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}
}
  • D. Eppstein
  • Published 9 July 2003
  • Mathematics
  • ACM Trans. Algorithms
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… 

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References

SHOWING 1-10 OF 49 REFERENCES

Intersection graphs of curves in the plane

Geometric Thickness of Complete Graphs

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

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

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

On Coloring Unit Disk Graphs

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.

On representations of some thickness-two graphs