disks

- Mathematics, Computer ScienceComput. Geom.
- 2015

### Geometric Biplane Graphs I: Maximal Graphs

- MathematicsGraphs Comb.
- 2015

It is shown that two maximal biplane graphs—in the sense that no edge can be added while staying biplane—may differ in the number of edges, and an efficient algorithm is provided for adding edges to a biplane graph to make it maximal.

### Mexican Conference on Discrete Mathematics and Computational Geometry Geometric Biplane Graphs I : Maximal Graphs ∗

- Mathematics
- 2014

We study biplane graphs drawn on a finite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and can be decomposed into two plane graphs. We…

### A Coloring Solution to the Edge Crossing Problem

- Computer Science2009 13th International Conference Information Visualisation
- 2009

This work defines a "closeness" metric on edges as a combination of distance, angle and crossing, and uses the inverse of this metric to compute a color embedding in the L*a*b* color space and assign "close" edges colors that are perceptually far apart.

### Graph-Theoretic Solutions to Computational Geometry Problems

- Computer ScienceWG
- 2009

The art gallery problem, partition into rectangles, minimum-diameter clustering, rectilinear cartogram construction, mesh stripification, angle optimization in tilings, and metric embedding are surveyed from a graph-theoretic perspective.

### Listing all spanning trees in Halin graphs - sequential and Parallel view

- Mathematics, Computer ScienceDiscret. Math. Algorithms Appl.
- 2018

This paper presents a sequential and parallel algorithm to enumerate all spanning trees in Halin graphs and proves that the number of spanning treesIn graphs is $O((2pd)^{p})$.

## References

SHOWING 1-10 OF 49 REFERENCES

### Finding Connected Components of an Intersection Graph of Squares in the Euclidean Plane

- Mathematics, Computer ScienceInf. Process. Lett.
- 1982

### Finding the Connected Components and a Maximum Clique of an Intersection Graph of Rectangles in the Plane

- Mathematics, Computer ScienceJ. Algorithms
- 1983

### Geometric Thickness of Complete Graphs

- MathematicsJ. Graph Algorithms Appl.
- 1998

We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straight-line edges and assign each edge to a layer so that no two…

### Bounded-Degree Graphs have Arbitrarily Large Geometric Thickness

- MathematicsElectron. J. Comb.
- 2006

This work proves that there exists Delta-regular graphs with arbitrarily large geometric thickness, and proves that for all Delta >= 9 and for all large n, there exists a Delta- regular graph with geometric thickness at least c Delta^{1/2} n^{1 /2 - 4/Delta - epsilon}.

### Graph Treewidth and Geometric Thickness Parameters

- MathematicsGraph Drawing
- 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⌉.

### Some Simplified NP-Complete Graph Problems

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1976

### On Coloring Unit Disk Graphs

- MathematicsAlgorithmica
- 1998

It is shown that the coloring problem for UD graphs remains NP-complete for any fixed number of colors k≥ 3, and a new 3-approximation algorithm for the problem is presented which is based on network flow and matching techniques.