Testing bipartiteness of geometric intersection graphs

  • Authors:

  • David Eppstein

  • Affiliations:

  • University of California, Irvine, CA

  • Venue:
  • ACM Transactions on Algorithms (TALG)
  • Year:
  • 2009

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Abstract

We show how to test the bipartiteness of an intersection graph of n line segments or simple polygons in the plane, or of an intersection graph of balls in d-dimensional Euclidean space, 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. Our algorithms for these problems return either a bipartition of the input or an odd cycle in its intersection graph. We also consider lower bounds for connectivity and k-colorability problems of geometric intersection graphs. For unit balls in d dimensions, connectivity testing has equivalent randomized complexity to construction of Euclidean minimum spanning trees, and for line segments in the plane connectivity testing has the same lower bounds as Hopcroft's point-line incidence testing problem; therefore, for these problems, connectivity is unlikely to be solved as efficiently as bipartiteness. For line segments or planar disks, testing k-colorability of intersection graphs for k 2 is NP-complete.