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A Clique-Based Separator for Intersection Graphs of Geodesic Disks in [Formula: see text].

Boris Aronov1, Mark de Berg2, Leonidas Theocharous3

  • 1Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY 11201 USA.

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Intersection graphs of geodesic disks possess a clique-based separator, enabling efficient q-coloring algorithms. This also leads to a near-exact distance oracle for these graphs.

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Area of Science:

  • Computational geometry
  • Graph theory
  • Geometric intersection graphs

Background:

  • Shortest-path metrics and geodesic disks are fundamental in geometric analysis.
  • Intersection graphs of geometric objects are crucial in various computational fields.
  • Efficient algorithms for graph coloring and distance oracles are highly sought after.

Purpose of the Study:

  • To investigate the structural properties of intersection graphs of geodesic disks.
  • To develop efficient algorithms for q-coloring these graphs.
  • To construct a practical distance oracle for geodesic disk intersection graphs.

Main Methods:

  • Defining geodesic disks with respect to a shortest-path metric.
  • Proving the existence of a clique-based separator for the intersection graph.
  • Developing a q-coloring algorithm based on the clique-based separator.
  • Constructing a distance oracle utilizing the separator properties.

Main Results:

  • The intersection graph of geodesic disks has a clique-based separator with a bounded number of cliques.
  • An efficient algorithm for q-Coloring is presented, running in polynomial time.
  • A distance oracle with subquadratic storage and sublinear query time is developed, offering near-exact hop distances.

Conclusions:

  • The study extends the understanding of intersection graph properties.
  • The developed algorithms offer significant improvements for q-coloring and distance queries in geodesic disk intersection graphs.
  • This work provides a foundation for further research in geometric graph algorithms and data structures.