Loops in Reeb Graphs of 2-Manifolds

Authors:
Kree Cole-McLaughlin;Herbert Edelsbrunner;John Harer;Vijay Natarajan;Valerio Pascucci
Affiliations:
Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA;Department of Mathematics, Department of Computer Science, Duke University, Durham, NC 27708 and Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 9455 ...;Department of Mathematics, Department of Computer Science, Duke University, Durham, NC 27708, USA;Department of Computer Science, Duke University, Durham, NC 27708, USA;Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
Venue:
Discrete & Computational Geometry
Year:
2004

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Abstract

Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(n log n), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.